A simulated non-linear self-inductance

6

Annales de l'Assocmt~on internanonale pout" le Calcul analogique

N" 1 - - ]anvler 1971

A SIMULATED NON-LINEAR SELF-INDUCTANCE by D. B A E R T SUMMARY - - The paper gives the description of a circuit for the simulation of self-inductances. After a discussion of seveaal parameters influencing the quality-factor Q of the simulated self-inductance, it is shown that the circuit as also able, owing to a function genmator, to simulate a non-hnear self-inductance. Several records show the results which can be obtained with the non-linear circuit in different applications

1. Introduction. When direct simulation of phenomena is required, it is often necessary to have available large inductors. Large inductors are neither cheap, nor easy to vary in inductance value, thus it appears reasonable to simulate them with active circuits. Several circuits have already been proposed in the literature [2, 3, 4, 5, 6]. Some of these designs simulate excellent high-Q inductors; one of their drawbacks, however, is that it ~s very difficult to change them into non-linear inductors. The circuit that we propose here has the following advantages : 1. It is easy to simulate non-linear inductances with an arbitrary BH characteristic without hysteresis if a function generator is used. 2. Very large inductance values can easily be obtained. Changing the self-inductance is done by changing one resistor, one capacitor or an amplification factor in the circuit. This is very convenient for the problem scaling. 5. The current that flows into the simulated inductance can be monitored as a voltage in a point of the circuit. Besides, it becomes possible to monitor continuously the part of the BH curve that is used during an experiment with an XY oscilloscope. 4. By introducing positive feedback, a very low frequency oscillator can be constructed. Its frequency can be changed simply by varying one resistance. 5. Negative self-inductances or capacitances can also be simulated.

certain conditions are met. Others use gyrators or positive-immitance convertors. The approach given here is quite q natural >> and makes use of a current source and an integrator (fig. 1). The current source is

8

d~

Fig. 1. - - General circuit for the simulation of induction or capacitors; k il the transconductance of the current source, T is an amplifier with amphhcatmn factor A, and S is a sign indicator. S = -l- 1, no phase inversion of e'; S ~ - - 1, phase inversion of e'. derived from a circuit suggested by P.J. Baxandall and E.W. Shallow [1], and it delivers a current proportional to its input voltage. The voltage e' is because of integration or differentiation 90 degrees out of phase with respect to e. After amplification A with or without sign change (S = + 1 or - - 1 ) the signal SAe' is fed into the current source which delivers a current kSAe' at the input A : i =

+kSAI(e)

The different possibilities of the input impedance Z, of the circuit are shown in table 1.

One drawback of the circuit however is that the simulated inductance is earthed at one side. If desired, this disadvantage can be eliminated by connecting two identical circuits in ser~es, the series connexion being made between the earth terminals. The common earth terminal must than be floating and the total self-inductance is twice the self-inductance of a single circuit.

TABLE 1 + 1

G f edt 2. Principles of the circuit.

G

--

I

Z, =

pLl

Z, = - - pL:

L, =

I/kAG

L, = - - 1/kAG instable

stable

Most of the existing circuits depart from, a circuit that has a purely imaginary input impedance ] ~ L if * Manuscript received 12th December 1969. ** State University of Ghent - Belgium.

~ Te/

'I'~t

de

Zi =

:/pC,

,'lt

Cl = kAG

Zi = - - 1 / p C l Cl -

kAG

D. Baert : A aimu&ted non-linear sdf-inductance 3. Stability.

E,

D C. stability of the circuit is not always a fact : the integrator cannot give the required phase shift of 90 degrees for very low frequencies. At d.c. the integrator behaves as a sign invertor and this means that only positive inductors will allow the circuit to be stable at d.c. The lowest frequency range is thus determined by the integrator performance. At d.c., the input AB behaves like a small resistance %e = 1/kAoA where A, is the d.c. gain of the integrator. The simulated inductance can thus be allowed to carry d.c. current. One could limit the lowest frequency range by connecting a capacitor in the point C of the loop of figure 1, m order to obtain a stable negative inductance. In this case, however, the circuit has no longer d.c feedback and it becomes necessary to limit the integrator gain at d.c. In the case of the differentiator, one always obtains a stable d.c. s~tuation. At high frequencies however the circuit has a tendency to oscillate due to phase errors in the integrator and current source, combined with the very high gam of the differentiator. The circuit of figure 1 can thus not be recommended for the simulation of capacitances.

El

7 - - Ao R' CR' R (l + Ao)

ire + ~ (~ + ao)] [~ + p

R' + R (~ + a,,)

]

Consequently, the 3 dB point of the Bode plot lies at R ' + R(1 + ao) 2 7r CR' R (1 q- A,,) and the d.c. gain is - - A o R'

R' + R (~ + A,,) For the simulation of positive mductors one can take R' = co so that f~ = t/2 rr R (1 + A(,) C. The tangent of the phase error at a frequency f becomes then : Q, = tg~ = I/o, CR(1 + & )

= /,/I

Positive F.B. in the integrator circuit can increase the gain A o and decrease f, by at least one order of magnitude if needed.

4. Quality factor Q of the simulated inductance.

4.2. Ot&r faaors i~zfl, e,cmg the Q-factor.

4.1. If S = q- 1 and if an integrator is used we have L, = 1/kAG where G = 1/RC (fig. 3). The figure 2 gives the phasor diagrams for a losless coil

4.2.1. The non-zero output impedance Zo of the integrator has no influence on the integration. 4.2.2. The load impedance Z,. of the integrator has no influence since this impedance can be reduced to the output impedance Zo by a Th&enin equivalent circuit. This holds only for real Z~. 4.2.3. The amplifier input Impedance RD is unimportant as long as the d.c. gain Ao of the integrator amplifier is high.

~l.'r

a2 = ~{a

(0

4.2.4. With no voltage follower after the input AB, the effective input impedance Z~3 becom.es R in parallel with /,j Li, thus Q = R ~oL, and the actual quality factor Qt becomes : 1

1 --

Q~

oa~

Fig. 2. -- Quality factor Q of coils with series and parallel esistances. with a series or parallel resistance and the correspond•ng definitions of the quality factor Q. One can predict the Q-factor of the circuit ,as caused by the ira-

¢o Fig. %. - - Basic integrator circuit.

perfect integrator performance as follows. The transfer function of the circuit of figure 3 is, with p = j ~ :

1 _[-

q~

- -

q

If the current source is imperfect and has a finite internal resistance Re, then R is shunted by Rc and

Q-

R. R o R+Re

1 ~LI

In order to obtain high Q values it is absolutely necessary ~o have as large as possible values for R and R o . Since R is limited by the amplifier input impedance (R < < Rrj Ao) it appears necessary to use a voltage follower with very high common-mode impedance ROM at the input AB as shown in figure 1. Further on, Re, must be very large (say 50 Mr2). If needed, very high Q values can be obtained by regeneration at the input terminals AB of the circuit (fig. 4). From figure 4 we find the amplification factor : Eo R2 &--I + ~ E~ R~

8

Aunales de /'Assocmtlon internalionale pottr /e Calcul analogique

N ° 1 - - [anvier I971

i T F-1 ( 7 -) -- RC 5 vdt or

=

d RC i --dt L-f v-~ (Tll

d~h

-

dt

and thus RC Fig. 4. ~ Input circuit used to cancel the parallel tosses by regeneration. and the input impedance : i

I

I

i

1

+

I

Ai

RD [1 + A,/A~]

Rc~t

The damping of Re, RI~, R , , R a can be cancelled out if 1 R~

A,~

1

Ra

~T

F-~ (T)

N o w it is possible to simulate an arbitrary B H curve without hysteresis, since after a few elementary calculations one can relate the ~ (i) curve to the B H curve for a given coil. 6, Circuits.

1

+

-

i

1

1

1

R~

Re

ReM

6.t. Figure 6 shows the circuit of the current source, This current source can generate a current of about. + 1 mA to - - i mA. It is a requirement to have a

1 OUT

q~ For an easy adjustment of the regeneration tt is convenient to change R~ and to have high values for Ra and R 4 . 4.2.5. InfIuence of the operational amplifier characteristic. The operational amplifier characteristic can be approximated by a first order characteristic

h (i ,o) -

.

400)~

-d8¥ ÷~mv

U-"

0

4oo~: 8,

Ao 1 + jo,/~,,

It can be shown that, due to this characteristic, sermus errors in integrating performance appear at the frequency Ao ~oo. This means that the highest frequency f that can be integrated must sahsfy the inequality

f < < a.l,,. A.

Iv,

_ d I

Fig. 5. - - General circuit for the simulation of non-linear inductances ; F.G is the function generator.

-o -qfV

Fig. 6. - - Circmt of the current aource. Q~, Q:, Q., Q~ are FNP transistors (BC 158) , Ql, Q~, Q,, Q7 are NPN tranalstors (BC 147). The transconductance k of the current source is about 0,1 mA/V. symmemcal veraon of the current source proposed by P.]. Baxandall and E W. Shallow, because it is important to have no d.c, component in the input current when e" is zero. The internal resistance is about 30 M~2, but this can be increased up to 1000 M ~ as indicated in their pubhcation, with a corresponding loss in temperature stabihty as a result.

5. N o n - l i n e a r inductances (fig. 5). I f a function generator F.G, is used after the integrator, the following equations hold : e" =

F(Te') T d

~

6.2. If one feeds back the output voltage of the function generator to the input AB via a resistance R t , it is sometimes possible to simphfy the circuit. The regeneration circuit of figure 4 must now be included if high Q inductances are required. Indeed, RL cannot be very large if the inductance must deliver some minimum current. If e.g, i,,,L, = ± 0.1 m A

D. Baert .' A s~mulated non-linear self.inductance and the maximum F.G. output voltage is ± then Re,,~ = 150 K

9

15 V,

We have to remark that if regeneration is used by any method, the circuit has a greater tendency to oscillate and becomes difficult to adjust If e.g. there is any change in A~ or R,, we have d

1

1 -

-

Z,

R,

dA1 - - ~

1

dR~

Rl ~

It follows that Ra and R4 must be large if d (I/Z,) has to be small. 6.3. At higher frequencies (> 1 Ko) the integrator can be replaced by an RC passive integrator. Since at high frequencies the inductance that must be simulated has in general a smaller value than at low frequencies, we must include some additional gain A after the integrator to obtain reasonable values of L, = 1/kAG.

1\

2V IA A

Photogiaph I. - - Response of the filter of fig, 7 to a square wave.

7. Applications. 7A. Simulation of non-linear filtering : e.g. the filter of figure 7. Photograph 1 gives the response of this filter to a square wave signal.

- - ----,.A A

/ ~hl

i i i i-

............... fill

iiii

fill

iiii

1,

III~

Fig, 7. - - Noo-hnear fdter with a square wave generator.

7 2. Simulation of ferroresonant conditions (fig. 8). Photograph 2 : resonant current in L~ at the generator frequency f. With the circuit of figure 8, the generation of subharmonic of the generator frequency

V---l--

A

It

o

Photograph 2. - - Resonant current in Lt at the fundamental *esonant con&don (upper trace) and generator voltage (lower trace) according to fig. 7.

/-: B a

Fig. 8. - - Circuit for the simulation of ferroresonant conditions.

f is possible under certain conditions. This can be seen in the following photographs, where a component with frequency f/3 is superimposed upon the component with frequency f. Photograph 3" current in Ll (lower trace) compared with the generator voltage. Photograph 4 : voltage AB compared with the generator voltage. Photograph 5 : q~(i) curve corresponding to the situation of photographs 3 and 4.

7.3. Simulation of rectifiers with RLC or LC filters; swinging choke-input filters. 7.4. Time-varying inductances can be realized if the loop gain is changed during the course of the process. Parametric amplifiers at very low frequencies can be simulated. 7.5. If a condensor Cl is placed across the input terminals AB the circuit behaves like a parallel tuned circuit. By increasing the regeneration at the input, the circuit can oscillate at the resonance frequency

f =

1/2=VL~C~

of the circuit. Since Ll is dependent of A (fig. 1), f can easily be &anged by one resistance if an operational amplifier is used as amplifier A.

A n n a l e s de l'Associalmn #n/ernatmnale pour le Calcul analogiquv

l0

--

1971

-I

/ / '"

E Ii

[

III

D,E

/

N" 1 --[anz,ier

III

I'I

"'"

I

V

II

/1 F

Photograph 3. - - Current iu L, (lower trace) in the case of subharmonic generatmn and generato, voltage (upper trace) according to fig 7.

Phntograph 5. - -

N(i)

curve in the case of subharmonic &vision.

REFERENCES

\ i i i -

-

?v

y dll

i 1 , ,

ii

-

Jill

,,,'"I

A

llll

II~I

~ A

~I",, J

A

AAII/ Pi/t//1t ljI/I

//X1,1 v V

V

-:V

V

-I Photograph 4. - - VoRage between AB (upper trace) m the case of subharmonic generation and generator voltage (lower trace) accordmg to fig. 7.

[1] BAXANDALL p.J., SHALLOW E W., <~Constant current source with unusually high internal resistance and good temperature stability>>, Electromcs Letters, September 1966, ',7ol. 2, N '* 9, pp 351-352. [2] RIORDAN R.H.S., <, Electronlcs Letters, February 1967, Vol. 3, n" 2, pp. 50-5I. [3] KEEN A.W., PETERS J L, <>, Electronics Letters, April 1967, Vol. 3, N ° 4, pp 136-137, [4] KEEN A W., PETERS J.L., (~ Nonreciprocal representahon of the floating inductor with grounded-ampl[fler realisations ~>, Electronics Letters, August 1967, Vol. 3, n ° 8, pp. 369-371 [5] GORSKI-POPIEL J., <{RC-active synthesis using posmveimmitance convertors >>, Electronics Letters, August 1967, Vol. 3, N" 8, pp. 381-382. [6] BERNDT D F, DLITTA ROY S C., <~Inductor simulation using a single unity gain amphfler >>, IEEE Journal of Sohd-State Circuits, Jura 1969, Vol SC-4, n" 3, p 161-162.