CN205657592U - Single switch tube of three winding coupling inductance double -pressure direct -current converter that steps up - Google Patents

Abstract

The utility model discloses a single switch tube of three winding coupling inductance double -pressure direct -current converter that steps up belongs to power electronic converter's technical field. The converter includes: drive unit, first voltage -multiplying unit, second voltage -multiplying unit and output unit, control has been simplified to drive unit, first voltage -multiplying unit, switch tube of second voltage -multiplying unit sharing, has improved converter power density to the less duty cycle and the coupling inductance turn ratio have realized higher voltage gain, have expanded the range of application of converter.

Description

A kind of three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter

Technical field

The invention discloses a kind of three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter, especially one is applicable to The voltage boosting dc converter of high voltage gain occasion, belongs to the technical field of converters.

Background technology

At uninterrupted power source (UPS, Uninterruptible Power Supply) system, high-voltage gas discharging light (HID, High Intensity Discharge) system, in the grid-connected power generation system such as fuel cell and solaode, need relatively low direct current Voltage transformation is that higher DC voltage is for use.Traditional boost changer can use in theory limit dutycycle with Obtain the biggest voltage gain, but in actual applications, due to the impact of device parasitic parameter, use relatively big space rate (big In 0.8) time, its conversion efficiency is greatly reduced.Typical normal shock and anti exciting converter can realize high voltage by the regulation turn ratio and increase Benefit, but bigger due to voltage spikes occurs when the leakage field of transformator can cause switching tube to turn off, and also leakage field energy is not by effectively Utilize.The most traditional boost changer and typical normal shock anti exciting converter are not all suitable for having the low electricity that high-gain requires Pressure high input voltage output occasion.

For improving the gain of booster converter, the patent of Application No. 201310377481.3 discloses a kind of two winding coupled inductance Voltage-multiplying type Single switch high-gain converter, by cascade coupled inductance voltage doubling unit after tradition boost unit, it is suppressed that open Close due to voltage spikes when pipe turns off, reduce the voltage stress that switching tube bears, reclaimed leakage inductance energy, improve voltage gain. Changer disclosed in this patent, when switching tube turns on by the energy storage of DC source in vice-side winding and the second electric capacity, secondary The electric energy that limit winding and the second electric capacity store powers to the load when switching tube turns off, it is seen then that DC source is not directly to load Power supply, power utilization rate is the highest.Voltage gain big not under certain turn ratio and duty cycle condition, and increase turn ratio meeting Make changer volume and weight increase and the coupling inductance linearity can be made to be deteriorated, increase dutycycle and conduction loss can be caused to increase.Former Limit leakage inductance energy is only by the second capacitive absorption being connected with vice-side winding, and the second electric capacity bears larger current impact at short notice, It is unfavorable for the reliability service of changer.

Summary of the invention

1. invention to solve the technical problem that

It is an object of the invention to the deficiency for above-mentioned background technology, it is provided that a kind of three winding coupling inductance voltage-multiplying type Single switch Guan Sheng Straightening current converter, realizes higher voltage gain with less dutycycle and the coupling inductance turn ratio, solves existing high gain boost Converter voltage lifting scarce capacity, the technical problem that changer conversion efficiency is the highest, operational reliability is the best.

2. technical scheme

For solving the problems referred to above, the technical scheme that the present invention provides is:

A kind of three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter, including: driver element, the first voltage doubling unit, Second voltage doubling unit and output unit, wherein,

Described driver element includes: the first winding and switching tube, and the first terminal of the first winding is connected with direct voltage source positive pole, Second terminal of the first winding is connected with the drain electrode of switching tube, and the source electrode of switching tube is connected with direct voltage source negative pole,

Described first voltage doubling unit is the loop that the branch road comprising the second winding is formed with described switching tube, and the second voltage doubling unit is bag The loop that branch road containing the tertiary winding and described switching tube are formed, the second winding and the turn ratio of the first winding and the tertiary winding and the The turn ratio of one winding is identical,

Described output unit: its input and direct voltage source, the first winding, the second winding, the branch road phase of tertiary winding composition Connecting, outfan provides the unidirectional current after boosting when switching tube turns off to load.

Further prioritization scheme as described three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter:

In first voltage doubling unit, the branch road comprising the second winding includes: the second winding, the second electric capacity, the 3rd electric capacity, the two or two Pole is managed, and the first terminal of described second winding and the second electric capacity one pole connect, and another pole of the second electric capacity is connected with switching tube drain electrode, Switching tube source electrode and the 3rd electric capacity one pole connect, and the 3rd another pole of electric capacity is connected with the second diode anode, the second diode cathode It is connected with the second terminal of the second winding, the first terminal of the second winding and the first terminal Same Name of Ends each other of the first winding.

In second voltage doubling unit, the branch road comprising the tertiary winding includes: the tertiary winding, the 4th electric capacity, the first electric capacity, the four or two Pole is managed, and the first terminal of the described tertiary winding and the 4th electric capacity one pole connect, and the 4th another pole of electric capacity is connected with switching tube source electrode, Switching tube drain electrode is connected with the first electric capacity one pole, and another pole of the first electric capacity is connected with the 4th diode cathode, the 4th diode anode It is connected with the second terminal of the tertiary winding, the second terminal of the tertiary winding and the first terminal Same Name of Ends each other of the first winding.

Further, in described three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter, the first voltage doubling unit also wraps Including the first diode, described second voltage doubling unit also includes the 3rd diode, the first diode anode and the second electric capacity and switching tube The junction point of drain electrode is connected, and the junction point of the first diode cathode and the second diode anode and the 3rd electric capacity is connected, the three or two pole The junction point of tube anode and the first electric capacity and the 4th diode cathode is connected, the 3rd diode cathode and the 4th electric capacity and switching tube source The junction point of pole is connected.

Further, in described three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter, output unit includes: defeated Go out diode, output capacitance, described output diode anode and the second winding the second terminal and junction point of the second diode cathode Being connected, output diode negative electrode connects output capacitance one pole, another pole of output capacitance and the tertiary winding the second terminal and the 4th diode The junction point of anode is connected.

As the further prioritization scheme of described three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter, described three winding coupling Close the inductance leakage inductance that by three winding ideal transformer, is folded to the former limit of three winding ideal transformer and and receive the magnetizing inductance on former limit Equivalently represented.

3. beneficial effect

The technical scheme using the present invention to provide, compared with prior art, has the advantages that

(1) utilize the first winding and the switching tube composition driver element of three winding coupling inductance, utilize the second winding, the tertiary winding With electric capacity, the voltage doubling unit of diode two symmetrical configuration of composition, achieve higher with less dutycycle and the coupling inductance turn ratio Voltage gain, driver element and two voltage doubling unit circuit altogether and only one of which switching tube, simplify control, improve change Parallel operation power density；

(2) two voltage doubling units respectively form the branch road of an absorbing coupling inductance leakage inductance energy when switching tube turns off, it is suppressed that open Close due to voltage spikes when pipe turns off, reduce the voltage stress of switching tube, reduce the voltage stress of other power device simultaneously, Improve transducer reliability；

(3), when switching tube turns off, direct voltage source directly powers to the load through three windings, improves Power convert rate, and Further increase gain；

(4) less dutycycle can make that input current peak value is low, input current ripple is little, reduces conduction loss simultaneously, less The turn ratio avoids the problem that magnetic core causes the linearity to be deteriorated because the turn ratio is too high, and due to the effect of coupling inductance, has both added defeated Go out voltage gain, the problem alleviating again diode reverse recovery, reduce loss.

Accompanying drawing explanation

Fig. 1 is the main circuit topological structure figure of changer of the present invention.

Fig. 2 is the voltage/current oscillogram of main devices in one switch periods of changer of the present invention.

Fig. 3 (a) is one of changer operation mode figure.

Fig. 3 (b) is the two of changer operation mode figure.

Fig. 3 (c) is the three of changer operation mode figure.

Fig. 3 (d) is the four of changer operation mode figure.

Fig. 3 (e) is the five of changer operation mode figure.

Fig. 3 (f) is the six of changer operation mode figure.

Fig. 4 be changer of the present invention with Flyback changer, the disclosed changer of aforementioned patent voltage gain with change in duty cycle Graph of a relation.

Label declaration in figure: V_inFor direct voltage source, N₁、N₂、N₃Be respectively the first winding of coupling inductance, second around Group, the tertiary winding, 1 is the first terminal, and 2 is the second terminal, L_mFor magnetizing inductance, L_kFor leakage inductance, S is switching tube, C₁、 C₂、C₃、C₄It is respectively the first electric capacity, the second electric capacity, the 3rd electric capacity, the 4th electric capacity, D₁、D₂、D₃、D₄It is respectively first Diode, the second diode, the 3rd diode, the 4th diode, D_oFor output diode, C_oFor output capacitance, R is negative Carry, C_DoFor the equivalent parallel electric capacity of output diode, C_D2And C_D4It is respectively the equivalence of the second diode and the 4th diode also Connection electric capacity.

Detailed description of the invention

For further appreciating that present disclosure, in conjunction with drawings and Examples, the present invention is described in detail.

Changer disclosed by the invention is as it is shown in figure 1, include: driver element, the first voltage doubling unit, the second voltage doubling unit and Output unit.

Driver element includes: the first winding N₁With switching tube S, the first winding N₁The first terminal 1 and direct voltage source V_inJust Pole connects, the first winding N₁The second terminal 2 be connected with the drain electrode of switching tube S, the first winding N₁Two ends are also connected to excitation electricity Sense L_m, the source electrode of switching tube S and direct voltage source V_inNegative pole connects.

First voltage doubling unit is for comprising the second winding N₂The loop that formed of branch road and switching tube S, comprise the second winding N₂Branch road Including: the second winding N₂, the second electric capacity C₂, the 3rd electric capacity C₃, the second diode D₂, the second winding N₂The first terminal 1 With the second electric capacity C₂One pole connects, the second electric capacity C₂Another pole is connected with switching tube S drain electrode, switching tube S source electrode and the 3rd electricity Hold C₃One pole connects, the 3rd electric capacity C₃Another pole and the second diode D₂Anode connects, the second diode D₂Negative electrode and second Winding N₂The second terminal 2 connect, the second winding N₂The first terminal 1 and the first winding N₁The first terminal 1 the most of the same name End.

Second voltage doubling unit is for comprising tertiary winding N₃The loop that formed of branch road and switching tube S, comprise tertiary winding N₃Branch road Including: tertiary winding N₃, the 4th electric capacity C₄, the first electric capacity C₁, the 4th diode D₄, tertiary winding N₃The first terminal 1 With the 4th electric capacity C₄One pole connects, the 4th electric capacity C₄Another pole is connected with switching tube S source electrode, switching tube S drain electrode and the first electricity Hold C₁One pole connects, the first electric capacity C₁Another pole and the 4th diode D₄Negative electrode connects, the 4th diode D₄Anode and the 3rd Winding N₃The second terminal 2 connect, tertiary winding N₃The second terminal 2 and the first winding N₁The first terminal 1 the most of the same name End.Second winding N₂With the first winding N₁Turn ratio and tertiary winding N₃With the first winding N₁Turn ratio identical, first The circuit structure of voltage doubling unit and the second voltage doubling unit is symmetrical.

The input of output unit and direct voltage source V_in, the first winding N₁, the second winding N₂, tertiary winding N₃Propping up of composition Road is connected, direct voltage source V when switching tube S turns off_inElectric energy directly feed load R.Output unit includes: export two poles Pipe D_o, output capacitance C_o, output diode D_oAnode and the second winding N₂The second terminal 2 and the second diode D₂Negative electrode Junction point be connected, output diode D_oNegative electrode connects output capacitance C_oOne pole, output capacitance C_oAnother pole and tertiary winding N₃ The second terminal 2 and the 4th diode D₄The junction point of anode is connected.

Three winding coupling inductance is by three winding ideal transformer, leakage inductance L that is folded to the former limit of three winding ideal transformer_kWith and receive The magnetizing inductance L on former limit_mEquivalently represented.

The voltage/current oscillogram of the main devices in a switch periods of changer shown in Fig. 1 is as in figure 2 it is shown, switching tube S Driving voltage V_gs, magnetizing inductance L_mElectric current i_Lm, leakage inductance L_kElectric current i_Lk, the second winding N₂Electric current i_N2, the tertiary winding N₃Electric current i_N3, the electric current i of switching tube S_ds, the first diode D₁Electric current i_D1, the second diode D₂Electric current i_D2, Three diode D₃Electric current i_D3, the 4th diode D₄Electric current i_D4With output diode D_oElectric current i_DoA switch week Interim waveform.

Fig. 3 (a) to Fig. 3 (f) is the different operating mode in a switch periods of the changer shown in Fig. 1, and concrete condition is as follows:

1) mode 1 [t₀,t₁]: the driving voltage V of switching tube S_gsBecoming high level from low level, switching tube S turns on, excitation electricity Sense L_mContinue through coupling inductance secondary and provide energy, the second winding N to load₂Electric current i_N2With tertiary winding N₃Electric current i_N3Hurry up Speed reduces, leakage inductance electric current i_LkLinear increase, switching tube S electric current i_dsLinear increase.Shown in current flow paths such as Fig. 3 (a), when Second winding N₂Electric current i_N2With tertiary winding N₃Electric current i_N3When being reduced to zero, output diode D_oTurning off, this mode terminates.

2) mode 2 [t₁,t₂]: at t=t₁Time, the second winding N₂With tertiary winding N₃Electric current be reduced to zero, output diode D_o Equivalent parallel electric capacity C_DoStart to leakage inductance L_kRelease energy, output diode D_oReverse recovery energy store leakage inductance L_k In, the second winding N₂Electric current i_N2With tertiary winding N₃Electric current i_N3Start to inversely increase, direct voltage source V_inStart to excitation electricity Sense L_mCharging, shown in current flow paths such as Fig. 3 (b)；Work as t=t₂Time, the Reverse recovery energy quilt completely of output diode Do Leakage inductance L_kAbsorbing, this mode terminates.

3) mode 3 [t₂,t₃]: switching tube S keeps open-minded, shown in current flow paths such as Fig. 3 (c), direct voltage source V_inContinue To magnetizing inductance L_mCharging, the second diode D₂With the 4th diode D₄Turn on owing to bearing forward voltage drop, leakage inductance electric current i_Lk、 Second winding N₂Electric current i_N2, tertiary winding N₃Electric current i_N3With switching tube S electric current i_dsThe most linearly increase, the second winding N₂With Three electric capacity C₃To the second electric capacity C₂Electric discharge, tertiary winding N₃With the first electric capacity C₁To the 4th electric capacity C₄Electric discharge, works as t=t₃Shi Kai Closing pipe S to turn off, this mode terminates.

4) mode 4 [t₃,t₄]: the driving voltage V of switching tube S_gsBecoming low level from high level, switching tube S turns off, electric current stream Shown in path such as Fig. 3 (d), the second winding N₂Electric current i_N2With tertiary winding N₃Electric current i_N3Quickly reduce, leakage inductance electric current i_LkLine Property reduce, the second diode D₂With the 4th diode D₄Still turn on, the first diode D₁With the 3rd diode D₃Just bear Beginning to turn on to pressure drop, leakage inductance energy is recycled in electric capacity, as the second winding N₂With tertiary winding N₃Electric current be reduced to zero Time the second diode D₂With the 4th diode D₄Turning off, this mode terminates.

5) mode 5 [t₄,t₅]: the equivalent parallel electric capacity C of the second diode D2 and the 4th diode D4_D2And C_D4To leakage inductance L_k Release energy, the second winding N₂Electric current i_N2With tertiary winding N₃Electric current i_N3Start opposite direction to increase, current flow paths such as figure Shown in 3 (e), work as t=t₅Time, the Reverse recovery energy of diode is recovered completely, and this mode terminates.

6) mode 4 [t₅,t₆]: switching tube S is held off, shown in current flow paths such as Fig. 3 (f), magnetizing inductance L_mAnd leakage inductance L_kPart energy to the first electric capacity C₁With the 3rd electric capacity C₃Charging, leakage inductance electric current i_LkLinear reduction, output diode D_o Owing to bearing forward voltage drop conducting, direct voltage source V_in, the first winding N₁, the second winding N₂, tertiary winding N₃, second electricity Hold C₂With the 4th electric capacity C₄It is together in series to load R electric discharge, the second winding N₂Electric current i_N2With tertiary winding N₃Electric current i_N3Line Property increase, flow through the first diode D₁With the 3rd diode D₃Electric current linearly reduce, when switching tube S opens again, should Mode terminates.

Steady-state gain and power device voltage stress situation to changer shown in Fig. 1 are analyzed below.

Process to simplify the analysis, ignores time extremely short mode, is only analyzed mode 3 and mode 6.And do following Assume:

1, first to fourth electric capacity C₁、C₂、C₃、C₄With output capacitance C_oSufficiently large, the voltage V at the most each electric capacity two ends_C1、 V_C2、V_C3、V_C4、V_CoIn a switch periods, keep constant；

2, power device is ideal component, ignores the impact of device parasitic parameter；

3, the coefficient of coup k=L of three winding coupling inductance_m/(L_m+L_k), and turn ratio n=N₂/N₁=N₃/N₁。

When switching tube S opens, according to mode 3, have:

$v_{N1}^{III}=\frac{L_m}{L_m+L_k}={\mathrm{kV}}_{in}---\left(1\right)$

$v_{Lk}^{III}=\frac{L_k}{L_m+L_k}{V}_{in}=(1-k)V_{in}---\left(2\right)$

$v_{N2}^{III}=v_{N3}^{III}={\mathrm{nv}}_{N1}^{II}={\mathrm{nkV}}_{in}---\left(3\right)$

In formula, k is the coefficient of coup of three winding coupling inductance, and n is the turn ratio of coupling inductance, and this topology has symmetry, second Winding N₂With tertiary winding N₃The number of turn identical.

In switching tube S off-phases, by each winding of coupling inductance and leakage inductance application voltage-second balance principle, it is known that:

$v_{N1}^{VI}=\frac{Dk}{1-D}{V}_{in}---\left(4\right)$

$v_{N2}^{VI}=v_{N3}^{VI}=\frac{nDk}{1-D}{V}_{in}---\left(5\right)$

$v_{Lk}^{VI}=\frac{D(1-k)}{1-D}{V}_{in}---\left(6\right)$

As shown from the above formula:

$V_{C1}=V_{C3}=V_{in}+v_{N1}^{VI}+v_{Lk}^{VI}=\frac{1}{1-D}{V}_{in}---\left(7\right)$

$V_{C2}=V_{C4}=V_{C1}+v_{N3}^{III}=(nk+\frac{1}{1-D})V_{in}---\left(8\right)$

According to (4) (5) (6) and (8), output voltage can be expressed as:

$V_o=V_{Co}=V_{in}+v_{N1}^{VI}+v_{Lk}^{VI}+V_{C2}+v_{N2}^{VI}+v_{N3}^{VI}+V_{C4}=\frac{2nk+3}{1-D}{V}_{in}---\left(9\right)$

So voltage gain is:

$M=\frac{V_o}{V_{in}}=\frac{2nk+3}{1-D}---\left(10\right)$

K=1 when ignoring leakage inductance, now voltage gain is:

$M=\frac{V_o}{V_{in}}=\frac{2n+3}{1-D}---\left(11\right)$

The voltage stress that switching tube S bears:

$V_{ds}=V_{C1}=V_{C3}=\frac{1}{1-D}{V}_{in}=\frac{1}{2n+3}{V}_o---\left(12\right)$

Diode (D₁、D₂、D₃、D₄) voltage stress that bears:

$V_{D1}=V_{D3}=\frac{1}{1-D}{V}_{in}=\frac{1}{2n+3}{V}_o---\left(13\right)$

$V_{D2}=V_{D4}=\frac{1+n}{1-D}{V}_{in}=\frac{n+1}{2n+3}{V}_o---\left(14\right)$

$V_{Do}=\frac{2n+1}{1-D}{V}_{in}=\frac{2n+1}{2n+3}{V}_o---\left(15\right)$

Generally, traditional DC-DC boost converter, the steady-state gain such as basic boost changer and Flyback changer is 1/1-D and nD/1-D.Disclosed in the patent of Application No. 201310377481.3, transducer gain is n+1/1-D, during n=2, its Gain is 3/1-D, under ensureing the former limit number of turn and secondary total number of turns and above-mentioned patent same case, and the i.e. second winding N₂With the 3rd Winding N₃With the first winding N₁Turn ratio when being 1, transducer gain disclosed in this patent is 5/1-D, it is seen then that the present invention is special The carried changer of profit only can realize identical gain by less dutycycle, and this advantageously reduces conduction loss, improves changer effect Rate.

Fig. 4 shows the present invention and Flyback changer and Application No. when secondary total number of turns is with the ratio n=2 of the former limit number of turn 201310377481.3 patent disclosed in changer steady-state gain contrast situation.Can be seen that the steady-state gain of patent of the present invention Steady-state gain far above other two kinds of changers.

The embodiment of the above is only present pre-ferred embodiments, not limits the practical range of the present invention with this, therefore Fan Yiben The change that the shape of invention, principle are done, all should contain within the scope of the present invention.

Claims (5)

1. a three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter, it is characterised in that including: driver element, the first voltage doubling unit, the second voltage doubling unit and output unit, wherein,

Described driver element includes: the first winding and switching tube, and the first terminal of the first winding is connected with direct voltage source positive pole, and the second terminal of the first winding is connected with the drain electrode of switching tube, and the source electrode of switching tube is connected with direct voltage source negative pole,

Described first voltage doubling unit is the loop that the branch road comprising the second winding is formed with described switching tube, second voltage doubling unit is the loop that the branch road comprising the tertiary winding is formed with described switching tube, the turn ratio of the second winding and the turn ratio of the first winding and the tertiary winding and the first winding is identical

Described output unit: its input is connected with direct voltage source, the first winding, the second winding, the branch road of tertiary winding composition, and outfan provides the unidirectional current after boosting when switching tube turns off to load.

nullA kind of three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter，It is characterized in that，In described first voltage doubling unit，The branch road comprising the second winding includes: the second winding、Second electric capacity、3rd electric capacity、Second diode，The first terminal of described second winding and the second electric capacity one pole connect，Another pole of second electric capacity is connected with switching tube drain electrode，Switching tube source electrode and the 3rd electric capacity one pole connect，3rd another pole of electric capacity is connected with the second diode anode，Second terminal of the second diode cathode and the second winding connects，The first terminal of the second winding and the first terminal Same Name of Ends each other of the first winding，In described second voltage doubling unit，The branch road comprising the tertiary winding includes: the tertiary winding、4th electric capacity、First electric capacity、4th diode，The first terminal of the described tertiary winding and the 4th electric capacity one pole connect，4th another pole of electric capacity is connected with switching tube source electrode，Switching tube drain electrode is connected with the first electric capacity one pole，Another pole of first electric capacity is connected with the 4th diode cathode，4th diode anode is connected with the second terminal of the tertiary winding，Second terminal of the tertiary winding and the first terminal Same Name of Ends each other of the first winding.

A kind of three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter, it is characterized in that, described first voltage doubling unit also includes the first diode, described second voltage doubling unit also includes the 3rd diode, the junction point that first diode anode and the second electric capacity drain with switching tube is connected, the junction point of the first diode cathode and the second diode anode and the 3rd electric capacity is connected, the junction point of the 3rd diode anode and the first electric capacity and the 4th diode cathode is connected, 3rd diode cathode and the 4th electric capacity are connected with the junction point of switching tube source electrode.

4. according to three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter a kind of described in Claims 2 or 3, it is characterized in that, described output unit includes: output diode, output capacitance, the junction point of described output diode anode and second winding the second terminal and the second diode cathode is connected, output diode negative electrode connects output capacitance one pole, and another pole of output capacitance is connected with the junction point of the tertiary winding the second terminal and the 4th diode anode.

A kind of three winding coupling inductance voltage-multiplying type Single switch pipe voltage boosting dc converter, it is characterized in that, leakage inductance that described three winding coupling inductance by three winding ideal transformer, is folded to the former limit of three winding ideal transformer and and to receive the magnetizing inductance on former limit equivalently represented.