JP2007330091A - Method of configuring feedback circuit for stabilizing dc voltage generated from resonance circuit - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a configuration of a feedback circuit for stabilizing a DC voltage obtained by rectifying a high-frequency AC voltage outputted from a resonance circuit for a load in a wide range, and to provide a method for giving its circuit constant. <P>SOLUTION: An equivalent power source is introduced to clarify the condition that the feedback of a carrier is stabilized for driving the resonance circuit of the output voltage to the frequency. A virtual resonance circuit corresponding to the resonance circuit of an equivalent power supply receives the frequency-modulated carrier similarly to the resonance circuit, and outputs not the amplitude-modulated carrier but its envelope. A virtual rectifying/smoothing circuit corresponding to the rectifying circuit acts as a first-order lag filter on the envelope, and generates an output voltage of the equivalent power supply. The equivalent power supply is described by a simultaneous differential equation system, and a mathematical analysis is made to clarify the sufficient condition to stabilize the equivalent power supply. Under the sufficient condition, a circuit realizing a stable feedback is constituted and the stability is demonstrated by simulation. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

The present invention relates to stabilization of a DC voltage that is an output of a power supply in a stabilized DC voltage power supply that generates a voltage by using a resonance circuit, and a wide range of DC voltages that are the output of the power supply are applied to a wide range of loads. It stabilizes against.

As a power source that generates a voltage using a resonance circuit, for example, there is a power source that generates a DC high voltage using a resonance circuit of a piezoelectric transformer. The power source using this piezoelectric transformer has been used for applications that require a relatively high voltage output. In those applications, high-precision stability is not required for the output voltage. Therefore, in such a power source using a piezoelectric transformer, the stability of the output voltage with respect to fluctuations in the input voltage or load current is poor. For this reason, this type of power supply has a drawback that it is limited to applications that satisfy the conditions of fixed input and fixed load.
JP2002-359967 JP2005-137085

Patent Document 1 aims to provide a simple circuit configuration of an efficient direct current high voltage power supply device that provides a stabilized high voltage, and the direct current high voltage power supply is not an ordinary electromagnetic transformer, By adopting high voltage generation means using a piezoelectric transformer, the efficiency is improved, and the frequency dependence of the resonance characteristics of the piezoelectric transformer is used to stabilize the high voltage, thereby simplifying the circuit and reducing the number of parts. Solve the problem by measuring the decrease.

Patent Document 2 relates to a DC high-voltage power supply device, and for feedback that stabilizes the output voltage of the device, by implementing feedback with little delay independent of feedback with a large delay accompanying the generation of a high voltage, Improves voltage stabilization accuracy and speeds up response.

A method of providing a feedback circuit configuration and circuit constants for stabilizing a DC voltage obtained by rectifying high-frequency alternating current output from a resonant circuit against a wide range of loads.

In order to ensure the stability and accuracy of the output voltage against fluctuations in the input voltage or load current, a pole located at the origin is introduced into the transfer function of the feedback circuit. This pole causes a phase delay of π / 2. In order to achieve stabilization of the output voltage, it is necessary to compensate for the phase delay. Stable feedback is realized by introducing a plurality of zero points that compensate for the delay of the rectifying and smoothing circuit and the delay of the resonant circuit into the feedback circuit, and the output of the power supply is stabilized over a wide range of loads.

A resonant circuit having a high Q value is driven by a carrier wave having a frequency close to the resonant frequency having a locally constant amplitude. At this time, the DC voltage obtained by demodulating (rectifying) the high-frequency AC output from the resonance circuit depends on the frequency of the carrier wave. The DC voltage is stabilized by feeding back the DC voltage to the frequency of the carrier wave.

A stabilized DC voltage power source that outputs a stabilized DC voltage is composed of a voltage generation circuit and a feedback circuit, and the voltage generation circuit outputs a carrier wave that is a high frequency alternating current of locally constant amplitude that drives a resonance circuit. A driver circuit, a resonance circuit that receives the carrier wave, and a rectifying / smoothing circuit that generates a DC voltage from the output of the resonance circuit. A feedback circuit includes a DC voltage output from the rectifying / smoothing circuit and a reference voltage given in advance. Is composed of an error amplifier and a frequency modulation circuit that outputs a square wave having a frequency determined by the output of the error amplifier, and the output of this frequency modulation circuit controls and stabilizes the frequency of the high-frequency AC output from the driver circuit. The DC voltage output from the rectifying / smoothing circuit, which is the output voltage of the DC voltage power supply, is fed back to the frequency of the carrier wave and stabilized.

An equivalent power supply that approximates this stabilized DC voltage power supply that can be analyzed by mathematical analysis means is constructed, and a sufficient condition that the feedback is stable when the DC voltage output from the equivalent power supply is in the vicinity of the reference voltage. The problem is solved by finding the circuit constant so that the feedback of the error between the output DC voltage and the reference voltage to the frequency of the carrier wave satisfies the sufficient condition.

The power supply has a driver circuit that generates a carrier wave having a constant amplitude, a resonance circuit that is driven by the carrier wave that is the output of the driver circuit, and a DC voltage by rectifying the amplitude-modulated carrier wave that is the output of the resonance circuit. A voltage generating circuit including a rectifying / smoothing circuit to be taken out, an error amplifier for comparing a DC voltage output from the rectifying / smoothing circuit with a reference voltage given in advance to set an output voltage of the power supply, and an output of the error amplifier And a feedback circuit including a frequency modulation circuit for controlling the driver circuit. The output of the frequency modulator becomes an input to the driver circuit and controls the frequency of the carrier wave generated by the driver circuit. Is provided to stabilize the output DC voltage by feeding back to the frequency of the carrier wave.

In order to clarify the conditions under which this power supply operates stably, an equivalent power supply is introduced. The equivalent power source is composed of a virtual voltage generation circuit and a feedback circuit. The virtual voltage generation circuit includes a driver circuit, a virtual resonance circuit, and a virtual rectification smoothing circuit. Similar to the resonant circuit, the virtual resonant circuit receives a frequency-modulated carrier wave as input, and outputs the envelope instead of the amplitude-modulated carrier wave. The virtual rectifying / smoothing circuit receives this envelope as an input, acts as a first-order lag filter with respect to the envelope, and outputs a result equivalent to the output of the rectifying / smoothing circuit.

The equivalent power supply includes a driver circuit that generates a carrier wave having a constant amplitude, a virtual resonance circuit that is driven by a carrier wave that is an output of the driver circuit, and a virtual voltage generation circuit that includes a virtual rectifying and smoothing circuit to which the output is input. An error amplifier that compares a DC voltage that is an output of the virtual rectifying and smoothing circuit with a reference voltage; a feedback circuit that includes a frequency modulation circuit that generates a frequency determined by the output of the error amplifier and controls the driver circuit; The output of the device is input to the driver circuit, controls the frequency of the carrier wave generated by the driver circuit, and feeds back the DC voltage that is the output of the virtual rectifying and smoothing circuit to the frequency of the carrier wave, thereby making the output DC voltage constant. Means.

Since the operation of this equivalent power supply can be described by a system of simultaneous differential equations, mathematical analysis of stability is possible. A differential equation system is derived, and from this, sufficient conditions for the output voltage of the equivalent power supply to become stable near the reference voltage are clarified. An actual circuit that realizes stable feedback based on this sufficient condition and a method for giving circuit constants will be described.
Output voltage stabilization

A resonance circuit with a high Q value exhibits resonance characteristics such as sharp frequency characteristics and large load dependence. When a frequency-modulated carrier wave having a constant amplitude is input to the resonance circuit, the carrier wave is amplitude-modulated and output. When a load resistor is connected to the output of the resonant circuit, and the step-up ratio, which is the ratio of the amplitude voltage of the input carrier to the output carrier, is seen as a function of the drive frequency, the resonant circuit shows resonance and is near the resonance frequency. A large step-up ratio is shown. A power supply using a resonance circuit generates a high voltage by using this high step-up ratio. Further, the output voltage is stabilized by controlling the frequency by utilizing the fact that the step-up ratio depends on the frequency of the carrier wave.

When the frequency of the carrier wave that drives the resonance circuit is selected to be lower than the resonance frequency of the resonance circuit, for example, as shown in FIG. 1, when the output voltage is lower than the reference voltage, the frequency is increased to approach the resonance frequency, Further, when the output voltage is higher than the reference voltage, the voltage is stabilized by lowering the frequency and moving away from the resonance frequency.

When the carrier frequency is chosen to be higher than the resonant frequency, when the output voltage is lower than the reference voltage, the frequency is lowered to approach the resonant frequency, and when the output voltage is higher than the reference voltage, the frequency is increased to be higher than the resonant frequency. The voltage is stabilized by moving away.
Frequency modulation

The frequency modulator outputs a carrier wave frequency-modulated by an input voltage. The frequency of the output carrier is a function of the input voltage. Write u (t) as the output carrier wave as follows.
Here, w represents the time-invariant amplitude of the carrier wave, and the carrier wave input to the resonance circuit is the real part of Equation 63. φ is defined by the following equation.
Then, when v is the input voltage to the frequency modulator,
Holds. Here, f _self represents the free-running frequency of the frequency modulator, and k represents a proportionality constant between φ and v. In order to simplify future calculations, ω _c = ω ₀
Assuming However, ω ₀ is defined by Equation 69. From Equation 65, the following equation follows.

Resonant circuit

The resonance that the resonance circuit shows when connected to the load _RL can be written approximately as follows using its Q value, resonance frequency, and voltage boost ratio at the resonance frequency.
Here, a, b, and c are positive numbers that satisfy the following conditions.
Q is the resonance Q value, γ is the reciprocal of the Q value, ω _r is the angular velocity at the resonance frequency, and g _r is the boost ratio of the resonance circuit at the resonance frequency.

Let α and β be the roots of the equation s ² + as + b = 0 made from the denominator of the transfer function. α and β can be written using δ and ω ₀ as follows.
From Equation 68 and Equation 69

At this time, δ and ω ₀ are positive numbers.
The Q value is about several tens and the resonance is sharp.
Holds.

A transfer function h ₀ (s) having a Q value of 30, a step-up ratio of 100, and a resonance frequency of 120 kHz can be written as follows.
Assuming that δ and ω of h ₀ (s) are δ ₀ and ω ₀ respectively, the values are as follows.
In the discussion developed below, one transfer function h (s) of the resonance circuit is fixed, and the condition that the DC voltage generated from the resonance circuit having this transfer function is stable is obtained.
Transfer function near resonance frequency

The transfer function in Equation 67 can be approximated by a first-order lag with complex poles in the vicinity of the resonance frequency. Using the operator method with the input of the piezoelectric transformer as u in Equation 63 and the output as f,

here

Since the frequency of the carrier wave is in the vicinity of the resonance frequency, it can be seen that there is a positive constant M from the left side of Equation 65, and that the following condition is satisfied for an arbitrary t (0 ≤ t).
At this time, the following evaluation holds.

As well

Perform integration on A.

If you ask for A,

A can be evaluated as follows.

Similarly, integration is performed on B.

B can be evaluated as follows.

The upper limit of the band in which the frequency modulator modulates the output frequency is M. M is about the same as δ or at most 10 times as large. On the other hand, the resonance frequency ω ₀ is about 100 times δ. Further, | α | = | β |. It can be seen that even if B is ignored, the error is at most 10 percent. The transfer function h ^T ignoring B is as follows.
At this time, the output g of the resonance circuit is defined as follows.
The difference between the output f and g of the resonant circuit is at most several percent. This shows that the transfer function of the resonant circuit can be approximated by h ^T near the resonant frequency.
Comparison of transfer function h ₀ (s) and its approximation h ^T ₀ (s)

Seeking h ^T ₀ (s) that approximates the contrast transfer function being provided h ₀ (s) in Equation 73 is compared with the h ₀ (s) and h ^T ₀ (s). In FIG. 2, h ₀ (s) and h ^T ₀ (s) are plotted simultaneously for the boost ratio and in FIG. As can be seen from this figure, h ^T ₀ (s) is a good approximation of h ₀ (s).
Differential equation and resonance envelope near resonance frequency

As shown in Equation 97, g satisfies the following differential equation.

Here, g is converted as shown in Equation 101 using variables p and q that take real numbers. Constants r _r and r _i are defined by Equation 102.

Here, if w is a real number, r _r and r _i are as follows.
It can be seen that p and q satisfy the following equation.

Here, θ is the phase of the carrier wave output from the frequency modulator.
The following equation is obtained from Equation 100 and Equation 64.
Substituting this equation into Equation 104 and Equation 105 yields the following normal differential equation system.

The absolute value of the output g of the resonance circuit is an envelope, and when this is y, y is given by the following equation.

Amplitude of steady output from resonant circuit

Here, φ is assumed to be constant. Also, let p (t) and q (t) be the solutions of Equation 107 and Equation 108. Then

Here r is defined as follows.
The steady amplitude of the carrier wave output from the resonance circuit is
Given by. From this, the steady amplitude is maximum at φ = 0, and the amplitude is
It can be seen that is given by.
Changes in steady amplitude with changes in φ

The steady amplitude of the carrier wave output from the resonance circuit is
Therefore, by differentiating this by φ, the change in amplitude caused by steady amplitude by changing φ by Δφ is
It can be expressed as.
Evaluation of constant r

From Equation 102
That is, r depends on the carrier amplitude | w | To emphasize this dependency, we will write r _{| w |} instead of r. Then, for any a
Holds.

Evaluating r against h ^T ₀ (s) defined for h ₀ (s) given by Equation 73

It becomes. from now on
Follow that.
Q value of resonant circuit

The Q value of the resonance circuit varies depending on the load connected to the output of the resonance circuit. The Q value increases as the load decreases. In many resonant circuits, the output amplitude at the resonant frequency is approximately proportional to this Q factor. That is, the step-up ratio at the resonance frequency is proportional to the approximate Q value. In many resonance circuits, the resonance frequency also changes depending on the load. That is, the resonance frequency also depends on the Q value. However, this dependence is small in many resonant circuits.

With respect to the piezoelectric transformer having the frequency characteristic shown in FIG. Fig. 5 shows the Q factor dependence of the boost ratio, and Fig. 6 shows the load dependence of the resonance frequency. It can be seen from FIG. 5 that the boost ratio is almost proportional to the Q value. From Fig. 6, when the load changes from 150 kΩ to 300 kΩ, the resonance frequency shifts by about 600 Hz. This shift is small enough for the resonance frequency of 120 kHz.

If the transfer function of a resonance circuit to which a certain load is connected is represented by Equation 67, h (s), the coefficient c is given by Equation 68 as follows.
Here, g _r is the step-up ratio of the resonance circuit at the resonance frequency, and is approximately proportional to the Q value. Further, ω _r is an angular velocity at the resonance frequency, and is considered to be approximately constant regardless of the load. From this, it can be seen that the coefficient c does not approximately depend on the Q value. In other words, c is unique to the resonant circuit and can be considered to be approximately constant regardless of the load.
Extraction of δ and r from resonant circuit

From Equation 117 and Equation 103

Q is dozens, so
Since the coefficient c is a constant inherent to the resonance circuit and w is the amplitude of the carrier wave, it can be seen that r is a number that hardly depends on the Q value and hence δ. This r can be directly obtained by measuring the output from the resonance circuit as follows.

It is widely practiced to measure the Q value of resonance. Let Q be the Q value of resonance when the resonance circuit is driven by a carrier wave of a resonance frequency having a constant amplitude. If angular velocity ω _r is the resonance frequency,
Can obtain δ.
If the maximum amplitude of the carrier wave output from the resonant circuit is R, then Equation 114
From this, r can be obtained. This r is almost independent of δ from Equation 125.
Equivalent power supply configuration with envelope

Here, the power supply subject to stability analysis is a driver circuit that generates a carrier wave having a locally constant amplitude, a resonance circuit driven by a carrier wave that is an output of the driver circuit, and an output of the resonance circuit. A voltage generating circuit including a rectifying and smoothing circuit that extracts a DC voltage by rectifying the amplitude-modulated carrier wave, an error amplifier that compares the output voltage of the power source with a reference voltage that is an output of the rectifying and smoothing circuit, and A feedback circuit including a frequency modulation circuit that generates a frequency determined by an output of the error amplifier and controls the driver circuit, and an output of the frequency modulator is input to the driver circuit to control a frequency of a carrier wave generated by the driver circuit; The output DC voltage is made constant by feeding back the DC voltage that is the output of the rectifying and smoothing circuit to the frequency of the carrier wave. Figure 7 shows the configuration of this power supply.
Virtual resonance circuit and virtual rectification smoothing circuit

The output of the resonance circuit is a carrier wave amplitude-modulated by resonance as shown in FIG. 8, and this carrier wave is converted into a DC voltage by the rectifying and smoothing circuit. In order to facilitate the analysis, a virtual resonance circuit and a virtual rectification / smoothing circuit are introduced, thereby approximating the resonance circuit and the rectification / smoothing circuit. The virtual resonant circuit receives the same frequency-modulated carrier wave as the resonant circuit, and outputs the envelope as shown in FIG. 9 instead of the amplitude-modulated carrier wave output from the resonant circuit. This envelope is input to a virtual rectifying / smoothing circuit. The virtual rectifying / smoothing circuit has a transfer function whose output is equivalent to the output of the rectifying / smoothing circuit. By converting the input of the virtual rectifying / smoothing circuit by this transfer function, an output equivalent to the output of the rectifying / smoothing circuit is realized.
Transfer function of virtual rectification smoothing circuit

The rectifying and smoothing circuit generates a DC voltage by rectifying and smoothing the amplitude-modulated carrier wave. For this reason, there is a large delay in the generation of the DC voltage. That is, the delay until the change in the amplitude of the carrier wave is reflected in the generated DC voltage is large. When the delay is large in this way, the transfer function can be approximated by a first-order delay. Therefore, a first-order lag transfer function is assumed in the virtual rectification smoothing circuit.

Let μ be the first-order delay time constant. When the rectifying / smoothing circuit generates a DC voltage that is ν times the amplitude of the high-frequency alternating current that is the input, the transfer function of the virtual rectifying / smoothing circuit corresponding thereto is
It becomes. If the first-order lag transfer function is canceled by inputting the output of the rectified and smoothed rectifying and smoothing circuit to the differentiating circuit, the output from the differentiating circuit is considered to be the envelope itself output from the virtual resonance circuit.

This ν is a multiplier associated with the double voltage rectification, and is related to the output impedance of the rectifying and smoothing circuit. Since the voltage output from the rectifying / smoothing circuit is determined by the output impedance and the resistance value of the load, the multiplier ν depends on the output impedance of the rectifying / smoothing circuit and the resistance of the load.
Equivalent power supply

The equivalent power source is an equivalent circuit of a power source, and is composed of a virtual voltage generating circuit including a driver circuit, a virtual resonance circuit, and a virtual rectifying / smoothing circuit, and a feedback circuit including an error amplifier and a frequency modulator. The DC voltage generated by the virtual voltage generation circuit is the output of the equivalent power supply, and the feedback circuit feeds back this DC voltage to the frequency of the carrier wave. The error amplifier includes a subtraction circuit and a phase compensation circuit. The output of the subtraction circuit that detects the voltage difference between the output voltage and the reference voltage is input to the phase compensation circuit. The output of this phase compensation circuit is the output of the error amplifier. The frequency modulation circuit generates a frequency determined by the output of the error amplifier and outputs a rectangular wave having this frequency. The output of the frequency modulator is input to the driver circuit and controls the frequency of the carrier wave generated by the driver circuit. Figure 10 shows a schematic diagram of the equivalent power source.
Equivalent power load

The load of the equivalent power source is reflected in δ of the virtual resonance circuit. As the load increases, the Q of the resonant circuit decreases and δ increases. On the other hand, when the load becomes lighter, Q becomes larger and therefore δ becomes smaller. Therefore, when analyzing the stability of the equivalent power supply, the load L is specified by δ _{L of the} virtual resonance circuit that drives the load. In other words, the stability of the equivalent power supply to which a certain load L is connected is understood as the stability of the power supply when δ of the virtual resonance circuit is at δ _L corresponding to the load.
Smart power supply stability

When the transfer function of the virtual rectifying and smoothing circuit is given by Equation 128, the equivalent power supply is called a smart power supply. Consider the stability of feedback for smart power supplies. In the smart power supply, the output voltage is fed back to the frequency of the carrier wave that drives the virtual resonance circuit. In order to consider the stability of feedback, a slight deviation of the output voltage is considered, and the change due to the feedback of this deviation is examined. If the feedback is stable, the deviation changes in a decreasing direction. If the deviation changes in the direction of expansion, the feedback is unstable. When considering the feedback of the slight deviation of the output voltage, it is assumed that δ, μ, and ν are constant because the deviation is very small.
An example of smart power supply and its differential equation system

The error amplifier includes a subtraction circuit and a phase compensation circuit. The subtraction circuit includes a dividing resistor and an amplifier, and the amplifier outputs a voltage difference between its inputs. The phase compensation circuit includes an amplifier and its peripheral circuits, and generates an error amplifier output from the output of the subtraction circuit. Since the amplifier operates sufficiently faster than the operation of the virtual resonant circuit or the virtual rectifying / smoothing circuit, it can be assumed that all delays caused by the error amplifier are caused by this phase compensation circuit. Figure 11 shows an example of a smart power supply. For the power supply in Fig. 11, if the output of the virtual resonant circuit is y and the output of the virtual rectifying and smoothing circuit is z,

A voltage dz obtained by dividing z by a dividing resistor and a reference voltage n are input to the amplifier of the subtraction circuit. Where d is the division ratio of the dividing resistor, and v is the input to the phase compensation circuit.

Can be written. If the input to the phase compensation circuit in Fig. 11 is v and the output is v _o , then v and v _o are
It is tied in a relationship. This can be rewritten as:
here

From Equation 130,
from now on

Applying Equation 66 to Equation 139 gives

In other words, this smart power supply is described by the following differential equation system.


Smart power supply loop gain

If the output of the virtual resonance circuit is y and the output of the virtual rectification smoothing circuit is z,
A voltage dz obtained by dividing z by a dividing resistor and a reference voltage n are input to the amplifier of the subtraction circuit. Where d is the division ratio of the dividing resistor, and v is the input to the phase compensation circuit.

Can be written.

At this time, if the output z of the virtual rectifying / smoothing circuit changes to z + Δz, the input to the error amplifier changes by Δv. Where Δv is
It is. When the gain of the error amplifier is L, the output of the error amplifier changes by −L Δv (= −L d Δz). As a result, the frequency of the carrier wave output from the frequency modulator changes by Δφ. Where Δφ is
Can be written. At this time, the amplitude output from the virtual resonance circuit is
The output of the virtual rectifying / smoothing circuit is
You can see that it only changes. In other words, the loop gain is
It becomes. When A is the DC gain of the error amplifier
The DC loop gain is
It is expressed. In general, a large DC loop gain is mounted on a stabilized DC voltage power source as a feedback circuit in order to increase the stability of the output voltage.
Near δ

Figure 12 shows the DC loop gain as a function of φ. As can be seen, the loop gain has a maximum value when φ = δ / √2, and is almost 1/4 of the maximum value when φ = 3δ. For convenience of later discussion, if we define the interval [δ / √2, 3 δ] as the neighborhood of δ in the domain of φ,
Holds.
Resonance circuit delay and rectifying and smoothing circuit delay

Consider the delay of the resonant circuit and the delay of the rectifying and smoothing circuit in the smart power supply. If the pole delay of the resonant circuit is f sec and the pole delay of the rectifying and smoothing circuit is μ sec,
It is. The delay of the resonant circuit depends on δ
Can be evaluated. from now on
It can be seen that
Smart power supply and differential equation system

The transfer function of the virtual rectifying and smoothing circuit is given as follows.
Therefore, the output y of the virtual resonant circuit and the output z of the virtual rectifying and smoothing circuit are differential equations.
Meet. Figure 13 shows a schematic diagram of a smart power supply that takes into account the zero point that offsets the delay caused by this virtual resonance circuit. Consider the stability of feedback for this smart power supply. The error amplifier includes a subtraction circuit and a phase compensation circuit.
When the drive frequency is higher than the resonance frequency

When the frequency of the carrier wave driving the resonance circuit is higher than the resonance frequency, that is, φ ≧
In the case of 0, the transfer function of the phase compensation circuit can be written as
Here, E, A, and B are positive constants. From formula 161
From this equation, the second derivative of z is calculated using Equation 107 and Equation (108.

After all, it can be seen that the smart power supply is described by the following differential equation system.


Stability near the equilibrium point

The equilibrium point of the differential equation system equations 166 to 169 is the solution of the following equation system for p, q, and φ.

At the equilibrium point
Therefore, the equilibrium point is the solution of the following simultaneous linear equations.

Formula 177 is combined with Formula 177 to obtain p, q, and z as a function of φ, and this is substituted into Formula 173 to obtain Formula 178. The equilibrium point that is the solution is denoted by p _e , q _e ,
When z _e , φ _e , p _e , q _e , z _e and λ can be expressed as a function of φ _e as follows:

The stability of the differential equation system near the equilibrium point is investigated by Lyapunov's method. p, q, z, and φ are expanded in the vicinity of p _e , q _e , z _e , and φ _e , respectively, as follows.

By substituting this equation into the differential equation system 166 to 169 and ignoring higher-order terms, the following normal and linear differential equation system relating to Δp, Δq, Δz, Δφ is obtained.

here

At this time, the elements of the matrix M are functions of p _e , q _e , z _e , and φ _e , but can be considered as a function of φ _e by Equation 182. Therefore, let M (h) be the eigen polynomial of M,
Write. Coefficients a ₀ , a ₁ , a ₂ ,
a ₃ and a ₄ are functions of φ _e as follows.

Here, r is defined as follows.

When the drive frequency is lower than the resonance frequency

When the frequency of the carrier wave driving the resonance circuit is lower than the resonance frequency, that is, φ ≦
When 0, φ is set to -φ. For example, when the output voltage is higher than the voltage set by the reference voltage, the drive frequency moves away from the resonance frequency, so the transfer function of the phase compensation circuit is
It can be written with the same formula as Formula 162. When φ is set to −φ, Equations 166 and 167 are as follows.

When the second derivative of z is calculated from Equation 163 using Equation 199 and Equation 200, the second derivative of z does not include φ.

And the same formula as formula 165.

It can be seen that for a drive frequency lower than the resonant frequency, the smart power supply is described by the following differential equation system with φ ≧ 0.

The equilibrium points p _e , q _e , z _e and λ can be expressed as a function of φ _e as follows:

M (h) is an eigen polynomial of the differential equation system equations 202 to 205 linearized in the vicinity of this equilibrium point.
And
As coefficients a ₀ , a ₁ , a ₂ ,
Since a ₃ and a ₄ are again given by Equations 192 to 196, both when the drive frequency is higher and lower than the resonance frequency,
Coefficients a ₀ , a ₁ ,
It can be seen that a ₂ , a ₃ , and a ₄ are given by Equations 192 to 196 with φ ≧ 0.
Redefining the eigenpolynomial

In Equation 192 to Equation 196, the terms k, d, ν, r appear together with A, B, or E, so
And then B ’and E ′

That is

And B 'and E' are written again as B and E. The coefficients a ₀ , a ₁ ,
a ₂ , a ₃ and a ₄ are functions of φ _e as follows.

The addition Z _e
In this case, Equations 217 to 221 can be written as follows.

Using the coefficients a ₀ , a ₁ , a ₂ , a ₃ , a ₄ redefined in Eqs. 217 to 221 or Eqs. 223 to 227, the intrinsic polynomial m (h) of M defined in Eq. Define again as follows.

Root of eigenpolynomial

In a differential equation system describing a smart power supply, if the output voltage is near the reference voltage, in other words, the behavior of the output voltage near the equilibrium point is given by the poles of this differential equation system. The differential equation system describing the smart power source The poles of Equation 202 to Equation 204 are given by the roots of the inherent polynomial Equation 228 of this differential equation system.

The stability of the feedback is determined by the roots of the intrinsic polynomial. In order for a differential equation system to be stable in the sense of Lyapunov, it is necessary and sufficient that it be negative in the real part of all roots of the eigenpolynomial. It is a necessary condition that the power supply must satisfy that the differential equation system describing the power supply is stable in the sense of Lyapunov. Stability in the sense of Lyapunov is not enough for power supply stability. For example, there is a case where the output voltage of a stable power source vibrates in the meaning of Lyapunov. In some cases, the output voltage oscillates in the vicinity of the output voltage set by the reference voltage, or may stabilize slowly while oscillating over a long time.

The desired characteristics as a power source are realized by the proper root placement of the eigenpolynomials. In other words, the desired characteristics are realized by selecting the values of E, B, and N according to the values of δ, μ, and ν of the power source and realizing appropriate root placement. Among the roots of the eigenpolynomial, the root whose absolute value of the real part is the minimum is called the characteristic representative root. The characteristic representative root is a root whose real part is closest to the origin among the roots of the polynomial. The characteristic of the output voltage of the power supply stabilized by feedback strongly depends on this characteristic representative root. As will be described later, when the characteristic representative root is a real root separated from an imaginary root, the power supply realizes a desirable characteristic. In the next section, we find the condition that the characteristic representative root becomes a real root separated from the imaginary root under practical assumptions.
Decomposition of eigenfunction m (h)

The eigenfunction m (h) can be written as
The second term on the right side of Equation 229 matches the numerator of the transfer function of the feedback circuit. From this, it can be seen that the second term is a contribution from the feedback circuit. Also, it can be seen that the first term is a contribution from the high voltage generation circuit because it does not depend on the transfer function of the feedback circuit. The first term h corresponds to the pole located at the origin of the feedback circuit. The term (μ h + 1) corresponds to the rectifying / smoothing circuit, and indicates that the delay time constant is 1 / μ. The term ((x + δ) ² + φ ² ) corresponds to the resonant circuit.

From Equation 229, if the transfer function of the high-voltage generating circuit is F _w and the transfer function of the feedback circuit is B _k , F _w except for the difference between CF _w and (1 / C) B _k for constant C and the common constant C about
And about B _k
And calculating the closed-loop transfer function F _w / (1 + F _w B _k )

Therefore, it can be seen that the eigenfunction m (h) matches the denominator of the closed-loop transfer function F _w / (1 + F _w B _k ) except for a constant multiple.

From Equation 229, m (h) when φ = 0 is
It becomes. And when h = -1 / μ, m (h) is
Given by.
Introduction of poles near the origin by feedback

A smart power supply is described by a differential equation system, and an intrinsic polynomial is determined by this differential equation system. The smart power supply is composed of a high voltage generation circuit and a feedback circuit. That is, the intrinsic polynomial depends on the transfer function of the high voltage generation circuit and the transfer function of the feedback circuit. The high voltage generation circuit includes a pole generated by the rectifying and smoothing circuit. This pole corresponds to the delay of the rectifying and smoothing circuit and is at a low frequency. In the smart power supply, by introducing the pole located at the origin in the feedback circuit, the real root located near the origin is generated in the eigenfunction, and at the same time the real root corresponding to the delay of the rectifying and smoothing circuit is shifted to a higher frequency. A wide frequency range that can be approximated by a first-order lag is secured in the frequency range from a nearby pole to a pole that has moved to a higher frequency.

For example, by increasing N corresponding to the loop gain, the root corresponding to the delay of the rectifying and smoothing circuit can be shifted to a higher frequency. This movement of roots shifts the roots near the origin of the natural polynomial and the roots other than those corresponding to the delay of the rectifying and smoothing circuit to a lower frequency as a result. In order to secure a wide frequency range that can be approximated by first-order lag, it is necessary to select E, B, and N appropriately according to the parameters of the power supply.
Two functions

Functions f ₁ (h) and f ₂ (h) are defined as follows from the expression of m (h) given by Equation 229.

At this time, the proper polynomial m (h) is
Can be written.
Function f ₁ (h)

Considering the function graph y = f ₁ (h), from equation 229 this graph intersects the h axis at the origin and -1 / μ. Considering that −δ << − 1 / μ, it can be seen that the function takes a positive value at h <−1 / μ. FIG. 14 shows a graph of y = f ₁ (h).
Function graph y = f ₂ (h)

The graph y = f ₂ (h) of the function f ₂ (h) is an upward convex quadratic curve when φ and N are given. Consider a condition where m (h) = 0 has a real root near the origin regardless of φ and N. If the graph y = f ₂ (h) has no intersection with the h axis, the graph moves in the -y direction when N or φ increases, so graph y = f ₁ (h) and graph y = f ₂ (h ) Cannot have a real root near the origin.

When the graph y = f ₂ (h) has an intersection with the h axis, all roots of f ₂ (h) = 0 are negative real roots from the definition of the function f ₂ (h). Of these real roots, the real root having the smallest absolute value is set as h _a , and the other roots are set as h _b . At this time, the graph y = f ₂ (h) passes through h _a and h _b regardless of φ and N. Of these real roots, h _a is −1 / μ on the h axis ≦
When h <0, m (h) = 0 has a real root near the origin regardless of φ and N. Furthermore, h _b satisfies h _b <−1 / μ, and it can be seen that root-1 / μ corresponding to the delay of the rectifying and smoothing circuit moves to a higher frequency as shown in FIG.

When real roots h _a and h _b are both in the range of -1 / μ ≤ h <0 on the h axis, m (h) = 0 is -1 / μ ≤
There will be 2 real roots in the range of h <0. If both real roots h _a and h _b are not in the range of -1 / μ ≤ h <0 on the h-axis, m (h) = 0 must have a real root near the origin when N or φ increases. I can't. If it has real roots, it has two real roots in the range of -1 / μ ≤ h <0.

When the eigenpolynomial has a real root near the origin and at the same time shifts root-1 / μ corresponding to the delay of the rectifying and smoothing circuit to a higher frequency, h _a is −1 / μ ≦ h <0 on the h axis. When it is in range. Graph y = f ₂ (h) is h axis and -1 / μ ≤
The necessary and sufficient condition with one intersection in the range of h <0 is

Is true at the same time. By the way, Equation 239 is automatically satisfied from E> 0. From formula 238
Follow. from now on
Since m (0)> 0, m (h) = 0 is -1 / μ ≤
It can be seen that there is one real root in the range of h <0. Also, from Equation 240, the graph y = f ₂ (h) is -1 / μ with h axis
To have one real root in the range ≤ h <0
It turns out that it is necessary. Since f ₂ (h) = 0 has different real roots
It turns out that it is.
Intersection of graph y = f ₁ (h) and y = f ₂ (h)

In order for feedback to be stable in the sense of Lyapunov, it is necessary and sufficient that the real part of all roots of the eigenpolynomial be negative. Assuming that Equation 240 holds and the eigenpolynomial m (h) = 0 has one real root in the range of -1 / μ ≤ h <0, the real part of all roots of the eigenpolynomial is negative. To clarify. Since the eigenpolynomial has real roots, the eigenpolynomial may have four real roots, or two real roots and two imaginary roots. Since the eigenpolynomial m (h) = 0 is m (h)> 0 when h> 0, it can be seen that all four real roots are negative and stable in the sense of Lyapunov.

When the eigen polynomial m (h) has two imaginary roots h ₃ and h ₄ in addition to the real roots h ₁ and h ₂ , the imaginary roots h ₃ and h ₄ have a common real part h _r . At this time, the intersections h _a and h _b between the real roots h ₁ and h ₂ and the graph y = f ₂ (h) and the h axis are
Are in a relationship. From the relationship between roots and coefficients

And
Therefore, it becomes possible to evaluate the outline of the real part of the imaginary root from the evaluation of the real root.

To make h _r negative from Equation 247
Is a necessary and sufficient condition. The sufficient condition of Equation 248 can be expressed in various ways. For example
If
Since h ₁ is located near the origin,
It can be seen that the roots are arranged in the order close to the origin, h ₁ , and then h ₂ and the real part of the imaginary root. In the same way
If
It can be estimated that the roots are aligned with h ₁ , h ₂ , and the real part of the imaginary root in the order closer to the origin. Equation 249 is
Because you can write
Shows that this is a sufficient condition of Equation 249.
Limit to N by B

As can be seen from Equation 247, h ₂ is a value near -2δ,
At the time of
Therefore, the real part h _{r of the} imaginary root is near the origin,
Meet. That is, as h ₂ decreases toward −2δ, h _r increases from the negative side toward the origin. If h ₂ becomes smaller than -2δ, h _r becomes positive beyond the origin, and feedback is not stable in the sense of Lyapunov.

If h _b is greater than -2δ, no matter how large N, h ₂ will always be greater than -2δ and h _r will never be positive. However, when h _b is to increase the N in the case sufficiently smaller than -2Deruta, feedback because h ₂ becomes smaller than the -2Deruta it becomes unstable. That is, when h _b is sufficiently smaller than −2δ, it can be seen that N has an upper limit in order to realize stable feedback.

The function f ₂ (h) is a function of h
It can be written as, for h _b because h _a is in the vicinity of the origin
You can see that Therefore, as B decreases, h _b decreases beyond -2δ. From this, in order to realize stable feedback in the sense of Lyapunov, it is understood that N has an upper limit when B is smaller than 1 / (2δ).

In summary, if Formula 240 holds:
If B is sufficiently larger than 1 / (2δ), the feedback is stable in the sense of Lyapunov regardless of N, but is stable in the sense of Lyapunov when B is smaller than 1 / (2δ). In order to realize feedback, an upper limit is required for N.
Eigenfunction m (h) as a function of φ

From Equation 229, the eigenfunction m (h) can be written as


h ≤ -1 / μ and Bh ²
+ h + E ≥ 0

As you can see from Equation 262
Then the first term is positive,
The second term is also positive if
Always holds regardless of φ.
For example, if B <μ, Bh ² + h + E = at h = -1 / B
Because it becomes E
Holds regardless of φ.
h ≤ -1 / μ and Bh ²
+ h + E <0

As can be seen from Equation 262, the intrinsic polynomial m (h) depends not only on h but also on φ. m (h) considered as a function of φ with h fixed is represented as E _h (φ). Ie
It is defined as First, the derivative of E _h (x) with respect to x is calculated.

If we find E ′ _h (x) at x = 0,
Because
It can be seen that it is. Also clearly where x is large
So,
It can be seen that there exists x satisfying.

If we calculate the second derivative of E _h (x) with respect to x,

It becomes. From this equation and assumption, if x ≥ 0
Holds. This shows that E _h (x) is convex downward as a function of x. From Equation 270 and Equation 271, there is only one x that satisfies Equation 272. Writing x satisfying Equation 272 and phi _h, the function E _h (x) has a minimum at x = phi _h. E ′ _h
From (φ _h ) = 0

The N using the future φ _h
It can be expressed as.

Using Equation 276 to find the minimum value E _h (φ _h ),

It becomes. From now on, when Bh ² + h + E <0 is satisfied at h = -δ, that is,
Then -δ <-1 / μ
It turns out that it is.

From Equation 277
When is satisfied, E _{h (φ h)} a positive satisfy φ _h> 0 φ _h is present. The domain of φ _{h where} the minimum value E _h (φ _h ) is positive can be obtained as follows. G _h
It is defined as Where g _h (φ) is defined as follows.

Here, Equation 282 is a partial term of Equation 277.

G
It is defined as g _h (x) is x ≥
0 is a monotonically decreasing function of x. Calculating the value of g _h (x) at x = δ / √2
-2 δ ≤ h ≤
-1 / μ
If h ≤ -2 δ
It can be seen that
Φ _h and E _h (φ _h ) determined from N

When δ (> 0) is given, a function ml (x) of x whose domain is x ≧ 0 is defined as follows.
By definition, ml (0) = 0 and when x increases monotonically to infinity, ml (x) monotonically transitions to infinity. Therefore, for a positive number T given arbitrarily,
X that satisfies the condition uniquely exists.

N is a constant by definition and therefore does not depend on h. Rewrite Equation 276 as follows:
On the right side of Equation 289, assuming that δ and μ are constants and N, E, and B are constants given in advance, the right side is a function of h. Equation 289 indicates that φ _h is uniquely determined for an arbitrary h <−1 / μ. Any h ≤
Φ _h is uniquely determined for −1 / μ, and therefore E _h (φ _h ) is determined. Then h ≤
If we write F (h) as a function that makes E _h (φ _h ) correspond to _h satisfying -1 / μ, F (h) becomes

Meet simultaneously h ie
Defined for
This function depends on N, E, and B. When N is increased, the range of h where F (h) <0 widens, and when it is decreased, the range of h becomes narrower. Therefore, the range of h where F (h) <0 can be controlled by N.

If F (h)> 0, then m (h)> 0 for any φ> 0, so the eigenpolynomial m (h) has a real root in the region of h that satisfies F (h)> 0 regardless of φ There is nothing.
Evaluation of real roots near the origin

In order for feedback to be stable, from Equation 240 and Equation 243

It is necessary to hold at the same time.
When holds, the feedback is stable regardless of N.

In an actual circuit, a ripple is superimposed on the output voltage. N is related to the amplification of this ripple, and B is related to the differentiation of the ripple. As N increases, the ripple increases, so N is actually limited by the ripple. As a result, B can be used without sacrificing feedback stability.
It becomes possible. In the actual circuit in relation to ripple

Is a practical choice of constants. When E and B follow Equation 298 and Equation 299, it can be seen that the eigen polynomial m (h) defined in Equation 228 has a real characteristic root in the vicinity of -E.

In other words, in the schematic diagram of the smart power supply shown in FIG. 13, the transfer function of the phase compensation circuit is expressed by E and B that have sufficient DC loop gain implemented in the power supply and satisfy the conditions 298 and 299. Is given by
The eigenpolynomial has a real characteristic root near -E.
Value of eigenpolynomial m (h) Value of eigenpolynomial m (h) at h = -E / 2

For example, consider the value of the eigen polynomial m (h) at h = −E / 2.

Thus, when the equation 301 is rewritten, the following equation is obtained.

E «is a δ, is a δ <Z _e. Also because E μ <1.
Holds. On the other hand, B is approximately equal to 1 / δ, and E << δ
It becomes. φ _e N / Z _e ³ is the DC loop gain at the equilibrium point φ _e , and there are at least several tens.
It turns out that it becomes.
The value of the eigenpolynomial m (h) at h = -2 E

Also consider the value of m (h) at h = -2E.

When Equation 307 is rewritten, the following equation is obtained.

Based on similar discussion

Because
It turns out that it becomes. This shows that m (h) has at least one root in the interval from -1/2 E to -2E. As shown below, it can be seen that m (h) has a root in the vicinity of -E by the same method.
Value of eigenpolynomial m (h) at h = -σ E

The value of m (h) at h = -σ E is

It becomes. σ satisfying m (-σ E) = 0 is obtained from Equation 306 and Equation 312.
It turns out that it is. Also

In the same way
Also because μ E <1
Can be evaluated. φ _e N / Z _e ³ is a direct current loop gain at the equilibrium point φ _e , and there are at least several tens. ^{On the} subterm σ ² EB-σ + 1
Holds.

So a quadratic equation for σ
And approximate the root of Equation 318 with this root. Let σ ₁ and σ ₂ be the roots of the quadratic equation 319, and σ ₁ ≤
σ _{2 is} assumed. Since EB << 1, σ ₁ is in the vicinity of 1. And if we define the function of σ as
The slope of this function at σ = 1 is
Can be evaluated. Since the direct current loop gain N φ _e / Z _e ³ is large, the slope of the straight line becomes steep, and the eigen equation m (h) = 0 has a root in the vicinity of h = −E.
When EB <1/4

From Equation 243 B and E are conditions
Meet. At this time, the functions f ₁ (σ) and f ₂ (σ) of σ are defined as follows based on Equation 313.

m (-σ E) =
Σ that satisfies 0 is the equation f ₁ (σ) -f ₂ (
It can be obtained from σ) = 0. Let the roots of f ₁ (σ) = 0 be σ ₁ , σ ₂ and σ ₁ ≦ σ ₂ . The roots of f ₂ (σ) = 0 include the roots of σ = 0 and σ = 1 / (μ E). σ is [0,1 / (μ
E)]
Because

Therefore, when σ is in the range of [0, 1 / (μ E)], the function f ₂ (σ) can be approximated by a quadratic function f ₃ (σ) relating to σ defined below.
Equation _{f 1 (σ) - f 2} (σ) = equation f ₁ instead of 0 (sigma) -
Find an approximate solution of σ that satisfies m (-σ E) = 0 from f ₃ (σ) = 0.

Consider the intersection of graph y = f ₁ (σ) and graph function y = f ₃ (σ). By the way, if f ₁ (σ) at σ = 1 / (μ E) is calculated,
It becomes. On the other hand, 0 ≤ f ₃ (σ)
In the range of σ σ ≤ 1 / (μ E)
Holds. Therefore, σ at the intersection of graph y = f ₁ (σ) and graph function y = f ₃ (σ) is smaller than σ ₁ . E
Since B <1/4, σ ₁ is
It is evaluated.

conditions

For the root h ₁ near the origin of the eigenpolynomial m (h) = 0 without assuming the magnitude of the DC loop gain
Can be evaluated.
Two functions

The functions f ₁ (h) and f ₂ (h) are defined as follows using the coefficients a ₀ , a ₁ , a ₂ , a ₃ , a ₄ defined in Expression 191.

At this time, the proper polynomial m (h) is
Can be written.

The function f ₁ (h) is not only a function of h but also a function of φ _e . Under the assumption of the condition μ δ >> 1, the function f ₁ (h) is h <0 and φ _e ≧
It becomes a monotonically decreasing function as a function of h when δ / √8, also phi _e
It can be seen that when ≧ δ / √2, a downward convex function is obtained as a function of h as follows.
Function f ₁ (h)

The condition μ δ >> 1 is assumed. A ₀ = as defined in Equation 191
1.
It becomes.

Also, since μ δ >> 1
from now on
And therefore
from now on
Follow,

It becomes. From now on for all h
It turns out that it becomes. Considering the function graph y = f ₁ (h), Equation 349 shows that this graph has a double root at the origin, and always takes a positive value except at the origin.

From now on, Equation 349
And write

It becomes.

On the other hand, using Equation 350, f ₁ (h) is multiplied as follows.
If we calculate the derivative of f ₁ (h)
Regardless of h
When the holds, f ₁ becomes the differential negative f ₁ (h) in h ≦ 0 (h) becomes decreasing function. by the way

from now on
Then, the inequality Equation 355 holds regardless of h, and when h ≦ 0, the first derivative of f ₁ (h) is negative, and f ₁ (h) is a decreasing function.

If we calculate the second derivative of f ₁ (h)
Regardless of h
When the holds, f ₁ becomes twice differentiated positive f ₁ (h) (h) is a convex function below. by the way

from now on
If the inequality equation 361 holds regardless of the h, second order differential of f ₁ (h) is positive, f ₁ (h) is a convex function below.

φ _e is the following condition
Y = f ₁ (h) is a downward convex function that decreases monotonically at h <0. So the straight line y =
There are at most two intersections with f ₂ (h).
Function graph y = f ₂ (h)

The graph y = f ₂ (h) of the function f ₂ (h) is a straight line. The slope of the straight line is -a ₃ , the intersection with the y-axis is -a ₄ , and the intersection with the x-axis is -a ₄ / a ₃ . Here, a ₃ and a ₄ are given by the following formulas 223 to (227.

The slope of the straight line is
The slope of the slope becomes steep as φ _e increases. Also

It turns out that it is. from now on
It turns out that it is.

Considering that enough DC loop gain is built in, from Equation 156
Therefore, Formula 370 is as follows.
In other words, the intersection of the graph y = f ₂ (h) with the h axis can be approximated by -E.
Realization of stable feedback Characteristic root of characteristic polynomial

If the real root near the origin of the eigenpolynomial is e ₀ , e ₀ is the characteristic root of the eigenpolynomial, so the output voltage is approximately
Stand up over time. In other words, the rise time constant is 1 / e ₀ . Also, in a practical setting in an actual circuit, e ₀ is in the vicinity of E, so the rise time constant is approximately 1 / E. As E decreases, e ₀ decreases, and as E increases, e ₀ increases. If E is increased, the start-up becomes faster.

If E is further increased, E will eventually fail to satisfy Equation 240 or Equation 243. As a result, an overshoot appears in the output voltage. This corresponds to the characteristic representative root e ₀ being changed to an imaginary root. This is discussed based on the functions y = f ₁ (h) and y = f ₂ (h) in Fig. 15. The function y = f ₁ (h) does not depend on E, but y = f ₂ (h) depends on E. The slope of y = f ₂ (h) does not depend on E, but the intercept with the y-axis is a function of E. Therefore, when E increases, y = f ₂ (h) translates away from the origin. . As a result, the real roots h ₁ and h ₂ shown in Fig. 15 approach each other as E increases, and h ₁ = h ₂ when the function is in contact. Change.
Time constant μ and multiplier ν of rectifying and smoothing circuit

The time constant μ is introduced in Equation 128 as a time constant for approximating the rectifying / smoothing circuit with a virtual rectifying / smoothing circuit with a first order delay. Parameters describing the resonance circuit include a resonance frequency, a gain at the resonance frequency, and a half width of resonance. As a new parameter for describing the process by which the resonant circuit charges and discharges the rectifying and smoothing circuit, it depends on the time constant μ when the resonant circuit charges and discharges the rectifying and smoothing circuit, the output impedance, the load, and the double voltage of the rectifying circuit. The multiplier ν determined by Here, μ and ν are functions of the output voltage and output current, but they change slowly with changes in the output voltage and output current, so they should be approximated as constants in the vicinity of the specified output voltage and output current. Can do.
Charging and discharging of rectifying and smoothing circuit

In the smart power supply, the output voltage is generated by a rectifying and smoothing circuit. The rectifying / smoothing circuit includes a capacitor therein, and its output voltage is buffered by this capacitor. This buffering reduces the ripple contained in the output voltage. For example, when the output voltage is positive, the rectifying / smoothing circuit can pump the charge into the capacitor and raise the output voltage, but cannot draw the charge from the capacitor and lower the voltage. The output voltage of the rectifying and smoothing circuit cannot drop faster than the time constant determined mainly by the load resistance and the capacitance of the smoothing circuit. If the voltage input to the smoothing circuit drops earlier than this time constant, the output of the rectifier circuit cannot charge the capacitance of the smoothing circuit. That is, no current flows into the rectifying / smoothing circuit.
Energized state

Therefore, an energized state in which a current flows through the rectifying and smoothing circuit and a disconnected state in which no current flows are distinguished. In the disconnected state, the output voltage decreases. A state in which a constant voltage is output under a constant load is an energized state. In a state where a constant voltage specified by a reference voltage is output under a constant load, an increase in current flowing to the rectifying / smoothing circuit that realizes an increase in output voltage and a decrease in current that realizes a decrease are constant. Since this results in a change in current in an equilibrium state that outputs voltage, the operation of the power supply in the energized state can be described by the differential equation system equations 166 to 169, and μ in this differential equation system represents this equilibrium state. It can be seen that this is the time constant of the rectifying and smoothing circuit when the current to be realized flows. This shows that the time constant when the voltage rises matches the time constant when the voltage falls in the first approximation.

Since feedback works effectively in the energized state, the difference between the output voltage and the reference voltage is kept small in the steady state. Therefore, the stability of the differential equation system and the stability of the power source are equivalent. The condition for the power supply to be stable is that the real part of the characteristic representative root is negative and the real part is the same or larger in absolute value than the imaginary part from the condition that the differential equation system is stable.
Disconnected state

For example, when the output voltage is higher than the reference voltage and the output voltage is higher than the reference voltage even if the current flowing through the rectifying and smoothing circuit is reduced to zero, the disconnection state is established. Since feedback does not work effectively in the disconnected state, the difference between the output voltage and the reference voltage is a finite value. In the disconnected state, the output voltage drops by discharging the charge stored in the capacitor through the resistance of the load. The time constant for discharging is not smaller than the time constant determined by the resistance of the capacitor and the load. The disconnected state is switched to the energized state because current starts to flow through the rectifying / smoothing circuit when the output voltage becomes lower than the reference voltage.
Switch from disconnected to energized state

When switched to the energized state, the output voltage increases. In general, in order to increase the output voltage, it is necessary for the resonant circuit to charge the capacitor of the rectifying and smoothing circuit and to supply a current to the load. The time constant for charging decreases with decreasing load and depends on the load. However, since feedback works effectively in the energized state, the time constant of the rise of the output voltage is controlled by feedback and is determined by the characteristic representative root of the eigen polynomial.
Overshoot at output voltage rise

If an overshoot of the output voltage is caused at the rise of the output voltage at this time, the output voltage becomes higher than the reference voltage and a disconnection state is caused again. As a result, a disconnection vibration that repeats the disconnected state and the energized state starts.

In order for the switching vibration not to rise, it is necessary that the increase in output voltage caused by switching from the disconnected state to the energized state does not cause overshoot, and if overshoot does not occur, feedback increases the output voltage. Since it works in the direction, the stability of the feedback results in the stability of the power supply described by the differential equation.

When the characteristic representative root is not a real root, the rise of the output voltage is accompanied by an overshoot, so that a disconnection vibration that repeats the disconnected state and the energized state starts. When the characteristic representative root is a real root, the differential equation is stable. Therefore, the power source described by this differential equation system is stable.

For example, when the output voltage accidentally becomes higher than the reference voltage due to noise, the voltage difference between the output voltage and the reference voltage is very small, so feedback that lowers the output voltage effectively works as described in the differential equation system. Even if the output voltage drops, the rectifying / smoothing circuit cannot lower the output voltage. There is no difference.
B value

For feedback to be stable, E and B are
And formula 243,
It is necessary to satisfy. For example
If you choose E to satisfy, you can see that there is a degree of freedom in choosing B. B is
h _b and formula 260
Therefore, Formula 244, that is,
Therefore, B constrains the position of h ₂ , and as a result, constrains N.

The dependence of h _b on E and B is described by the function F (h) defined in Equation 293. Ie
Satisfy h
For any φ (> 0)
It becomes. F (h) depends on E, B, and N. The range of h where F (h) <0 can be controlled by N. By obtaining F (h) for E, B, and N, h ₂ can be accurately evaluated.
Ripple by rectifying and smoothing circuit

It is the ripple superimposed on the output voltage of the power supply that restricts B and N in the actual circuit. In the power source described by the differential equation system, ripples are not superimposed on the output of the virtual rectifying / smoothing circuit. However, a high-frequency periodic ripple accompanying rectification is superimposed on the output of the actual rectifying / smoothing circuit. Differential equation system
166
In Expression 169, when the periodic ripple is superimposed on z corresponding to the output voltage, z oscillates around the output voltage z _e determined by the equilibrium point φ _e . In an actual power supply, the rise of the output voltage changes according to the differential equation system 166 to 169, but the rise and fall changes independently of this differential equation system. Therefore, the influence of the periodic ripple superimposed on the output voltage on the output voltage is complicated. Periodic ripple causes vibration around the z _e the output voltage.

Due to the periodic ripple, the output voltage oscillates in the vicinity of the voltage set by the reference voltage, and the amplitude of this oscillation increases as the amplitude of the ripple and the loop gain increase. Furthermore, when a large ripple is superimposed on the output of the error amplifier, this may cause a so-called feedback breakdown that lowers the drive frequency beyond the resonance frequency. Feedback breakdown due to ripple is undesirable for the power supply.

When E is given, the rise time constant of the output voltage can be approximated by 1 / E. A large DC loop gain can be realized by reducing the ripple outside the frequency band in which the feedback is effective with a filter. Patent Document 2 proposes a circuit effective for reducing the periodic ripple superimposed on the output voltage.
Selection of circuit constants

If stabilization can be achieved by introducing a pole at the origin, the stability of the output voltage against changes in the load or input voltage is improved. The transfer function of the error amplifier
As a result, a sufficient practical condition was obtained to stabilize the feedback. Parameters describing the resonance circuit include a resonance frequency, a gain at the resonance frequency, and a half width of resonance. To describe the process by which the resonant circuit charges the rectifying and smoothing circuit, the time constant μ is determined by approximating the rectifying and smoothing circuit with a first order delay, and also depending on the output impedance, the load, and the voltage doubler in the rectifying circuit. A multiplier ν was introduced.

When the resonance circuit is fixed, a rough value of δ is determined from the boost ratio realized by the resonance circuit with respect to the load. In addition, the time constant μ for charging the capacitance of the rectifying and smoothing circuit can be estimated from the output impedance of the resonance circuit. The time constant μ for charging the capacitance varies greatly depending on the load. When the maximum value of the time constant μ is μ _max
Choose E so that

A high-frequency ripple accompanying rectification is superimposed on the output of the actual rectifying / smoothing circuit. Since N amplifies ripple, the size of N is limited. From the viewpoint of the differential coefficient of ripple, it is desirable to select B small, and B = 0 is one of the options. From the viewpoint of stability, choosing a smaller B will limit the size of N. In an actual circuit, N is limited by ripple amplification,
Is a practical choice for B. That is, the selection range is from a fraction of 1 / δ to several times.

At this time, the rise time constant of the output voltage is approximately -1 / E, and the DC loop gain can be increased by filtering the ripple outside the frequency band where the feedback is effective.
Right half plane zero

As shown in FIG. 16, when the piezoelectric transformer is driven at a frequency higher than the resonance frequency, it is necessary to drive at a frequency closer to the resonance frequency in order to increase the output voltage, and the drive frequency is lowered. As a result, the frequency of the high-frequency alternating current, which is the output of the piezoelectric transformer, decreases, and the output direct-current voltage obtained by rectifying the high-frequency alternating current may also temporarily decrease. The change in the output voltage caused by the change in the driving frequency becomes large when the load is heavy.

When the drive frequency of the piezoelectric transformer changes, the change in the amplitude of the high-frequency alternating current that is output changes as the standing wave rises in the piezoelectric transformer, so the frequency of the high-frequency alternating current first changes, and then the amplitude of the high-frequency alternating current changes. When the drive frequency is higher than the resonance frequency, the DC voltage obtained by rectifying the high-frequency alternating current may drop temporarily when the drive frequency is lowered, and then rise with an increase in amplitude. There is sex. The degree to which the DC voltage drops depends on the load coupled to the output of the piezoelectric transformer.

In other words, in a power supply that feeds back the output voltage to the drive frequency of the piezoelectric transformer and stabilizes the output DC voltage, if the drive frequency range is selected to be higher than the resonance frequency, the output voltage May go down and then up. This is one of the features of the control system that has a zero point on the right half plane.
How to check the zero point of the right half plane

To confirm the zero point of the right half plane, the direct current voltage is stabilized by feeding back the output voltage to the drive frequency of the piezoelectric transformer instead of directly looking at the change of the direct current voltage that is output when the drive frequency is changed. The stability is compared by simulation for the case where the drive frequency is higher and lower than the resonance frequency. In order to stably return a control system having a zero point on the right half plane, it is necessary to carefully select parameters. When the load is heavy and the loop gain is increased, the existence of the zero point on the right half plane becomes obvious. As a result of simulation, there is a clear difference in stability between the case where the drive frequency is lower than the resonance frequency and the case where the drive frequency is lower, and the case where the drive frequency is lower than the resonance frequency is more stable. In an actual power supply using a piezoelectric transformer, the implementation of the driver circuit that drives the resonance circuit is related to the zero point of the right half plane, but the range of parameters that realizes stable feedback when the drive frequency is lower than the resonance frequency Is wide.
Resonant circuit

When the resonant circuit is a piezoelectric transformer, it is more stable when the drive frequency is lower than the resonant frequency, but the situation where the zero point is generated on the right half plane is common to the piezoelectric transformer and the general resonant circuit. Therefore, even in the case of a general resonance circuit, the case where the drive frequency is lower than the resonance frequency is more stable than the case where it is high.

The present invention provides a feedback circuit that stabilizes a DC voltage obtained by rectifying high-frequency alternating current output from a resonant circuit, and a circuit constant thereof. As a result, for example, a high-voltage power supply using a piezoelectric transformer as a resonance circuit can stabilize the output voltage with high accuracy against fluctuations in the input voltage and changes in the load current. Enables supply of variable output voltage to the load.
In order to stabilize the output voltage with high voltage accuracy, it is necessary to introduce a pole located at the origin to the feedback circuit that stabilizes the output voltage.
In the present invention, the transfer function of the error amplifier is
As a result, a sufficient practical condition was obtained to stabilize the feedback.
Consider an eigen polynomial m (h) of a linear differential equation system that approximates the differential equation system linearly in the vicinity of the equilibrium point of the differential equation system that describes the power source. When the eigenpolynomial m (h) has a real root located near the origin and a real root conjugated to it, stable feedback is realized by determining E and B so that the real part of the other root is negative. . If a sufficient DC loop gain is implemented in the power supply and E is not large, the eigenpolynomial has a real root near -E. A real root conjugate with a real root in the vicinity of -E can be roughly evaluated as -1 / B, and 1 / δ is a reference B value.
For example, when the output voltage is positive, the rectifying and smoothing circuit can increase the output voltage, but cannot decrease the output voltage. The feedback works effectively only when the output voltage is in the direction away from the ground, and the characteristic representative root of m (h) needs to be a real root. It is always possible to select E and B so that the characteristic root is the real root, and at this time, the time constant of the rise of the output voltage can be approximated to 1 / E.
When the voltage generation circuit of the power supply is given in advance, a transfer function satisfying a sufficient condition having a pole located at the origin and two zero points is obtained from the parameters of the voltage generation circuit, and an error amplifier that realizes this transfer function is obtained. Including feedback becomes stable. In addition, the characteristics of the power supply thus realized can be predicted in advance.

As an embodiment of the present invention, the transfer function of the phase compensation circuit is determined from the parameters of the high voltage generation circuit according to the method of the present invention for the DC stabilized high voltage power supply comprising the high voltage generation circuit using the piezoelectric transformer as the resonance circuit and its feedback circuit. And show that the feedback is stable.

A direct current stabilized high voltage power source using a piezoelectric transformer as a resonance circuit is composed of a high voltage generation circuit and a feedback circuit. The high voltage generation circuit is composed of a driver circuit, a piezoelectric transformer used as a resonance circuit, and a rectifying / smoothing circuit. The feedback circuit includes an error amplifier and a frequency modulation circuit. The driver circuit converts a DC voltage supplied from an external power source to the driver circuit into a high-frequency alternating current having the same frequency as the rectangular wave output from the frequency modulation circuit, and drives the piezoelectric transformer by the high-frequency alternating current.
The piezoelectric transformer boosts this high frequency alternating current to a high voltage high frequency alternating current. The rectifying / smoothing circuit converts the output of the piezoelectric transformer into a DC high voltage, supplies this to the load as the output of the high-voltage power supply, and inputs it to the feedback circuit. The error amplifier detects the deviation by comparing the output voltage input to the feedback circuit with the reference voltage, and inputs this deviation to the frequency modulation circuit. The frequency modulation circuit outputs a rectangular wave having a frequency proportional to the input to the driver circuit. In this way, the output voltage is fed back to the frequency of the high-frequency alternating current that drives the piezoelectric transformer and stabilized.

As shown in FIG. 1, the frequency of the high-frequency alternating current that drives the piezoelectric transformer is selected to be higher than the resonance frequency of the piezoelectric transformer. Therefore, when the output voltage is higher than the reference voltage, the frequency is increased to move away from the resonance frequency, and in the opposite case, the frequency is decreased to approach the resonance frequency.

This DC stabilized high-voltage power supply can supply 2 kV to 4 kV DC voltage to a 20 MΩ or 200 MΩ load. This DC stabilized high-voltage power supply will be described with reference to FIG. Next, a simulation circuit for simulating this high voltage power supply is constructed. Furthermore, circuit constants that realize stable operation of the high-voltage power supply are obtained from parameter measurements based on this simulation. Finally, simulations show that this circuit constant power supply operates stably over a wide range of loads and output voltages.
Piezoelectric transformer

Piezoelectric transformers utilize the piezoelectric effect of previously polarized piezoelectric elements. When an external force is applied to the piezoelectric element and deformed, a voltage is generated. Conversely, when a voltage is applied, a stress is generated and deformed. A piezoelectric transformer uses this effect to transmit electrical energy by converting electrical vibrations into mechanical vibrations on the primary side and transmitting them to the secondary side, and returning them to electrical vibrations on the secondary side. . Thus, in the piezoelectric transformer, the electrical energy is converted into mechanical vibration on the primary side, and this is converted back to electrical energy again on the secondary side. The secondary side is a capacitance, and a voltage is generated when charges are injected through mechanical vibration.

Mechanical vibrations are gradually attenuated due to energy dissipation by the load. The decay time constant increases with load R. When the output is high voltage, the value of R is generally large. In this sense, a piezoelectric transformer that stores energy by mechanical vibration is suitable for a DC high-voltage power supply.

The piezoelectric transformer includes a resonance circuit inside. For this reason, unlike a normal electromagnetic transformer, the piezoelectric transformer exhibits sharp frequency characteristics and large load dependence. This piezoelectric transformer is used in a high voltage generation circuit. A load resistance is connected to the output of the piezoelectric transformer, and a step-up ratio that is a ratio between the input voltage and the output voltage is considered. FIG. 19 shows a graph in which the step-up ratio is actually measured as a function of frequency for each load resistance. From this graph, it can be seen that the piezoelectric transformer exhibits a large step-up ratio near the resonance frequency.
Rectifier smoothing circuit

As shown in FIG. 18, the rectifying / smoothing circuit includes a step-up rectifying circuit including a three-stage Cockloft-Walton circuit in which capacitors and diodes are connected in cascade, and an output capacitance for the purpose of reducing ripples. The output high voltage is generated by a Cockloft-Walton circuit. When the value of the load connected to the output of the Cockroft-Walton circuit is very large, an ideal n-stage Cockloft-Walton circuit will produce a DC of 2nE voltage when an alternating current of amplitude E is applied to the input. Output. Although the voltage boost ratio in this boost rectification depends on the load, when the load connected to the output of the Cockloft-Walton circuit is viewed from the input of the Cockloft-Walton circuit, the magnitude of the load is 2 of this voltage boost ratio. Means inversely proportional to the power. That is, the Cockloft-Walton circuit works as a resistance converter together with the boost rectifier circuit.

The output of the Cockloft-Walton circuit is buffered by the output capacitor. This buffering reduces the ripple contained in the output voltage. If the output voltage is positive, the Cockroft-Walton circuit can pump charge into the capacitance and generate a high voltage, but cannot draw charge from the capacitance and lower the high voltage. The electric charge stored in the piezoelectric transformer is pumped up in one cycle of the carrier wave. Considering the capacitance of a piezoelectric transformer that stores electric charge on the output side of the piezoelectric transformer
Therefore, if the combined capacitance of the Cockloft-Walton circuit and the capacitance of the buffer is estimated to be 100 nF, approximately 6000 cycles are required to reach equilibrium. Since the period required to reach equilibrium is considered to be determined by the ratio of the capacitance of the piezoelectric transformer and the rectifying / smoothing circuit, the first approximation does not depend on the high voltage of the load or output.

Considering that the carrier frequency is around 120 kHz, it can be seen that at least 50 msec is required to reach equilibrium. In other words, the rise time constant of the high voltage that is the output does not become shorter than 50 msec. In order for the output high voltage to actually rise, several hundreds of milliseconds are required. The time to reach equilibrium is considered to be approximately equal to the time constant μ of the rectifying / smoothing circuit.
Can be estimated.

When the output voltage is positive, considering the change in the output voltage, in order for the voltage to rise, it is necessary to charge the capacitor of the Cockloft-Walton circuit and the output capacitor at the same time that the piezoelectric transformer supplies current to the load. . The output high voltage is lowered by discharging these capacitors through the resistance of the load. The time constant when the voltage rises becomes smaller as the load becomes lighter. The time constant when the output high voltage drops is determined by the resistance of the capacitor and the load.
Driver circuit

The capacitance can be seen when the piezoelectric transformer is viewed from the input terminal. The use of a sine wave is essential to drive the capacitance efficiently, and by resonating with the inductance, the driver circuit produces an approximate sine wave.

The driver circuit is composed of two sets of resonance circuits composed of two inductances L1 and L2 and two MOSFETs Q1 and Q2, and an FET drive circuit. The output of the frequency modulation circuit is input to the FET drive circuit. The FET drive circuit alternately turns on and off two sets of FETs in synchronization with the pulse wave that is the output of the frequency modulation circuit. The inductance value is determined so that the resonance frequency determined by this inductance and the capacitance of the piezoelectric transformer is substantially equal to the resonance frequency of the piezoelectric transformer. As a result, the so-called zero volt switching, which is performed when the FET is turned on and off when the voltage applied to the FET is approximately 0 V, is realized.
Error amplifier

The error amplifier includes a dividing resistor, a subtraction circuit, and a phase compensation circuit. The subtraction circuit compares the output voltage divided by the dividing resistor and input to the terminal X with the reference voltage supplied from the outside to the terminal Y to set the output voltage, and the difference between the voltages is input to the terminal Z. The reference operating voltage is added and output.
The division ratio of the dividing resistor is defined by Equation 131. Accordingly, when the division ratio of this dividing resistor is d ₀ , the following is obtained.

The phase compensation circuit is an inverting amplifier whose reference operating voltage is the ground potential, and this transfer function has a pole arranged at the origin and two zero points. The input of the phase compensation circuit is converted by this transfer function and output. This output is clamped to a limited range of voltages from the reference operating voltage by diode D1.
Frequency modulation circuit

The frequency modulation circuit includes a voltage controlled oscillator and a frequency divider. A TLC555 used as a timer is used as a voltage controlled oscillator. A rectangular wave having a frequency determined by the voltage input to terminal A is output from terminal B. This rectangular wave is input to a terminal CLK of a frequency divider composed of a flip-flop 74HC73, and a rectangular wave with a duty ratio of 50% divided by 1/2 is output from the Q and bar Q terminals. A rectangular wave from 85 kHz to 139 kHz output from the frequency divider is the output of the frequency modulation circuit, and this is input to the drive circuit.

When the voltage input to terminal A changes by 1V, the frequency output from the Q and bar Q terminals changes by about 40 kHz. When k defined by Equation 65 is calculated and this is k ₀ ,

Auxiliary power

The auxiliary power supply is a stabilized power supply for supplying a reference operating voltage to the error detection circuit.
Circuit for simulation

Figure 20 shows a simulation circuit for this DC stabilized high-voltage power supply. The parameters are measured by simulating a DC stabilized high-voltage power supply, and N, E, and B that realize stable feedback are obtained based on this, and the simulation shows that the feedback is stable.
The high voltage generation circuit in the simulation circuit is a faithful reproduction of the DC stabilized high voltage power supply circuit except that the piezoelectric transformer is replaced by an equivalent circuit. The feedback circuit in the simulation circuit is basically a linear circuit. For this reason, a simple circuit that reproduces the relationship between the input and output of the feedback circuit is employed in the simulation circuit.
Equivalent circuit of piezoelectric transformer

Fig. 21 shows the equivalent circuit of the piezoelectric transformer used in this high-voltage power supply and its parameters. An equivalent circuit of the piezoelectric transformer includes an ideal transformer. Consider an ideal transformer in which the winding ratio of the primary coil and the secondary coil is n as shown in FIG. Primary voltage E ₁ and primary current I ₁ Secondary voltage E ₂ and primary current
Satisfies the following relational expression. For the simulation of the ideal transformer, a circuit shown in FIG. 23 is used which realizes an ideal transformer using a voltage-controlled voltage source and a current-controlled current source.

Frequency modulation circuit model

Circuit elements called behavior models that can specify the relationship between input and output using mathematical relational expressions can be used for simulation. The simulation circuit of the frequency modulation circuit is configured as shown in FIG. 24 by combining two behavior models A and B and an amplitude limiter. The amplitude limiter can specify an upper limit HI and a lower limit LO of the amplitude and a gain GAIN. Behavior model A outputs the integral of the input. In behavior model B, an equation corresponding to Equation 63 is set. As a result, a sine wave having a frequency proportional to the voltage input to the behavior model A (that is, the output of the error amplifier) is output from the behavior model B. The amplitude limiter amplifies this sine wave and clips its amplitude. In this way, a 50% duty rectangular wave having a frequency proportional to the input voltage is output.
Error amplifier simulation circuit

Figure 25 shows the error amplifier simulation circuit. The error amplifier is composed of a subtraction circuit and a phase compensation circuit. The subtraction circuit consists of a 2-input 1-output behavior model and an amplifier. The behavior model is set so that the input difference becomes the output. The output of the behavior model is amplified by an amplifier and input to a phase compensation circuit.

The phase compensation circuit includes an integrator, a gain 1 amplifier, an integrating circuit attached thereto, a differentiator, an integrating circuit attached thereto, and two 2-input adders.
The integrator is composed of a gain E amplifier and a behavior model. The behavior model set with the function SDT (x) outputs the time integral of the input. The output from this behavior model is input to an amplifier having a gain E, and the output from this amplifier becomes the output of the integrator.

The differentiator is composed of a gain B amplifier and a behavior model. The behavior model with the function DDT (x) is set to output the time derivative of the input. The output from this behavior model is input to an amplifier having a gain B, and the output from this amplifier becomes the output of the differentiator.

The integration circuit attached to the gain 1 amplifier and the differentiator is intended to suppress high frequency noise by integrating it. When the integrator gain is E and the differentiator gain is B, the transfer function of the phase compensation circuit is
Given by. In this figure, S is a voltage source that generates a reference voltage. The gain set for the amplifier of the subtraction circuit is C. The DC loop gain can be controlled by changing the gain set in the amplifier of the subtraction circuit. In the simulation circuit of FIG. 20, E = 10, B = 0.0002, and C = 30 are set.
Approximation of resonance characteristics

Various factors that determine the characteristics of the resonance circuit, such as the amplitude of the carrier wave input to the resonance circuit of the high-voltage power supply, are fixed, and this determines the parameter r of the resonance circuit. The parameter δ of the resonant circuit is determined by the load connected to the power source. The output voltage at the resonance frequency of the resonance circuit to which this load is connected is
Given in. When the load is light, δ decreases and the output voltage at the resonance frequency increases. When the load corresponds to δ, the output voltage is a function of φ. When φ is φ _e, the output voltage is
Given in. As φ _e decreases, the output voltage increases, and as φ _e increases, it decreases.

In circuits where feedback works effectively, the characteristics are mainly determined by feedback. In the case where feedback works effectively for a DC stabilized high voltage power supply, the characteristics of the power supply do not greatly depend on the characteristics of the high voltage generation circuit or the rectifying / smoothing circuit. In this sense, the characteristics of the power supply do not depend sensitively on, for example, r of the high voltage generation circuit or μ or ν of the rectifying / smoothing circuit. So, consider a simple way to roughly estimate r.
Rough estimate of r

For the rectifying and smoothing circuit connected to the load, the transfer function from the input voltage to the output voltage is
, Ν depends on the load or output voltage. The change in load is reflected in δ of the resonance circuit, and the change in output voltage is reflected in φ. In this sense, μ and ν are considered to be functions of δ and φ. However, it is not easy to obtain specific expressions for these functions.

Therefore, consider a high voltage generating circuit composed of a piezoelectric transformer and a rectifying / smoothing circuit. When a load given in advance is connected to the output of the high voltage generation circuit, the output high voltage depends on the frequency of the carrier wave input to the high voltage generation circuit. When plotting the high voltage output against frequency, this graph shows the resonance characteristics. The first approximation of this resonance characteristic is
It can be seen that is given by. By this approximation, the half-value width regarding the frequency of the resonance characteristic can be obtained. When rewriting this half-value width to angular velocity, δ is set, and the maximum value of high voltage is v _max
R can be obtained from It is obvious that ν = 1 when r is obtained directly from the output of the high voltage generation circuit.
Approximate μ estimate

The time constant μ is a time constant when the rectifying and smoothing circuit is linearly approximated with a first-order lag, and its upper limit is given in Equation 389. The delay of the rectifying / smoothing circuit is considered to be sufficiently larger than the delay of the resonance circuit. In other words, between δ and μ
Holds. In addition, since the root in the vicinity of -E of the eigenpolynominal hardly depends on μ, an accurate value of μ is not required for selection of circuit constants.
Measurement of parameters of high voltage generation circuit

Figure 26 shows the measurement circuit for resonance characteristics. The resonance characteristic obtained from this measurement is the resonance characteristic of a high voltage generation circuit composed of a driver circuit, a piezoelectric transformer, and a rectifying / smoothing circuit. The resonance characteristics measured in 1 kHz steps with different load resistances are shown below. Calculate r from the resonance characteristics of each load according to Equation 398. This r depends on the load. The r calculated from the resonance characteristics below is a rough approximation, but takes a narrow range regardless of the load because the first approximation of the resonance characteristics is expressed by Equation 397.
Resonance characteristics with a load of 20 MΩ

If the output voltage at the resonance frequency is 7.5 kV and the half width is 980 Hz in frequency, δ is 6158, and from this, r ₂₀ can be estimated as 0.46 × 10 ⁸ .
Resonance characteristics of load 30 MΩ

If the output voltage at the resonance frequency is 8.5 kV and the half width is 930 Hz in frequency, δ is 5843, and from this, r ₃₀ can be estimated to be 0.49 × 10 ⁸ .
Resonance characteristics of load 40MΩ

If the output voltage at the resonance frequency is 9.1 kV and the half width is 910 Hz in terms of frequency, δ is 5718. From this, r ₄₀ can be estimated to be 0.52 × 10 ⁸ .
Resonance characteristics of load 50MΩ

If the output voltage at the resonance frequency is 9.1 kV and the half width is 910 Hz in terms of frequency, δ is 5718. From this, r ₄₀ can be estimated to be 0.52 × 10 ⁸ .
DC loop gain

When the gain of the error amplifier subtraction circuit is written as C, N is expressed by Equation 154.
Can be written. Where ν = 1 and r, k, d are replaced by r ₀ , k ₀ , d ₀ respectively,
It becomes. At this time, as shown in Fig. 12, the maximum value of the DC loop gain is
Given in. This value is 134 when C = 1 and δ = 6000.
About E

About the time constant of the rise of the output high voltage, E from the evaluation of Formula 389
Get. Therefore, E = 10. B ≥ for practical ranges of φ, δ, μ
It can be seen that 0.0002 is sufficient. Therefore, B = 0.0002.
Stable feedback simulation example

The use of the simulation circuit 20 shows that the feedback thus constructed is stable. Figures 31 and 32 show the simulation results for the case where a voltage of 4 kV is output to a load resistance of 20 MΩ and the case of a voltage of 2 kV being output to a load resistance of 200 MΩ. In each figure, the output of the error amplifier, the output of the subtraction circuit, and the time course of the reference voltage are plotted with the horizontal axis as the time axis, the vertical axis 1 as the error amplifier output, the vertical axis 2 as the subtraction circuit output, and the vertical axis. 3 is shown as a reference voltage.

Resonance frequency and drive frequency range Comparison of step-up ratio between transfer function h ₀ (s) and its approximation h ^T ₀ (s) Comparison of phase of transfer function h ₀ (s) and its approximation h ^T ₀ (s) Frequency dependence of step-up ratio Q factor dependence of step-up ratio Load dependence of resonance frequency Power supply Resonant circuit and rectifying / smoothing circuit Virtual resonance circuit and virtual rectification smoothing circuit Equivalent power supply Example of smart power supply Φ dependency of DC loop gain Schematic diagram of smart power supply The real root near the origin and the intersection of the graph y = f ₁ (h) and y = f ₂ (h) Intersection of graph y = f ₁ (h) and y = f ₂ (h) Piezoelectric transformer resonance characteristics and drive frequency range Block diagram of DC stabilized high voltage power supply using piezoelectric transformer DC stabilized high voltage power supply using piezoelectric transformer Frequency dependence of step-up ratio by measurement Circuit for simulation Equivalent circuit of piezoelectric transformer Ideal transformer Ideal transformer simulation circuit Frequency modulation circuit simulation circuit Error amplifier simulation circuit Resonance characteristic measurement circuit Resonance characteristics of load 20MΩ Resonance characteristics of load 30MΩ Resonance characteristics of load 40MΩ Resonance characteristics of load 50MΩ When outputting 4 kV to a load of 20 MΩ When outputting 2 kV to a load of 200 MΩ

Claims (6)

When the resonance frequency of the resonance circuit is ω _r , the Q value is Q, and the boost ratio at the resonance frequency is g _r , δ, ω ₀ and c are

And a frequency-modulated carrier wave with the constant w as the amplitude and the time function ψ as the phase.
The transfer function of a resonant circuit where the modulation band of the carrier wave is sufficiently narrow compared to the resonant frequency of the resonant circuit.
And the mapping from ψ to φ in Equation 4
By defining the frequency of the frequency-modulated carrier wave described in Equation 4 as
And r _r and r _i

When the carrier wave given by Equation 4 is input to the resonance circuit whose transfer function is given by Equation 5, the amplitude of the carrier wave output from the resonance circuit is the simultaneous differential equation.

P, q satisfying
Z represents the temporary delay with the DC voltage generated by rectifying and smoothing the carrier wave output from the resonant circuit, the time constant of the rectifying and smoothing circuit is μ, and the multiplier of the amplitude in the rectifying and smoothing circuit is ν. Differential equation with respect to
As a solution of
Further, the output voltage z from the rectifying and smoothing circuit is compared with the reference voltage λ, and the transfer function of the feedback circuit that feeds back the voltage error to the carrier frequency φ is φ defined in Equation 6, z in Equation 13, and the reference voltage λ And positive numbers k, d, E, A, B, and φ ≥ 0
Therefore, by connecting this transfer function with the differential equations of Equation 10, Equation 11, and Equation 13, the following normal differential equation system is derived,

The equilibrium point of this differential equation system is expressed as (p _e , q _e , z _e ,
φ _e ), p _e , q _e , z _e and λ are functions of φ _e

Let m (h) be the eigenpolynomial of the differential equation linearized in the vicinity of this equilibrium point, and m (h)
And
As coefficients a ₀ , a ₁ , a ₂ ,
a ₃ and a ₄ are

Given by
When φ ≦ 0, by setting φ to −φ, the transfer function of the feedback circuit that feeds back the voltage error to the carrier frequency is expressed by Equation 14. In Equations 10 and 11, φ is −φ. If you put

So that the range of φ is expressed as φ ≧
By deriving as a differential equation set to 0, a normal differential equation system corresponding to Equations 15 to 18 is derived, and an eigen polynomial of the differential equation linearized in the vicinity of this equilibrium point is m (h), and m (h h)
And
As coefficients a ₀ , a ₁ , a ₂ ,
Since a ₃ and a ₄ are again given by Equation 25 to Equation 29, in each case, the coefficient
It can be seen that a ₀ , a ₁ , a ₂ , a ₃ , a ₄ are given by Equation 25 to Equation 29 with φ ≧ 0,
Furthermore, in Equation 25 to Equation 29, the term kd ν r appears together with A, B, or E, so
And then B ’and E ′

That is

And B 'and E' are written again as B and E. The coefficients a ₀ , a ₁ ,
a ₂ , a ₃ , a ₄ are as follows

redefined as a function of φ _e
Using the coefficients a ₀ , a ₁ , a ₂ , a ₃ , and a ₄ of Equation 39 to Equation 43, the intrinsic polynomial m (h) defined in Equation 23 and Equation 32 is used.
And the eigenfunction m (h) can be written as
Assuming that the transfer function of the high-voltage generation circuit is F _w and the transfer function of the feedback circuit is B _k , except for the difference between CF _w and (1 / C) B _k for constant constant and the common constant C from Equation 45, F About _w
And about B _k
The closed-loop transfer function F _w / (1 + F _w
B _k )

Therefore, it can be seen that the eigenfunction m (h) matches the denominator of the closed-loop transfer function F _w / (1 + F _w B _k ), except for a constant multiple, and that all roots of the eigenpolynomial m (h) are real. By selecting E, B, and N so that the part is negative, stable feedback is realized in the sense that the real part of all roots of the denominator of the closed-loop transfer function consisting of the high-voltage generator and feedback circuit is negative A method characterized by that. From equation 45 of claim 1, the eigenfunction m (h) is

Since this first term is positive when h <-1 / μ, the region S of h is
For any h belonging to S, m (h)> 0 regardless of φ, so the real root of m (h) = 0 is included in the complement of S.
E _h (φ) where m (h) is considered to be a function of φ for any h belonging to R assuming that R is not an empty set.
Defining and, uniquely determined for any h belonging to R phi _h in E _h (phi) is the minimum value E _h (phi _h) That
Therefore, a function F that uniquely corresponds E _h (φ _h ) to any h belonging to R, that is,
About this F
Since h (m)> 0 regardless of φ, F is E,
A method characterized by limiting the range of real roots of m (h) = 0 by appropriately selecting E, B, and N by utilizing the dependence on B and N. A DC stabilized high-voltage power supply rectifies a driver circuit that generates a carrier wave that drives a resonance circuit, a resonance circuit that receives the carrier wave, and a high-frequency alternating current that is the output of the resonance circuit, thereby generating a DC voltage that is output from the power supply. Generates a frequency of a carrier wave that drives a resonance circuit and an error amplifier that receives a voltage generation circuit including a rectifying and smoothing circuit to be generated, and an output voltage of the voltage generation circuit and a reference voltage that sets an output voltage supplied from outside The frequency modulator controlled by the error amplifier has means for controlling the frequency of the carrier wave generated by the driver circuit, and modulates the frequency of the carrier wave driving the resonant circuit. The output of the resonant circuit that converts the carrier wave from frequency modulation to amplitude modulation is a carrier wave that is amplitude modulated by the output of the error amplifier, and is a rectifying and smoothing circuit Therefore, the feedback that stabilizes the output voltage realized by inputting the output voltage demodulated from the amplitude-modulated carrier wave to the error amplifier works only in the direction in which the output voltage of the rectifying and smoothing circuit is away from the ground. It is characterized by stabilizing the output voltage for a wide range of loads connected to the power supply by realizing feedback such that the characteristic representative root of the intrinsic polynomial equation (44) is a real root separated from the imaginary root. Circuit In the DC stabilized high voltage power supply given in claim 3, the transfer function of the phase compensation circuit is expressed by using positive numbers E and B.
And E and B are

Is satisfied, the characteristic representative root of the proper polynomial m (h) defined in Equation 44 is a real root located near the origin, and F (h) defined in claim 2 is N, E, B By selecting N, E, and B appropriately using the dependence on F, and realizing F (h) that separates the characteristic representative root from the imaginary root, the real root separated from the imaginary root is returned as the characteristic representative root. A method characterized by realizing. Transfer function of phase compensation circuit using positive numbers E and B
In the DC-stabilized high-voltage power supply according to claim 3, the ripple is superimposed on the output voltage, and N is related to the amplification of the ripple or B is related to the differential of the ripple. When constraining, E and B are

Becomes a realistic choice, and the characteristic representative root near the origin is -ξ
The rise time constant of the output voltage is at least longer than 1 / (2 E), so the ripple outside the frequency band where feedback is effective is attenuated by the filter to increase the DC loop gain of feedback. A method characterized by: A DC stabilized high-voltage power supply rectifies a driver circuit that generates a carrier wave that drives a resonance circuit, a resonance circuit that receives the carrier wave, and a high-frequency alternating current that is the output of the resonance circuit, thereby generating a DC voltage that is output from the power supply. Generates a voltage generator circuit including a rectifying / smoothing circuit to be generated, an output voltage of the voltage generator circuit, and a reference voltage supplied from the outside for setting the output voltage, and a frequency of a carrier wave driving the resonance circuit The frequency modulator controlled by the error amplifier has a means for controlling the frequency of the carrier wave generated by the driver circuit, and a resonance circuit for the DC voltage that is the output of the voltage generating circuit. In the power supply that stabilizes the output voltage by feeding back to the frequency of the carrier wave that drives the By driving the resonance circuit with a carrier wave having a frequency, the amplitude of the carrier wave output from the resonance circuit is increased or decreased when the resonance circuit is driven with a carrier wave having a frequency higher than the resonance frequency of the resonance circuit. A circuit that avoids changes in the frequency accompanying a decrease or increase in frequency from temporarily acting to prevent changes in the output voltage of the rectifying and smoothing circuit due to changes in amplitude.