Applications of describing functions to estimate the performance of nonlinear inductance

S.C.Chung, S.R.Huang, JSHuang and E.C.Lee

Abstract: Passive filter design is traditionally considered to be very important for many power electronic circuits and power systems. It is is very critical in the design of these power electronic circuits and power systems to know how to obtain accurate corner frequencies. It is known that when the load is large, the inductance of an inductor is known to always change. This phenomenon is due to the fact that when the current load is high, the inductance always suffers from saturation or hystersis. It is extremely difficult to estimate the actual value for the nonlinear inductance in large currents because the inductance is nonlinear. The paper investigates the saturation phenomenon for a nonlinear inductance, The describing function method can be used to linearise the nonlinear inductor and then estimate the inductance in large current situations. Hence, the corner frequency for the low- pass filter can also be calculated accurately. It is shown that, when the current is very large, the corner frequency drifts to a larger value in the low-pass filter. The drift value of the corner frequency can be easily calculated by the describing function. Simulation and experimental results verify this phenomenon. However, it should be stressed that the method used in the paper is restricted to low frequencies. The higher frequency effects are neglected throughout the paper.

1 Introduction

Passive filter design is traditionally considered to be very important for power electronic circuits and power quality in power systems [l-51. It is essential in the design of these power electronic circuits and power systems to know how to obtain accurate corner frequencies. It is extremely diffi- cult to estimate the actual value for the thyristor protection or harmonic elimination under large currents, because the inductance is nonlinear.

On the other hand, the describing function method has been widely used to determine the limit cycle and the dynamic behaviour for the nonlinear systems [&lo]. An advantage of the describing function method is that it can be applied in large signal situations, which means in power systems, this method is still valid even in large current or voltage situations. Moreover, the describing function method can be viewed as another kind of harmonic balance method [9, lo]. This paper explores the saturation phenomenon for a nonlinear inductance. The describing function method can be used to linearise the nonlinear inductor and then estimate the inductance during the large current situations. The corner frequency of the low-pass filter can then be determined. Moreover, it should be noted that the method used here is restricted to low frequencies. The higher frequency effects are neglected throughout the paper.

0 IEE, 2001 IEE Proceedings odne no. 20010368 DOI: 10.1O49~p~mt2oO10368 Paper fmt received 4th October 2000 and in revised form 26th February 2001 S.C. Chung is with the Department of Automatic Control Engineering, Feng- Chia University, Tai-Chung City, Taiwan, Republic of China S.R. Huang is with the Department of Electrical Engineering, Feng-Chia Uni- versity, Tai-Chung City, Taiwan, Republic of China J.S. Huang and E.C. Lee are graduate students from Feng-Chia University

10%

2 Preview of describing function

The describing function method has been extensively used to determine the limit cycle and dynamic behaviour for nonlinear systems [&lo]. According to Fig. 1, a nonlinear element exists in the feedback loop described by qb(.).

L Fig. 1 A conventwnal Lw’eproblem

Consider a sinusoidal input to the nonlinear element, of amplitude A and frequency w, such as i(t) = A sin(@, as displayed in Fig. 1. The output of the nonlinear element &t) = Ql($ is frequently periodic. By using Fourier series, this periodic function &t) can be expanded as follows:

where the Fourier coefficients U, and tions of A and w, determined by

7T

1 a0 = - / A(t )d(wt)

?r

b, are generally func-

-7r

a, = - j. A ( t ) cos(nwt)d(wt)

bn = - 7T 1 1 A( t> sin(nwt)d(wt)

7f -7r

(2) --7F

IEE Proc.-Sci Meas. TechnoL. Vol. 148, No. 3. May 2001

If the nonlinearity is an odd function, then a. = 0. Further- more, if the transfer function has the low-pass properties [&lo], then

IG(jw)I >> IG(jnw)l for n = 2 ,3 ,4 . . . (3) This assumption is called the ‘filtering hypothesis’. In this case, the fundamental component Al(t) must be considered, which can be described by

X(t) x XI ( t ) = a1 cos(wt) + bl sin(&) = M sin(wt + 0) (4)

where

M ( A , w ) = d ( u : + b?) and 0(A ,w) = tan-’(al/bl)

The describing function of the nonlinear element is the complex ratio of the fundamental component of the non- linear element as defined by the input sinusoidal waveform, such as

(5 )

= [[ $ ( A sin(cJt)) sin(wt)d(wt) AT

1 + j 1 $(A sin(&)) cos(wt)d(wt) -75,

=q AT $(A sin(wt))e-jwtd(wt) ( 6 ) -7r

2. I Example: Describing the function of a hardening spring The characteristics of a hardening spring are given by 9 = i + i3, with i being the input and $(t) being the output. The input is i(t) = A sin(@. The output $(t) = A sin(wt) + A3 sin(wt)/2 can be expanded into a Fourier series, with the fundamental being $I = ( A + 3/8 A3) sin(@, and the describing function of this nonlinear component is

(7) 3 8

N(A ,w) = N ( A ) = 1 + -A2

where A is the amplitude for the sinusoidal input. Remark 1: The describing function method is valid for the case of the feedback loop where the linear transfer function possesses low-pass filter property. According to Fig. 1, if the linear transfer function G(s) possesses low-pass filter properties, then the high-order harmonic terms in the Fou- rier series can be ignored.

3 Inductance of inductors

This Section attempts to derive the inductances for the lin- ear and nonlinear inductors. For an inductor, the induct- ance is defined as follows:

L = X / i ( 8 ) where il = N y denotes the flux linkage, N represents the winding numbers, y is the magnetic flux, i denotes the cur- rent and L represents the inductance. Consider a magnetic circuit with mean length 1 and cross-sectional area a. For

IEE Proc.-Sci. Meas. Technol., Vol. 148. No. 3, May 2001

the linear magnetic material (which means the current is small), the magnetic flux y can be expressed as

(9)

where 1 and a are the length and cross-sectional area of the magnetic circuit and ,U is the permeability of the magnetic material. Substituting eqn. 9 into eqn. 8,

Now, consider the large current case, in which the flux link- age is no longer a linear function of current i and can be expressed as

where y denotes the magnetic flux in webers (Wb) passing across an arbitrary surface, and il represents the flux link- age in weber-turns given by the product of a number of turns N and the flux y linking N. Now, consider the case where the current is described by

If the high-order harmonic terms are ignored, applying the definition of inductance in eqn. 8 and the describing func- tion defined in eqn. 6 produces

Le, = N(A,w) = - j 1 $(A sin(wt))eOwtd(wt)

X = $(i) = N $ (11)

i(t) = A . sin(wt) (12)

AT ?r

(13) where Le, denotes the equivalent inductance of the inductor for large currents. In eqn. 13, the equivalent inductance L obviously depends on the amplitude of current A and the frequency w. However, if the nonlinear function $(.) is odd symmetric, the inductance L depends only on current A, i.e. Le7 = N(A). In general, the magnetic flux will be satu- rated if the current load is high enough. Hence, consider the nonlinear function $(.) being a saturated function [8, 91 as follows:

i > isat 7 Li,at

$(i) = sat( i ) = isat 2 i 2 -isat , Li (14)

i < -isat r - l i s a t

flux

i where isal denotes the saturation current and L represents the initial inductance (small current) as displayed in Fig. 2.

flux

1’7 I n I A W)

21 Isat burrent I l3 \L1 the slope L

I I(t)= A sin(wt)

Fig. 2 Saturation characterhtics with sine-wave input

109

1.Of

-I

0.6

Using the describing function method, the equivalent inductance Le can then be obtained as follows [8, 91, as shown in the Appendix

I:\ -

L

Also, another saturated function [lo] is considered as follows:

ai #( i ) = {

where a, b > 0. In eqn. 16, the saturated flux linkage for large currents such as i + h, can be obtained as kalb. Meanwhile, for cases involving small currents, such as i -+ 0, the initial inductance L is found to be

With eqn. 13, the equivalent inductance Le, becomes

Le, = N ( A )

tan (i) + bA

] tan2 (i) = lim - -tanh-'

- tanh-I [ - tan-' ban (:)I ~TTGT

- tan-' I a n (;)I d m t a n 2 (;) + tanh-' [ I/-lb+Ab2A2]

+tanh-' [ bA ] tan2 (i) + bA d_l+b2Az t an2

t+r- " { A [ 4-1 + b2A2]

tan (5) + bA -1 + b2A2

4-1 + b2A2

/ { nb2A-J-l+b2AZ

Eqns. 15 and 18 demonstrate that the equivalent induct- ance will decrease as the current increases, as shown in Figs. 3 and 4, respectively.

0.4

0.2

t < 01 I 0 5 10 15 20 25 30

current Visat

Fig.3 I = saturation current fi initial inductance

Equivalent inductance for the saturation function in eqn. 14

1

25 30 0 5 10 15 20 0'

current VIsat

Fig. 4 Isat = !/! = saturation current L = initial inductance

Equivalent inductance for the saturation,function in egn. 18

Remark 2: In eqn. 11, the flux linkage A is only a function of the current i, which is described by A = $(i). However, in some situations, the flux linkage will be influenced by fac- tors like skin, proximity, gap effects etc. In this paper, the authors neglected these effects and assumed that the flux linkage il is only a function of the current i.

4 Low-pass filter

4.1 L-R filter This Section applies the describing function method to analyse the L-R low-pass filter circuit. The L-R and L-C low-pass filter circuits are widely employed in power systems and power electronics. Fig. 5 displays an L-R low- pass filter circuit in which a voltage source V, = A . sin wt is employed. The transfer function is

such that, the corner frequency is

Conventionally, the inductance L in eqn. 20 is considered a constant, and thus the corner frequency is also constant. However, with eqns. 15 and 18, the inductance L in eqn. 20 is no longer a constant under a large current. The induct- ance L will decrease with increasing current, meaning the corner frequency will drift to larger values under a large current.

I I

L I +

I- Fig.5 L R low-passfilter circuit

4.2 Block diagram for L-R filter Section 3 proved that the describing function method can be applied to estimate the inductance of an inductor under a large current. However, the describing function method is only valid for when the loop transfer function possesses the low-pass filter property. Otherwise, the inductance indi- cated in eqns. 15 and 18 will be invalid. In this Section, it

IEE Proc.-Sci. Meas. Technol., Vol. 148, No. 3. May 2001 110

will be shown that the L-R filter can be transformed into a standard Lur'e problem. This result is very close to that in [ll]. An L-R fdter circuit is shown in Fig. 5. According to Kirchhoff s voltage law, the following equations hold:

dX dt

V, = i R + -

V, = iR (21) where V, is the output voltage.

Taking its Laplace transform, eqn. 21 can be rewritten as 1

X(s) = -(Vs - V0) S

where n(s> is the Laplace transform of the flux linkage qt ) . In eqn. 11, the relationship between the flux linkage and

current can be rewritten as

2 = +-I@) (23) where @;'(.) is the inverse function of N.).

Fig. 6 Control block diagram f o r the L R low-passjlter civcuit

According to eqns. 21-23, the block diagram of the L-R filter can be represented by Fig. 6, which is a Lur'e prob- lem. As displayed in Fig. 6, the linear transfer function is

G(s) = l /s (24) which clearly possesses low-pass filter property. Thus, the describing function method can be applied in this system. However, in Fig. 6, the nonlinear function is i = $'(A). As for the linear inductors, the relation between the flux link- age and current can be expressed as

1 L

i = - X

where L is the linear inductance. In the following, the authors will show that the describ-

ing function of q?-l(.), denoted as N-'(B), is equivalent to l/Leq, where A = B sin(ot) and Leq is defined in eqn. 13. The authors will prove this result by the harmonic balance method.

First, consider i(t) = A . sin(ot), and assume that the function @(.) is an odd symmetric function. The definition of the describing function method produces

X ( t ) = 4( i ) = AN(A) sin(wt) + a3 sin(3wt) +a5 sin(5wt) + . , . (26)

Meanwhile, eqn. 26 produces

i = d-'(+(i)) = +- ' (AN(A) sin(wt) + a3 sin(3wt) +a5 sin(5wt) + . . .) (27)

Then, based on Taylor series expansion, eqn. 27 can be rewritten as

i = $-'(+(i)) = + - I ( A N ( A ) sin(ut)) + b3 sin(3wt) +b:, sin(5wt) + . . . . (28)

where b3, b5, ... are coefficients which are appropriately defined and related to the function @-l(.), its derivative and the coeficients a3, a5, ... . Meanwhile, let B = A N @ ) and N-l(.) be denoted as the describing function of @(.). Then, eqn. 28 becomes

IEE Proc.-Sci. Meas. Technol.. Vol. 148, No. 3, May 2001

i = qF1 ( B sin(&)) + 133 sin(3wt) + b5 sin(5wt) + . . . that is A sin(&)) = N-'(B)B sin(&) + e3 sin(3wt)

+e5 sin(5wt) + . . . (29) where c3, c5, ..., like b3, b5, ..., are some coefficients which are appropriately defined. Equating the coefficients in eqn. 29, and comparing the coefficients on the left and right sides of eqn. 29, produces

From the definition of equivalent inductance in eqn. 13, eqn. 30 becomes

(31) 1 N - y B ) = -

Le, The above equations demonstrate that the G R low-pass filter circuit can be transformed into a Lur'e problem, as displayed in Fig. 6. In this Lur'e problem, the transfer func- tion possesses the low-pass fdter property. However, it should be noted that the nonlinear function of the Lur'e problem is qF-'(.), rather than N,). Meanwhile, it demon- strates that the describing function of this inverse function @(.) is identical to 1/Le , meaning the equivalent induct- ance Leq defined by the iescribing function eqn. 13 can be employed to analyse the G R low-pass filter circuit.

Fig. 7 L C - R lowpmsjlter circuit

4.3 L-C-R filter and its block diagram Fig. 7 displays an L-C-R low-pass filter circuit, to whch a voltage source V, = A sin ot is applied. The transfer func- tion is

( 3 2 ) V 1 v, L C S ~ + L / R S + ~

H ( s ) = - =

and the natural frequency is

(33) 1

w, = - m Similar to the L-R filter, the natural frequency in eqn. 33 will drift under a large current. With a similar procedure as with the G R filter, it can be shown that the system of L-C-R filter can be also transformed into a Lur'e problem. The block diagram is shown in Fig. 8. According to Manson's rule [ll], the linear transfer function can be obtained as

R s(RCs + 1)

G(s) = (34)

which clearly possesses the low-pass filter property. The describing function method can be applied to this problem. As in eqn. 34, the natural frequency becomes

(35) 1

w, = ~ m I l l

In eqn. 35, it can be observed that the natural frequency drifts under a large current.

Fig.% Control block diagram for the LC-R low-pmsfilter circuil

5 results

Comparison of theoretical and experimental

This Section illustrates the phenomenon that, during large current situations, the inductance will decrease and then the corner frequency of an G R filter will drift. This phenome- non can be confirmed by experimental results.

Consider an G R low-pass filter circuit where R = 11.4Q and the initial inductance is 31.8mH. Then, the transfer function for the linear inductor is

(36) vow - 11.4 Vs(s) 11.4 + ~0,0318

H&) = - -

From eqn. 20, the corner frequency is w, = 358.5rads.

7-

-1.5 -1.0 -0.5 0 0.5 1.0 1.5 current, A

Hystersis andsaturation m e s f o r the inductor (experimental results) Fig. 9

Fig. 9 presents the experimental results of the hystersis and saturation curves for a nonlinear inductor. The Figure shows that the maximum flux linkage Lax is 0.04817 Wb- turns and saturation current Imax is 0.85A. The experimen- tal results from Fig. 9 reveal that, with this inductor, the saturation model of eqn. 16 can be expressed by

0.0318i A = { 1 + 0.661il (37 )

where il is the flux linkage and a = 0.0318, b. = 0.66.

function for the linearised system becomes Then, from the describing function method, the transfer

(38 ) VAS) - 11.4 V3(s) 11.4 + sLeq H&) = - -

and the corner frequency is

(39) 1 l . Y

wc1= - Le,

where Leg is the equivalent inductance defined in eqn. 18. Consider, first, small current situations and an applied

voltage of 5.2V. Fig. 10 displays the results. From the experimental results, it can be observed that the corner fre- quency is 365rad/s.

112

io-’ 1 00 I O ’ 102 I( frequency, radls

3

Fig. 10 a pled voltage is 5.2 dwll current situation) ({Nonlinear; (ii) describing function; (iii) linear

Bode plot or the inductor of L A circuit in the example where

Based on the describing function method, it can be found that the corner frequency for the linearised system of eqn. 36 is 370radh. In this situation, the corner frequency calculated by the describing function method is quite close to that calculated by the conventional linear inductor coC = 358.5radh. Also, note that this is close to the experimental results value of 365radh. This phenomenon reveals that, during the small current situations, the performance of the nonlinear inductor is very close to the linear inductor.

Secondly, consider the large current situations where, as shown in Fig. 13, the applied voltage Vs is 20V. The corner frequency for the experimental results of the nonlinear inductor, the corner frequency predicted by the describing function method and the corner frequency of the conven- tional linear system, are 600, 610 and 358.5rad/s, respec- tively. It can be observed that the corner frequency predicted by the describing function method is rather close to the experimental results; therefore, based on this obser- vation, it can be concluded that the describing function method can be applied to accurately calculate the equiva- lent inductance and corner frequency of a nonlinear induc- tor.

I IO” 1 00 io’ IO‘! 1 03

-40

frequency, radh Fig. 1 1 applied voltage is 20V (large current situation) (i) Nonlinear; (ii) describing function; (iii) linear

Bode Plot for the inductor of L R circuit in the example where the

6 Simulation example

This Section presents a simulation example for the L-C-R low-pass filter. This simulation example reveals that the natural frequency will drift, because the equivalent induct- ance decreases in the large current situation. The simulation program is MATLAB 5.3 with SIMULINK.

IEE Proc.-Sci. Meas. Technol., Vol. 148, No. 3, May 2001

As displayed in Fig. 8, consider an GC-R low-pass filter circuit with resistance of 200Q capacitance of 30pF and inductance of 0.8168H. For the linear inductor, the transfer function is

0- U $ -10-

E - -20

V 1 H3(s ) = E = 2.4483 x 1 0 - 5 ~ ~ + 0.0041s + 1

(40)

-

The natural frequency for this linear system is 202.0139 rad/s.

Now, consider the saturation model of the nonlinear inductor expressed by Fig. 16, where a = 0.8168 and b = 13.613, i.e.

0.8168i 1 + 13,6131.11

According to the describing function method, the transfer fimction for the linearised system becomes

where Le, denotes the equivalent inductance defined in Fig. 18, and where a = 0.8168, b = 13.613.

First, consider the small current situation where the applied voltage is 5V. The simulation results are shown in Fig. 12. It can be observed that the corner frequencies for the conventional linear system, the linearised system by the describing function method, and the nonlinear system with a nonlinear inductor, are 300, 320 and 330 radls, respec- tively. All three of these corner frequencies are rather close, because the nonlinear inductor behaves like a linear induc- tor in a small current situation.

-30 I I

10.1 1 00 IO’ 102 I o3 frequency, rad/s

Fig. $2 Bode plot for the inductor of LC-R circuit in the example where the applied voltage is 5 V (small current situation) V. = 5V L = 0.8168H; R = 200Q C = 3Op.F; linear resonant angular frequency = 262.013aradis (i) Nonlinear; (ii) describing function; (iii) linear

Secondly, consider the large current situation in which the applied voltage is 20V. The simulation results are shown in Fig. 13. This Figure shows that the corner fre- quencies of the nonlinear inductor, the linearised system predicted by the describing function method, and the con- ventional linear system are 640, 606 and 300 radk, respec- tively. Based on this observation, it can be concluded that the corner frequency will increase in a large current situa- tion, and the describing function method can be used to evaluate the equivalent inductance and the corner fre- quency very accurately. Remark 3: For the GC-R and G R filters, the authors have neglected the capacitative effects which may depend

IEE Proc.-Sei Meus. Technol., Vol. 148, No. 3, May 2001

on the frequency. When an inductor is operating in very high frequency, the capacitative effects cannot be neglected.

-40 1 00 101 102 I 03 I 04

frequency, rad/s Fig. 13 Bo& plot or the inductor of L C - R circuit in the example where the applied voltage is A V ([urge current situation) Initial frequency = 1 radis; end frequency = 2000radis; applied voltage = 20V, R = 200Q; C = 30pF; L = 0.8168H; resonant angular frequency = 202.0139radk (i) Nonlinear: (ii) describing function; (iii) linear

7 Conclusions

This investigation applied the describing function method to linearise the nonlinear inductor. Hence, the inductance can still be estimated with large currents. For the L-R and GC-R low-pass filters, the overall system can be trans- formed into a Lur’e problem. In this Lur’e problem, the linear transfer function possesses low-pass filter property. Experimental results prove that the corner frequency will increase for an L-R low-pass filter. Meanwhile, experimen- tal results also confirm that the describing function method can be applied to estimate the corner frequency for an L-R filter. Furthermore, the simulation results have proven that, for an L-C-R low-pass filter, the corner frequency will also increase for a saturated inductor.

8 Acknowledgment

The authors are grateful to the anonymous referees for their valuable comments which have contributed to the improved presentation of this paper.

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References

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HARADA, K., ISHIHARA, Y., and TODAKA, T.: ‘A novel high power factor converter using a magnetic amplifier’, IEEE Trans. Mugn., 1996, 32, (5), pp. 5013-5015 SEBASTIAN, J., and UCEDA, J.: ‘An alternative method for con- trolling two-output DC-to-DC converters using saturable core induc- tor’, IEEE Trans. Power Electron., 1995, 10, (4), pp. 419426 CHUNG, S.C., and LIN, J.L.: ‘A general class of sliding surface for the sliding mode control’, IEEE Trans. Autom. Control, 1998, 43, pp. 1509-1 5 12 CHUNG, S.C., and LIN, J.L.: ‘A transformed Lurk problem for the sliding mode control and chattering reduction’, IEEE Trarzs Autom. Control, 1999,44, (3) SLOTINE, J.E., and LI, W.-P.: ‘Applied nonlinear control’ (Prentice- Hall, 1991), p.159 VIDYASAGAR, M.: ‘Nonlinear system analysis’ (Prentice-Hall, 1994) COOK, P.A.: ‘Nonlinear dynamical systems’ (Prentice Hall, 1994) SARMA, M.S.: ‘Electric machines: steady-state theory and dynamic performance’ (West Publishing Co., 1994, 2nd edn.), p. 518

Sy~t.. 1981, PAS-100, (I), pp. 106-110

113

10 Appendix

Consider the input current of sine-wave form shown as

i ( t ) = A + sin(&) (43)

As seen in Fig. 2, the output wave may be described by

Li, 0 5 w t 5 0

$( i ) = s a t ( i ) = Lisat, /3 5 wt 5 T - ,D (44) 1 Li, 7 r - , D < w t 5 7 r

where p = sin-l(iSal/A). To find the coefficients of the fundamental frequency

terms in the Fourier series for the output of eqn. 4, the following equations hold:

LA sin w t cos wtdwt+ 7r 7r

0 P 71

o r + 4. / LA sin wt cos wtdwt (45) 7 r .

7r-p

from which al = 0, a 7r-a

bl = 2 / LA sin w t sin wtdwt + 7T 7r

0 P

+ 2 LA sin w t sin wtdwt (46) 7r

7r-P

From eqn. 46, the equivalent inductance Leq of eqn. 15 can be obtained.

114 IEE Proc-Sci. Meas. Technol.. Vol. 148, No. 3, May 2001