DE4205300C1 - Digital determination of phase and amplitude of periodic signal in phase locked loop - sampling periodic signal using sampling period to give constant whole number of measured value samples and phase estimates per period - Google Patents

Abstract

The method involves performing synchronised digital sampling of the periodic signal (u(t)) and varying the sampling period (TA) such that a whole number of samples are taken per signal period. The samples are Fourier analysed and the resulting coefficients of the cosine terms of the fundamental are used to determine the phase of the fundamental. The coefficients of the sine terms of the fundamental are used to determine its amplitude. The coefficients of the cosine terms are fed as a phase error signal a controller in the phase locked loop. The sine and cosine functions for a sampling phase angle can be stored in a look-up table. USE/ADVANTAGE - E.g. for digital control of power converters. Allows simple pre-estimation of input signal.

Description

The invention relates to a method according to the preamble of claim 1.

The basic structure of a phase-locked loop (PLL) is shown in FIG. 2. The PLL consists of a phase detector 1 , a filter 2 and a controlled oscillator 3 (VCO). A periodic input signal u is linked via the phase detector 1 to a signal v generated by the oscillator 3 , the frequency of which in the steady state is the same as that of the input signal u. This consists, for example, of a multiplier or a simple logic gate, which then only processes the signs of the two signals. The output signal d of the phase detector 1 essentially consists of a frequency component with the sum of the two input frequencies (stationary double input frequency) and a difference frequency element, which causes a constant component in a stationary manner. This stationary or at least only slowly changing DC component is a measure of the phase difference between the two signals u and v and can therefore be used after filtering via the filter 2 designed as a low-pass filter to control the oscillator 3 , the input signal of which is then an estimate of the input frequency ω, so that a phase-locked coupling between the signals u and v is achieved.

On a further presentation of the basics of PLL circuits is omitted here because with the book by R. Best "Theory and Applications des Phase-Locked Loops "AT-Verlag, Aarau (Switzerland), 1981, 2nd supplementary edition a good introduction is available (see in particular pages 11 to 13, 15 to 18).

The regulation of converters in digital mode is becoming more and more common carried out. Since such a control is sampling (time-discrete) works, the necessary control variables or voltages can be determined precisely for these sampling times. This means immediately that the phase and amplitude detection using such a scheme should work together in the same time grid as the regulation must work. If these digital controls work with a constant sampling time, one can digital PLL essentially according to that known from analog technology, above described procedures are built.

Various requirements (e.g. avoidance of beating symptoms) condition that the digital controls of converters are not constant Time cycle allowed to work, but with a fixed number of cycles synchronize the period of the fundamental oscillation (e.g. the mains voltage) have to. If the mains voltage varies in frequency, the frequency must also of the timing (sampling frequency) vary so that the desired number of Clocking (sampling intervals) per period remains constant. These special requirements have the consequence that the usual standard solution for phase detection, a phase locked loop or a PLL in the form described above is not easy to take over.

In the unpublished DE 40 32 441 C1, which, however, on a Older patent application is a method for measuring the phase position two analog signals of the same frequency are described, in which the Sampling rate based on the duration of the subsequent oscillation is constant in the range of ± 50%.  

A phase comparator is known from DE 35 12 216 C2, in which a Sawtooth signal is shifted when a strobe hits the sloping area of the sawtooth signal. This shift ensures that the sampling pulse always falls in the inclined rising edge of the sawtooth pulse.

The invention has for its object a method of the aforementioned Specify type that is easy to implement.

This object is achieved according to the invention by those characterized in claim 1 Process steps solved.

The advantageous peculiarity of this method lies in the digital  (Time-discrete) mode of operation, the samples themselves being phase-synchronized respectively. The sampling frequency of the control is therefore not temporal more constant. Rather, it takes place per period of the signal to be sampled a constant number of samples, so that the sampling frequency or the sampling period at variable frequency of the signal to be sampled also changed. This makes it possible to have a pulse pattern synchronized with the mains voltage to use which beats compared to asynchronous pulsing avoids. The synchronous sampling has an advantageous side effect: To determine the phase and also the amplitude are trigonometric Functions necessary. But because of the synchronization only the same angles corresponding to the number of samples can occur for each period, the function values of the sine and cosine functions for these angles are stored in a table that is easy to save in a computer.

Advantageous training and further developments of the method according to the invention are characterized in claims 2 to 13.

The method according to the invention will be explained below with reference to a structural diagram shown in FIG. 1:

In addition to the components of a phase detector 1 , which is designed here as a multiplier, a filter 2 and an oscillator 3 , which have already been explained in relation to FIG. 2, a scanning analog / digital converter (sample and hold) 7 is provided as an input element, to which an analog time-varying periodic signal u (t), for example an AC voltage, is supplied and which transmits a clocked, sampled signal u (k) to the filter 2 .

The phase position (the phase angle) and the amplitude of the fundamental oscillation are now to be determined from the signal u (t). Since a constant number of samples per period is to be used, it is therefore necessary to change the sampling period T _A when the frequency of the signal u (t) changes, and thus its period, and to provide a means of intervention for adjusting T _A.

Since all structural elements work with this time cycle, an adjustment of T _A affects every element.

Furthermore, a PI controller 4 and, within the "oscillator" 3 modified for the purposes of the invention, a memory element 5 and a cos characteristic curve generator 6 are available for carrying out the method according to the invention in accordance with the structure diagram of FIG. 1.

As previously described, a digital one (not shown here) is said to be Regulation at discrete times with the current values above Phase and amplitude of the periodic signal u (t) are supplied. However, the number of samples of the network period should be constant, so that this forces a synchronous operation of the control and PLL becomes.

In order to be able to carry out the synchronization between sampling and periodic signal, the phase angle ϕ of the periodic signal u (t) must be available. However, the PLL is responsible for determining the phase angle. This will provide an estimated or observation value for the angle ϕ, which must be used instead of the inaccessible phase angle ϕ. If one synchronizes the computing cycle of the PLL with the angle it supplies itself, the result is that the increment ϕ _{inc of} the phase angle between two samples is always constant:

(k + 1) = (k) + ϕ _inc ,

in which

and n _{A are} the constant number of samples per period, so that:

(k) = kϕ _inc .

The phase feedback control is carried out by varying the sampling period T _A (interval between two samples).

As already mentioned, such a synchronous scanning has an additional advantage: to determine the phase and also the amplitude - as will be explained in detail below - trigonometric functions are necessary. However, since only multiples of the increment ϕ _inc occur due to the synchronization, the function values of the sine and cosine functions for these angles can be stored in a table in the memory element 5 . Because of the symmetries of the trigonometric functions, for a number of, for example, n _A = 88 samples per network period, these are only 22 values that must be stored. Interpolations between these values are not required.

All the structural elements of Fig. 1 operate in discrete time (scanned), being triggered at intervals of the sampling period T _A, which provides the Regeler 4 at its output. In comparison with the basic structure according to FIG. 2, it is unusual that the frequency no longer appears; it is also not required directly. For a better understanding you can still use the frequency over the difference quotient

form. At the same time, this indicates how the output signals and T _{A of} the filter 2 in the structures of FIGS. 1 and 2 are related to one another.

The input of the filter 2 is fed from the multiplier is designed as a phase detector 1, that is the output signal of the phase detector 1 is:

d (k) = u (k) * v (k).

With

u (k) = a₁sin ϕ (k) and v (k) = cos (k)

this results

With a ₁ the initially unknown fundamental vibration amplitude of the input signal u (t) is designated. The filter 2 should now suppress the oscillating term sin (ϕ (k) + (k)) in the signal d (k) and, if possible, only provide the term of the differential phase angle as the signal e (k). This is used to correct the phase error via the controller 4 . In order to achieve this, low-pass filters of different orders are usually used in an analog implementation (cf. FIG. 2 "Best", p. 17), which depends on which suppression of the oscillating component is desired.

In the digital implementation provided here, other simple ones can be used Solutions are chosen that completely eliminate the harmonics Suppress: The Fourier coefficients of the fundamental oscillation become directly of the input signal based on the estimated phase, integrations being approximated by summations:

So you get in the stationary case, d. H. if ω≈, i.e. for the actual frequency ω of the input signal u (t):

â₁ (k) = a₁cos ((k) - ϕ (k)) = a₁cos Δϕ (k) ≈ a₁,

₁ (k) = -a₁ · sin ((k) - ϕ (k)) = -a₁ · sin Δϕ (k) ≈ -a₁Δϕ (k),

where the 1st order approximations for small phase errors

Δϕ = - ϕ «1

be valid.

In the transient case, the approximation applies

₁ (k) ≈ -a₁ / 2 (Δϕ (k) + Δϕ (k - n _A )).

This coefficient contains no further harmonics and can be fed to the controller 4 as a phase error signal. The summation over a period with n _A sampling steps can be carried out smoothly, or it is started again after each period, which is more advantageous since the effort is somewhat less. Deviating from the summation over an entire period, it is also only possible to sum over a half period, which likewise leads to the complete suppression of harmonics if it can be assumed that the input signal u (t) has symmetrical half oscillations. Then n _{A must be} replaced by n _A / 2.

The coefficient ₁ does not have to be calculated for a simple phase detection will. If, in addition to the phase, the amplitude of the Signal u (t) to be determined, ₁ gives an amplitude estimate the 1st harmonic of the input signal. In this case - over Going beyond structure 1 - a sine table is also provided be, but by the symmetries of the trigonometric functions is due to the cosine table.

This method can be used to determine the amplitude and phase position of the Harmonics in the signal u (t) by adding the Fourier coefficients higher order

such as

can be expanded without difficulty. The numerical effort is low because - as already mentioned - the trigonometric functions stands with recurring arguments that can be specified in tabular form are.  

The determination of the Fourier coefficients can also be applied to others signals synchronized with the input signal are expanded. For example are when determining the phase position and amplitude of a AC voltage the harmonics of the flowing due to the voltage Alternating current.

It is also possible to simultaneously acquire the phases and amplitudes of p signals u _l (l = 1... P) of a multiphase system, the Fourier coefficients then being determined for each signal

be determined. With the angles α _l (i), phase difference angles of the signals with one another are taken into account.

There are now two options in the further procedure:

• 1. You choose one of the signals u _l (l = l₀), set α _l0 (k) = 0 and use the coefficient _1l0 as an error _quantity for the phase _controller 4 , whereupon the phase-locked loop locks onto the phase of the signal u _l0 . â _1l0 gives the amplitude _{estimate of} the fundamental of u _l0 . The remaining coefficients _1l (l ≠ l₀) can now be used to influence the phase difference angle α _l (l ≠ l₀) by p-1 additional controllers so that the coefficients _1l are regulated to zero. Then the angles α _l (l ≠ l₀) set in this way indicate the phase angles of the _{fundamental oscillations of} the signals u _l (l ≠ l₀) based on the phase angles of the signals u _l0 and â _1l the estimated values of the amplitudes.
• 2. From the coefficients _1l , an average value is first formed for all l, which then serves the phase controller 4 as a phase error signal. Then p-1 additional controllers are used which influence the p angles α _{l in} such a way that the mean value of all α _l always remains zero and which, together with the “phase-mean value controller”, regulate the coefficients _1l to zero. Then, enter the angle α _l, the phase differences of the individual signals to an imaginary middle phase angle and _1l are estimates of the amplitudes of the fundamental vibrations of the individual signals.

As above, the method can also be used to detect harmonics multiphase signals are applied.

To implement the phase controller 4 , the time-discrete model equations of the input signal u (t) with the frequency ω and of the PLL must first be considered. The phase angle of the periodic signal progresses by ω (k) T _A (k) in one sampling period T _A (k). The angular frequency ω of the periodic signal is assumed to be approximately constant. It follows

ϕ (k + 1) = ϕ (k) + ωT _A (k).

The following applies to the phase-synchronous PLL:

The phase estimation error

Δϕ (k) = (k) - ϕ (k)

therefore follows the difference equation

Δϕ (k + 1) = Δϕ (k) - ωT _A (k) + ϕ _inc .

If the sampling period T _A (k) is only changed after a complete period of the periodic signal, the error equation can be continued via n _A steps:

Δϕ (k + n _A ) = Δϕ (k) - ωT _A (k) n _A + 2π.

If the index m of the period of the periodic signal is passed with k = n _A m, this becomes

Δϕ (m + 1) = Δϕ (m) - ωT _p (m) + 2π,

where n _A T _{A was} replaced by the period T _p . The filter 2 approximates the estimation error Δϕ in the form of the signal e as

e (m) = -b₁ (m) ≈ a₁ / 2 (Δϕ (m) + Δϕ (m -1))

to disposal. This allows an approach for the regulator law make a PI link:

T _p (m) = T _p (m - 1) + K (e (m) - βe (m - 1)),

where K is the controller gain factor and β is the controller zero. The sampling period T _A actually required follows from this

Transferring these equations to the z range gives:

By inserting one obtains for Δϕ (z)

It should be noted that the entire control loop gain is from the product ωKa ₁ depends not only on the controller gain K, but also determined by the frequency ω and the amplitude a₁ becomes.

If the amplitude or frequency change over a wide range, it makes sense to adapt the controller gain to this change. If the fundamental vibration amplitude is also determined, the adjustment can be made with â₁ and = ϕ _inc / T _A

can be achieved if K₀ for the amplitude a₁₀ and the frequency ω₀ is the specified controller gain.

It is cheap, as a reinforcement and controller zero point

and

β = 0.8 ± 0.1

to choose. This results in an even damping of all poles low overshoot achieved at the same time.

During the settling of the PLL, the phase estimates can still to be flawed. A control of an inverter relying on these values can result in significant current peaks cause. The PLL should therefore provide a signal which indicates when phase and amplitude values are reliable can apply.  

One determines from the determined estimated values and an estimate

û (k) = â₁ (k) sin - (k)

for the periodic input signal u, one can see the difference

Δu (k) = û (k) - u (k)

control the quality of the estimate. Δu (k) should lie within a suitable tolerance band for a certain time (a period is conceivable, ie n _A steps) before the phase position and amplitude of a downstream regulation are actually specified.

Conversely, there shouldn't be a single deviation to clear the signal cause measurement errors cannot be completely ruled out. Lie however, several values Δu (k) outside the tolerance will result in the signal removed and only set again when the condition described above is satisfied.

If, in addition to the fundamental, harmonics are also recorded, these should be included in û (k), for example by adding the third harmonic:

û (k) = â₁ (k) sin (k).
-

As a result, the tolerance band can be chosen to be correspondingly smaller.

It is also possible to estimate values (prediction values) leading q steps of the input signal

û (k + q) = â₁ (k) sin (k) + qϕ _inc ) + â₃ (k) sin 3 (k) + qϕ _inc ) + ₃ (k) sin 3 (k) + qϕ _inc )

to determine and a subordinate scheme to improve their Control properties to provide these prediction values.

Claims (13)

1. Method for digitally determining the phase position and the amplitude of a periodic signal, the frequency and amplitude of which can be subject to fluctuations in time, via a phase-locked loop (PLL (Phase-Locked Loop)) for specific points in time, characterized by a time-discrete sampling synchronized with the periodic signal of the periodic signal in such a way that a constant integer n _A of measured value samples is carried out per period and a constant integer n _A of phase estimates is made available by tracking the sampling frequency or the sampling period with a feedback of the fluctuating frequency of the input signal . 2. The method according to claim 1, characterized, that the samples are subjected to a Fourier analysis and off the coefficient of the cosine element obtained for the fundamental the phase position of the fundamental of the periodic signal is determined. 3. The method according to claim 2, characterized, that in addition from the coefficient of the sine element for the fundamental in the Fourier analysis the amplitude of the fundamental of the periodic signal is determined. 4. The method according to any one of claims 1 to 3, characterized, that the sampling period is constant over a period of the input signal is held. 5. The method according to any one of claims 1 to 3, characterized, that the sampling period only over a half period of the input signal is kept constant.   6. The method according to any one of claims 1 to 5, characterized in that the sampling period T _A to the predeterminable integer n _A of samples per period (or half-period) is corrected with a PI control that the transfer function T _p (m ) = n _A · T _A (m) = T _p (m - 1) + K (e (m) - βe (m - 1), comprising) eluting with e of the phase error as the control input, m is the index number of the period, K the controller gain, β the controller zero point and T _p the period of the input signal. 7. The method according to claim 6, characterized in that as a controller parameter and / or β = 0.8 ± 0.1 can be selected, with a₁₀ the expected fundamental vibration amplitude and ω₀ the expected frequency of the input signal. 8. The method according to claim 6 or 7, characterized, that the controller gain K is adjusted depending on the amplitude and frequency becomes.   9. The method according to any one of claims 1 to 8, characterized, that with the Fourier coefficients of a total of p input quantities of a multi-phase system, the method according to one of the claims 1 to 8 and other p-1 controllers the phase positions of the p input variables and determined the amplitudes of the fundamental vibrations will. 10. The method according to any one of claims 2 to 9, characterized, that in addition from the coefficients of the cosine and / or sine terms the phase positions for the harmonics in Fourier analysis and / or the amplitudes of the harmonics of the periodic Signal can be determined. 11. The method according to any one of claims 1 to 10, characterized, that during the settling process of the phase-locked loop the difference between the sample of the periodic signal Sampling time and the one obtained from the previous sampling time predetermined value of the periodic signal is monitored and the determined phase position and / or amplitude is then fed to a downstream regulation as a controlled variable, if the difference falls below a specified tolerance range. 12. The method according to any one of claims 1 to 11, characterized, that based on the phase and amplitude estimates of basic and Prediction values leading one or more steps in harmonics of the input signal can be determined. 13. The method according to any one of claims 1 to 12, characterized by the use of time discrete Regulation of four quadrant units.