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Analysis of a fast-response magnetic amplifier
Analysis of a Fast-response Magnetic Amplifier 1. NAGY Introduction
restriction may be made, because at a negative feedback from the output signal V c is always positive in the control range . (e) The amplifier is always examined with free even-harmonics. This means a good approximation, especially if the voltage arising on the control circuit resistance is negligible to Vc and to the feedback voltage.
The object of our investigation has been the analysis of the self-saturating magnetic amplifier (abbreviated in the following to s.m.a.) described by Scorgie l and Geyger2 . The basic connection of the amplifier is shown in Figure I (switch K open) and is derived from the d.c. input and output s.m.a. by its negative, rigid feedback from the output signal. By this connection-with certain circuit supplements-a s.m.a. operating during a cycle, being theoretically of infinite power gain and input impedance, may be realized. In the literature up to now the connection has been discussed in a qualitative way, whereas the quantitative analysis was
Operation of a Self-saturating Magnetic Amplifier with Feedback from the Output Signal First let us follow the phenomena taking place within a cycle. To facilitate comprehension the s.m.a. without feedback is examined (Figure I, switch K closed). The operation within a half-cycle may be described in three sections following each other in time, while the rectifiers, as well as the cores, may be replaced by linear circuit elements. Within the sections the values of the dependent, variable quantities and the range of validity of the sections may be determined in an elementary way. Let us connect to the control circuit, at the time of the supplyvoltage zero transition, the control voltage Vc' (the quantities reduced to the gate-circuit are marked by primes) [Figure 2 (a)] and suppose the fluxes of cores A and B to be <1>A(O) = <1>1 and <1>B(O) = +<1>s [Figure 2 (b)]. The first three columns of Table 1 comprise the relations describing the change in the characteristic quantities in the three sections of the first half-cycle. Figure 2 (b - d) have been plotted on the basis of these. Vc' goes in section I to the control winding of core A, and in sections II and 111 to that of core B (Table I) thereby changing the fluxes in the positive direction. Therefore in the following half-cycle the flux of core B has to change by flux
UA
iL~
t uct
~ic
uB
t
-I
II ~
I
2
1
L ________ --l
I
L ____________ J
Figure 1
-NI (j)
based on data obtained by evaluating the characteristics plotted on the basis of the experiments. A characteristic feature of the examination was the way of thinking in average values, not clearing up, however, the phenomena taking place in the amplifier within a half-cycle. In the interest of better comprehension, computability and design, a detailed analysis of the connection operation has been effected. This has led, on the one hand, to a more explicit physical picture, and on the other, to mathematical correlations clearing up the effect of the parameters on the operation of the amplifier and justifying the design, since measurement and calculation results agree approximately.
G
Jor" Vc' d(wt)
less up to the positive saturation than core A had to in th~ first half-cycle [Figure 2 (b)]. In the second half-cycle the values of the dependent, variable quantities are described by the same relations as in the first one, with the difference that : (a) cores A and B change places, (b) the initial flux value of core B is different from that of core A . In the second half-cycle, also, and in all succeeding ones-by an application of the equations of Table I-the values of the dependent, variable quantities may be determined. The effect of the control voltage Vc' is to bring about a steady change in the middle value of the output voltage, V L [see hatched areas Tl and T2 of Figure 2 (c)]. From Table 1 it may be demonstrated that the change in VL is in the first halfcycle after connecting Vc'
Assumptions We have made the following assumptions: (a) The dynamical hysteresis loop of the magnetic material
applied is square. (b) The rectifiers in the reverse direction are of infinite, in the forward direction of finite and constant resistance. (c) The load is a pure, ohmic one. (d) The control voltage, Vc, is a positive direct voltage. This
V Ll = D = D
13
1114
r~ [/I -
J~I
2lco(Ri
J:[[vo' -
+ R L )] d(wt) (1)
u
+ 2Ico(R; + R L )] dWf
1.
NAGY
effect of Uc' is to change UL by a constant 'speed' until it reaches its extreme value. By again making Uc' equal to zero, however, the change in UL is stopped. Consequently, with a rigid, negative feedback from the output signal, UL is changed until the voltage U,' does not equal Uc'. On the basis of the above, the steady-state operation of the s.m.a. fed back from the output signal (Figure 1, switch K open) is readily understandable. Since the current flowing on RL after the saturation may be several orders of magnitude higher than before, the feedback voltage, as compared with Uc', can be neglected in sections I and 11. A difference with respect to the s.m.a. without feedback may be discovered only in section Ill. This section must be divided into two further parts. In the first part the flux of the unsaturated core goes on changing in a negative direction (u,' > Uc'), whereas in the second part the flux change is already in the positive direction (u,' ~ Uc'). The direction of the change in the flux depends on the sign of the difference between the control voltage and the feedback voltage. The dependent, variable quantities in the two parts of section III are illustrated by the relations given in columns 4 and 5 of Table I. Figure 2 (f-h) were plotted with the aid of Table I. As compared with the ordinary s.m.a. 3 , considerable deviation appears in the change in time of the flux and of the control current. In steady-state conditions the change in
and in all succeeding ones, e.g. in the n
where D
=
RL
1 7TRi
+ RL
is the firing angle before connecting Uc', and r:t.n is the firing angle in the half-cycle n, calculated from the connection of Uc'. In general, we are interested only in the change with time of the output voltage (current) middle value. This may be illustrated by Figure 3 (a), or curve 1 of Figure 3 (b) (equation 2), neglecting the change of UL within the first half-cycle. (In the illustration, T denotes half of the cycle time.) Figure 3 (a) illustrates the change in UL where it would seem as if the total voltage change in the loading occurred suddenly at the beginning of the half-cycle. Curve 1 of Figure 3 (b) corresponds to a case in which it would seem as if the voltage change took place at a constant speed during the half-cycle. Of the two conceptions, the application of that leading to more simple relations in the case of feedback from the output signal is more advantageous. From our considerations it follows that the s.m.a.-should our assumptions be fulfilled-is an integrating element. The r:t.
Um
1+C05'"
.
(\
--1T- : sin r III
+---~-~--
(a)
: IW~G .r;,1TU~
I
iL
(c)
t
I I I
d(wt) :
I
I I
I
I ·r
le
(d) Yr_~L-
Figure 2
14
1115
_____
~~L-
______
uJt
~_
Table I Section I - - ---- -- ..
Section 11 - ~
Fulfilled inequalities
-
Vc' :> u
-- _.
Section III
0.;;; wt .;;; (31 here V' sin PI = --E..
I
Vc' .;;;
I I
I
11
(31 ';;; wt .;;;
I here
IX, .;;;
IXI
wt .;;;
VI
Ll
Current
iA
= =
N W
-firing angle
IXI
U
0
2Iro(Ri
11 -
It
Vc'd(wt) 1N1 (0
G
sin
f'
(i [11 1
+
2/"0( Ri
0
R1J
+
0
RIJ] d(wt)
Flux change LlB
Current
iB
2Iro
0
0
=
0
=
0
-11
+
Vc'
+
Ri
2Ieo(Ri
+
+
RL
Vc
RI
, RI
Ri RI
+
No
+ Ri + RL Ne 11 <
(3I1I
Pm .;;; wt .;;;
Vc'
TT
RL Vc Vrn
I
o
1
o
I
RI
+
,
I
+
r
RI)] d(wt)
Ri+RL Ji.
l eo
I I I I I
I
I r(iI ll(,
--
I uING• '"
Vc -
C
11
Rf
-
11
RINu/Ne R, + R i + RI,
RING/Ne) . d(ult) - I mNu + Ri + RL
f"
filII
(V' c
-11
RI
o
0 :
1
0
v'
,
I
C
-Vc'd(wt) wNu '"
I
=
PIlI = R f +
1
V'
R L)
If'
N fl 1 [-II+V/ (I) u + 2IeO(Ri 0
I Load current Control current ic'
wt .;;;
>
II
=
h,=
Nu -11 RE Nc
+
!
11
'B' core voltage liB
RI R i
here
V' c
Flux change
+
IX .;;;
TT
'A' core voltage =
RI
I
Um
11.4
Section 111 (b) ~ -
I
I Range of validity
Section III (a)
--~~
11
2/,0 -Ieo
Ri
I I
+
11
RL
RI + Ri I
1"0 -11'0
If feedback of the voltage reaches resistance RI in the expressions of section H, (R, + RJ must be written instead of Ri' leo = coercive current referred to the gate winding. Ri = sum of the resistances of the current source, the gate winding and the rectifier (in the forward direction).
1116
I
+ RE leo
RINu/Nc) d( wl ) + Ri + RI,
J. NAGY
flux of core A during a half-cycle is of the same magnitude, but in the opposite direction to that of core B. Substituting the corresponding values from Table I Uc' = (RfNclRLNc)UL = Ut'
the s.m.a. without feedback. The equations describing the static characteristics have been determined by using Figure 2 (g) and (h), Table 1, and equation 3. The final result is summarized in Table 2. The static characteristics plotted on the basis of the table are shown in Figure 4. The figures include also the static characteristics of the s.m.a. without feedback (dotted line). The deviation between the characteristics is obvious.
(3)
From the above physical considerations and from equation 3 it becomes apparent that the middle values of the control and feedback voltages are equal under steady-state conditions. The amplifier is of voltage-transforming character. From equation 3 the gain of the feedback is
Transfer Function-Equivalent Time Constant-Nyquist Diagram
(3a)
Af = R,NG /RI_Ne
For the dynamical examination of the regulating system, the transfer function of the elements appearing in the regulator is necessary. When the response is known [Figure 3 (a) and (b)], the transfer function of the s.m .a. without feedback may easily be determined and our results are given in the first line of Table 3. The last expression in the table refers to when time T with respect to the duration of the transient process is short and therefore- as in Figure 3 (b)-the response may be approached by line 2 of Figure 3 (b) . On the basis of these results, the transfer function of the s.m.a. with feedback-considering equation 3a- may be calculated in an elementary way (second line in Table 3). According to the elements with single time constant, the term
Static Characteristics The static characteristics of the s.m.a. with a negative feedback show an important deviation as compared with those of
R __ L U'
(a)
\
(4)
'R l '
characteristic for the operating speed of the s.m.a. with feedback, and being the so-called equivalent time constant, may be introduced. With all three Y*- for a step input- T,. equals, for unit final value, the area limited by the final value and the instantaneous values. While the response applying Y1 * may easily be determined, having a simple form [Figure 3 (c)] , the response based on Y 2 * is complicated. In the application of the three transfer functions, the following may be stated: with s.m.a. of 'fast' response Y1 * may be used, with that of 'slow' response Ya* is appropriate ; the use of Y 2 * is disadvantageous. In connection with s.m.a. of 'fast' and 'slow' response, the following must be considered. To characterize the speed of response, the time (T63) could also be introduced; this, calculated from the connection of the control voltage, is necessary for the middle value of the output voltage to reach 63 per cent of the final value. In Figure 5 the change in T63 relative to functions Uu(t) and that in Tr in function of the resultant gain AA, are plotted. Naturally, in a response of exponential form Tr = T63 . With small values of AA, this corresponds to our case, the transfer function being, with small loop gain, i.e. with slow response, Y3*' however, Tp and T63 at values AA, near to the unit differ considerably, and then the use of Y1 * is advantageous. Of the two time constants the choice of Tp is motivated, on the one hand, by the more simple way of determination, and on the other, by T63 being not interpretable (negative number) at values of AA, near to the unit. According to equation 4, increasing A, causes Te to decrease. By a suitable choice of A, it may be possible, by a step in Uc' in the first half-cycle, to let UL reach its constant value in the second half-cycle. The fastest speed of operation to be reached theoretically by the magnetic amplifier may be realized in this way. Then Te = T and (5)
I I
I
0
2",
"t
3,;
4,;
5'(
2 (b)
t-~
- - Uc ' 't
1:
2~
31:
4..:
5'[
Uu
1
4.0' AAf
(c)
o
or:
2'(
3'(
41:
The operation during a cycle is illustrated clearly in Figure 3 (c). The increase of the output voltage in the third and in all succeeding half-cycles is-because of equation 5-zero.
51:
Figure 3
16
1117
ANALYSIS OF A FAST-RESPONSE MAGNETIC AMPLIFIER
U'c
It tg~
--/-...-=--l·lt.,max
tg~
=Rc
_-====--,tL--4 U'C,L
=s
-le'
-Ic:o
l'cl
(a)
Ul,max
/ -U~
,/
/
/
/
- 1.
2 (c)
(b)
Ul
---:;;;--!-------------..._---/~-
leo
I
/ U'c U~,l
(d) Figure 4
Table 2
In the case of feedback, an approximation of Rc = 0 and I L,min = 0 was taken in the control range Ranlie ot'validity
Control current value
I
h ~1
Load current
Control VOltage Ve'
S(Jro
1/, [
< le'
+ 1/ )
--------
i
~i
N,. Rf N' S(/,o
,
+ 1/)
Vc'. 1 + R/(/e' - le', I)
--------
Ip i ~1
Input power
-1,,0';;; l e' ,;;; le', 1
(Vc',l -
1;' = 1
R/lc',l)lc'
+ Rc'Ic'2
1
Table 3 A = RL/CRt
+ R + R L) j
Accordinli to FiliI/re 3 Ca) (n
Transfer function of a s.m.a. without feedback Yn(p) =
I
---------------------------------1
=
I)
L[V",] RL e- PT L[V/] = Ri + RI. 1 - e-PT
A ccordinli to cl/I've I ot' F(fure 3 (b) (n = 2) L[ V L2 ] L[V/]
RI.
=
Ri +RL
e -pT
Pr
Transfer function of a s.m.a. with feedback
*
_
Yn(p)
e-PT
_
A --,---.,---;----::pT + AA,e jJT
Yn (p) - 1 + AY ( ) J "' P
17 2
1118
According to Cl/I've 2 of Filiure 3 Cb) (n = 3)
I. NAGY
Self-saturating Magnetic Amplifier as a d.c. Impedance Transformer The s.m.a. with feedback permits the development of an amplifier of high input impedance and, depending upon the manner of feedback, of high or low output impedance. A physical explanation for this can easily be given. The input impedance is increased because the control voltage must be balanced not only with the ohmic voltage drop but also with the feedback voltage; this is generally many times greater than the control voltage. The output impedance, in the case of feedback from RI or RL> is therefore high or low, respectively, because with constant Uc' the voltage reaching RI or RL is also constant. A constant current flows through the load in spite of RL being changed, consequently the amplifier is first a current generator, then a voltage generator and UL remains practically constant (Uc' = constant) even with a change between wide limits of the supply voltage. Table 4 presents simple relations for calculating the input and output impedances.
f :50 Hz
1)
T : 1 (log 0'37 63 2f log (l-AAf)
0'1
01
0'01
Figure 5
Increasing Figure of Merit
In Figure 6 are plotted the Nyquist diagrams of the s.m.a. with and without feedback.
All things considered, the quality of the s.m.a. connection is characterized by the figure of merit, J. J is a result of the power gain and of the equivalent time-constant ratio. By power gain the ratio Po,max/Pi,max is meant (the maximum output power is marked by Po,max, but the maximum input by
A A. f : 0·5
Pi,max), Real aXIs
0'6 0'5
.,
0'4
U 0'3 0'2 0'1 0
OD: c,1 (self-saturating magnetic amplifier without feedback)
0'4
Ri + RL Y3 (jttl't) RL
0'8
1'2
1'6
2'0
2'2
At Figure 7
Ri +RL RL
Ri + RL RL
'lf/15
'lf/15
Y,(jw't)
The value of J is higher for a s.m.a. with feedback than it is without feedback, and may be increased to a certain extent by increasing the gain of the feedback (Figure 7). A further, considerable increase in the figure of merit may be realized by certain circuit supplements. To understand and calculate the necessary complements, the following points must be considered. Figure 8 shows a block diagram of the s.m.a. with feedback; the voltage drop on the control circuit resistance is disregarded. In accordance with Table 3 and Figure 3 (c), it follows that, except for the first half-cycle following the change in Uc', the alteration in the output voltage middle value in any half-cycle as compared to that in the previous half-cycle is the product of the middle-value of the resultant control voltage supplied to the control circuit and of factor A. (This statement is valid also
Y2 (jW1:)
'If /15
Figure 6
18
1119
ANALYSIS OF A FAST-RESPONSE MAGNETIC AMPLIFIER
Table 4
Feedback
From R,
,
Input impedance
Zi =
oVe' -aI-e' ~
I
i
-------------------------Output impedance
~ Ri Re'lcO 1T
for the s.m.a. without feedback.) The middle value of the resultant control voltage in any half-cycle may be obtained by subtracting from Vc' not only Vt' but also Ic'Re' (Figure 8 (b». Knowing VI- and ID the value of le' can be determined from the equation of Table 2. (Between fe' and fL time delay is neglected.) The block diagram resulting is shown in Figure 8 (b),
U~
UL
+
At Vm
Figure 8 (c) indicates the way in which the intrinsic, negative feedback can be compensated. The s.m.a. must be provided with a bias - lcoNr; connected to a separate control winding as well as with a 'weak' positive feedback drawn from the output voltage, or current, and supplied to a third control winding. The control current will be--in the control range-zero when the gain of the positive feedback is Ai = 1/(SR1J. The circuit diagram of Figure 9 is based upon this information.
(a) R
( b)
Figure 9
Only the role of the constant voltage VI; is unexplained; its task is, where Vc = and VI;; = 0, whatever high value for current 12 may be chosen, le cannot be reduced to zero since the voltage Vt,min resulting from the minimum of IL will maintain a current (, of a certain value. To reduce to zero the value of I,. with the help of '2--also in the case of zero control voltage or near to it--the constant direct voltage VI; = Vt,mill is necessary. It must be emphasized that the circuit supplements described may easily be realized, so the dynamic behaviour--equivalent time constant--of the s.m.a. should not be subject to variations.
°
(c ) Figure 8
Appendix. Measurement Results
where, in addition to the extrinsic, negative feedback, an inherent, intrinsic, negative feedback also appears. Compensating this intrinsic, negative feedback the value of the output sign may be influenced, theoretically with zero control current, merely by changing the control voltage. This means that a S.m.a. of 'infinitely high' power amplification and input impedance may be realized.
To verify the correctness of our theoretical considerations and their suitability for numerical calculations, measurements have been carried out with a magnetic amplifier of 10 W, made of a magnetic material Permenorm 5,000 Z with toroidal core. During measurement the total output voltage was negatively fed back. 19
1120
r. NAGY
IL
fAJ u~
0·9
[vJ 2·25
08
15
0'3
075
0·2
05
0 ·1
0'25 I~
-2 ,8
o
-2
[mAJ
-2-8
o
-2
The dotted line refers to the case without feedb;!ck
Cb)
Ca)
u'c
[V] 275
2 '5 225
2 175 15
1' 25
0 ' 75
0·5 0·25
5
4
Cd) Figure 10
20
1121
3
2
o
ANALYSIS OF A FAST-RESPONSE MAGNETIC AMPLIFIER
measured h = hUe') are shown in Figure 11. The characteristics I, 2, and 3 measured in the order of enumeration by an ever-weakening positive feedback refer to case Uk = O. Therefore, in the environment of a zero control voltage le' is relatively great. For us characteristic 2 is of special interest: here, except for the zero control voltage and for its immediate
The characteristics
h = hUc'), Uc' = Uc'Uc'), Pi = PiUe')
= UL(Uc')
and UL
plotted with different values of RL are illustrated in Figure 10 (er Figure 4). It is obvious that calculation and measurement give qualitatively the same characteristics.
[1\] •- - - .
\/
:y:-
r' ! .
I·
\
\.
\ '.
\
i\ R.llkA ,
1 I
i i
5 \ 6-1 1'7
.\0'1 ,.
'., if 1/ , ..
f -2
j,.44kll.
'~
1/1243
/.
-N
"".{
k11kll. 02
..
\
A t curves 1.2.3 and 4 l2 ' 3'4 mA At curves 5.6 and 7 12' 2'3 mA
/
44k.fi--J
,,.~\/ .,
J
10'4
f
Supply voltage : 18 V Supply frequency : 50 Hz RL·, 20 ll.
/.
I
I
\;I
2·4
".
-1-6
• 1.2.3.
}.
4
;V!
/
°
-1,2 -08 -0·4
1c'
!
0·4
0·8
[mAl •
Figure 11 Table 5
:1[lJ I I RL
[.n] RI:
i L .mlia~
I;"I I
I
15
I-· ,~
;
-R l
!
reet.
!
I
Ig; ,5
i
5
~percentage error
MeaSUring Calculatio
!dlmension
I
:
[mm'] selenium
20
,
10 4 11 5
~mw] x
Iperc~n-
I,ma
50 x 50
0'564 .10 '
0'733 . 10'
I
23
1·7 45
I
43 x 43
0·564.10'
0679.10' i
17
1762
46
2 84
I
Z,
_ tage
252
I
2 305
308
1'095 .10'
I
13 7
i
149
1
tage
11 '49510' I
!
236
[.0.] Zo
5ec T.
sec-I
.
r, ~P
r5e c " ]J,'ca.. without feedbad )
Calculation Calculation C.lcu lati o
27
0·0 656
o 1204
19250
95
00812
o 0914
17100
6 77 0
I
I
I
1285
IPercen-
Measurin g k atcu,ation i error
Measur ing :calCulationi e rror
,
[fi]
. -- - . ~ -- -
4 3 x 43
0'442.10 3 0'463.10 3
1
3 26
1'706. 10 '
128
11'88'. 10'
I
- -- --
43 x 43 43 x 43
0'348.10 3
0368.10' 1 5 5
! 0,298 .10 3 , 0298.10 3
!
0
337 38
383 415
12
2 .10 '
1222510' 1 10 5
2-28510' ' 24110,1 • . I
84
00865 . . - - - -5 00941
00765 16350 - - - - _. --0·0699 14·900
5 930 ---- -
-
-
-
5·560 ~. -
-- ,
5015
vicinity, Ic' is approximately zero . (The value of l e' remained at ± 10 IlA.) When plotting curve 2, the value R = 22 kQ was applied (Figure 9) instead of R = SRLN1/N,. = 23'9 H2, which was necessary in principle. When introducing the voltage Uk = 2/,o RL = 0·134 V, the characteristic 4 was obtained. Here already the control current is approximately zero, even at low values of Uc' By changing the bias, the characteristics are shifted by themselves parallel to each other (see curves 5, 6, and 7).
For quantitative comparison some characteristic values of the connection have been calculated, on the one hand from the measurement results and on the other on the basis of the formulae derived in the study and summarized in Table 5. The difference between the results of calculations and of the measurements is partly attributable to the fact that the resistance of the selenium rectifier in the reverse direction is finite. This is verified by comparison of the measurement and calculation results obtained by rectifiers 50 x 50 mm2 and 43 x 43 mm 2-in the case of R L = 5Q-as well as by the fact that the calculated value of S is higher than that obtained by measurement (see Figure 4 (a». It is well known that with an increase in the current of the rectifier in the reverse direction the slope of the characteristic IL = I LU/) gets smaller. Measurements have been carried out on the same magnetic amplifier with the connection as in Figure 9. The characteristics
References 1
3
D. G. Fas t response with magnetic amplifiers. Trans. Amer. Inst. elect. Engrs 72, Pt. I (1953) 741 GEYGER, W. A. Magnetic amplifiers of the self-balancing potentiometer type. Trans. Amer. Inst. elect. En.l{rs 71, Pt. I (1952) 383 S TORM , H. F. Ma"fjnetic Amplifiers. 1955. New York; Wiley S CORGIE ,
Summary The object of our investigation has been the analysis of the selfsaturating magnetic amplifier. The basic connection of the amplifier IS derived from the d.c. input and output s.m.a. by its negative, rigid feedback from the output signal. In the interest of better comprehenSion, computability and design, a detailed analysis of the connection operation has been effected.
Firstly the s.m.a. with a negative feedback from the output signal is dealt with. It is shown that the change in time of the fluxes and of the control current deviates considerably from that of the s.m.a. without feedba ck. The equations of the characteristics (namely the load current, the control voltage and the input power as functions of the control 21
1122
1. NAGY
current) have ueen determined, having a most simple form. They reveal the basic difference between the static characteristics of the s.m.a. with and without feedback. The transient process of the amplifier is discussed and the possible expressions given for the transfer function, the response and the equivalent time constant. A relation is obtained for the operating conditions during a cycle. The Nyquist diagram is also plotted. The connection permits the realization of an amplifier of high input impedance and of low output impedance depending upon the manner of the feedback. In the course of examination, simple and relatively accurate relations have been found for the calculation of the input and output impedances. According to our considerations, the figure of merit is higher in
case of feedback than with the ordinary s.m.a" growing with the gain of the feedback. A most valuable conclusion may be drawn from the block diagram of the amplifier, namely, if completing the s.m.a. fed back by (a) a positive feedback taken from the output voltage, or current having a rate that can be read from the block diagram, (b) a constant d.c. bias, and (c) a small direct voltage compensating the VOltage drop on the feedback resistance caused by the minimum load current, it is possible theoretically to make the control current zero in the whole control range. To verify our quantitative results, measurements have been effected. They demonstrated the correctness of our results and their applicability for numerical calculations.
Sommaire On etudie les phenomenes transitoires dans I'amplificateur et !'on donne des expressions possibles de la transmittance, de la reponse et de la constante de temps equivalente. On obtient une relation reliant les parametres de fonctionnement au cours d ' un cycle. On trace egalement le diagramme de Nyquist. Selon les resultats obtenus, le facteur de merite est plus eleve avec contre-reaction que pour un amplificateur autosature ordinaire, ceci d'autant plus que le gain de la chalne inverse est plus grand .
L'objet de cette etude est un amplificateur magnetique autosature. Le montage de base de I'amplificateur derive de I'amplificateur magnetique autosature it entree et sortie continue, avec contre-reaction invariable du signal de sortie. Pour en faire mieux comprendre le fonctionnement, le calcul et la conception, on a effectue une etude detaillee du montage. On etudie d'abord I'amplificateur magnetique autosature avec contre-reaction du signal de sortie. On etablit que les variations des flux et du courant de commande different considerablement du cas OU il n'y a pas de contre-reaction.
Zusammenfassung fUr die Arbeitsweise wahrend einer Periode. Diese Angaben werden durch das Nyquist-Diagramm ergaznt. Die SchaItung macht den Aufbau eines Verstarkers moglich , der in Abhangigkeit von der Geg~nkopplung hohe Eingangsund niedrige Ausgangsimpedanz hat. Bei einem Magnetverstiirker des vorIiegenden Typs kann mit Hilfe von (a) einer Mitkopplung durch Ausgangsspannung oderstrom, deren Grad dem BIockschaItbild zu entnehmen ist, (b) einem konstanten Gleichstrom und (c) einer niedrigen Gleichspannung, die den durch den Mindeststrom i.iber dem Ruckopplungswiderstand verursachten Spannungsabfall kompensiert, der Steuerstrom im ganzen Regelbereich theoretisch zu Null gemacht werden. Die Ergebnissewurden durch Versuche bestatigt.
Die Grundschaltung eines MagnetversUirkers in Selbstsattigungsschaltung. Mit Gleichstromeingang und -ausgang kommt uber eine starre Gegenkopplung zustande. Beim Magnetverstarker mit Geg~nkopplung vom Ausgang her wird gezeigt, daf3 sich die zeitliche Anderung von Fluf3 und Steuerstrom von der des Verstarkers ohne Gegenkopplung erheblich unterscheidet. Die Gleichungen der wesentlichen Grof3en (d.h. Arbeitsstrom, Steuerspannung und SteuerIeistung als Funktionen des Steuerstromes) offenbaren den grundlegenden Unterschied im statischen Verhalten der Verstarker mit und ohne Gegenkopplung. .. Die Beziehungen fUr Ubertragungsfunktion , Frequenzgang und aquivalente Zeitkonstante werden angegeben . Es folgt eine Beziehung
DISCUSSIONS R. A.
LIPMAN,
(H. Storm and M. Rozenblat). The negative voltage feedback hardly changes the ampere-turns gain and, consequently, has no influence on the dynamic figure of merit of the amplifier. In the case of low time constants and 'high' frequencies in the control signal, it is necessary to take into consideration the discrete character of the transient response of the amplifier. In the case of a 100 per cent voltage feedback, such a magnetic amplifier represents a delay element of half a cycle of the supply frequency, i.e. it becomes fast acting. However, it is known that the dynamic figures of merit of ordinary and of fast acting magnetic amplifiers are nearly equal and they are practically equivalent, from the point of view of the dynamic as well as static characteristics. The statement of the author that the dynamic figure of merit of the amplifier is improved by negative feedback is not understood because, in the analysis of the circuit operation it is assumed that the side sections of the hysteresis loop have infinite slope and that the width of the hysteresis loop does not depend on the rate of change of the inductance, i.e. the influence of eddy currents and magnetic viscosity is not taken into consideration. However, as is shown in the book of M. Rozenblat, under such conditions an ordinary magnetic amplifier with self-saturation (without additional feed backs) will have an infinite ampere-turns gain and, consequently, it will have an infinite dynamic figure of merit. If there is an
U.S.S.R.
There are objections to one of the basic assumptions of the paper, namely, that the negative electrical voltage feedback improves the dynamic figure of merit of the amplifier. Usually, the dynamic figure of merit is understood to be the ratio of the power gain to the time constant. For magnetic amplifiers with a 'large' (compared to the half-cycle of the supply frequency) time constant, and also for magnetic amplifiers with any time constant, provided that 'Iow' control signal frequencies (Iow compared with the supply frequency) are considered, it is possible to disregard the discrete character of the transient response of the amplifier. Therefore, the magnetic amplifier can be considered as an inertia element of the first order with a time constant equalling
where K is the voltage gain, N e and N g are respectively the number of turns on the control and excitation windings. Under these conditions the negative feedback reduces to an equal extent the voltage gain and the time constant, thus the dynamic figure of merit remains unchanged. The correctness of this assumption also follows from the fact that the dynamic figure of merit is proportional to the ampere-turns gain. 22
1123
ANALYSIS OF A FAST-RESPONSE MAGNETIC AMPLIFIER
linear but slower. Since the time constant is proportional to the decrease At, the figure of merit will, indeed, increase with increasing At> see Figure 7 (in writing the above equation it was assumed that in using the static characteristics h = hUe') the control current lco/2 is located in the control range and also that all the assumptions made under point (d)* of the 'basic assumptions' of the paper are fulfilled). R. A. Lipman has in mind the third definition of the power amplification described in the book by Storm, namely, the product of the current gain and the voltage gain for small increments.
appropriate shift, such an amplifier will operate without taking any power from the control signal source. In reality, the high figures of merit for the experimental amplifier, described in the paper, were achieved by using external positive feedback. With this feedback it is possible theoretically to obtain an infinitely large ampere-turns gain and, consequently, also an infinite dynamic figure of merit. In practice, however, the degree of effectiveness of the positive feedback is limited by the instability of the amplifier characteristics during changes of ambient temperature, supply voltage and frequency, etc. I. NAGY in reply. I believe that basically the comments of R. A. Lipman result from a difference in the definition of the term 'power amplification'. In the paper this term is understood to be the ratio of the maximum output power (Po llIax) to the maximum input power (Pi max). If all the output voltages are transmitted by means of the feedback circuits, Po max will remain practically constant, whilst At (feedback gain) will increase. The value P,: lI1ax (the last row in Table 2) equals:
= 8"
I am sorry not to be able to agree with R. A. Lipman on the point that negative voltage feedback results in hardly any amplification with respect to the ampere-turns and, consequently, it does not influence the figure of merit of the amplifier. A detailed study of the Figures 4(a) and lO(a) of the paper shows clearly that the differential amplification with respect to the ampere-turns does change.
At
7f
Pimax
K =dh.dUL P dIe dUe
UnJco At
+1 +
Rj
* (Translator's note: The Russian text specifies d but, due to the difference between the sequence of the Cyrillic and the Latin letters, d in this case may well be e).
RL
This means that with increasing At the increase will not be
23
1124
restriction may be made, because at a negative feedback from the output signal V c is always positive in the control range . (e) The amplifier is always examined with free even-harmonics. This means a good approximation, especially if the voltage arising on the control circuit resistance is negligible to Vc and to the feedback voltage.
The object of our investigation has been the analysis of the self-saturating magnetic amplifier (abbreviated in the following to s.m.a.) described by Scorgie l and Geyger2 . The basic connection of the amplifier is shown in Figure I (switch K open) and is derived from the d.c. input and output s.m.a. by its negative, rigid feedback from the output signal. By this connection-with certain circuit supplements-a s.m.a. operating during a cycle, being theoretically of infinite power gain and input impedance, may be realized. In the literature up to now the connection has been discussed in a qualitative way, whereas the quantitative analysis was
Operation of a Self-saturating Magnetic Amplifier with Feedback from the Output Signal First let us follow the phenomena taking place within a cycle. To facilitate comprehension the s.m.a. without feedback is examined (Figure I, switch K closed). The operation within a half-cycle may be described in three sections following each other in time, while the rectifiers, as well as the cores, may be replaced by linear circuit elements. Within the sections the values of the dependent, variable quantities and the range of validity of the sections may be determined in an elementary way. Let us connect to the control circuit, at the time of the supplyvoltage zero transition, the control voltage Vc' (the quantities reduced to the gate-circuit are marked by primes) [Figure 2 (a)] and suppose the fluxes of cores A and B to be <1>A(O) = <1>1 and <1>B(O) = +<1>s [Figure 2 (b)]. The first three columns of Table 1 comprise the relations describing the change in the characteristic quantities in the three sections of the first half-cycle. Figure 2 (b - d) have been plotted on the basis of these. Vc' goes in section I to the control winding of core A, and in sections II and 111 to that of core B (Table I) thereby changing the fluxes in the positive direction. Therefore in the following half-cycle the flux of core B has to change by flux
UA
iL~
t uct
~ic
uB
t
-I
II ~
I
2
1
L ________ --l
I
L ____________ J
Figure 1
-NI (j)
based on data obtained by evaluating the characteristics plotted on the basis of the experiments. A characteristic feature of the examination was the way of thinking in average values, not clearing up, however, the phenomena taking place in the amplifier within a half-cycle. In the interest of better comprehension, computability and design, a detailed analysis of the connection operation has been effected. This has led, on the one hand, to a more explicit physical picture, and on the other, to mathematical correlations clearing up the effect of the parameters on the operation of the amplifier and justifying the design, since measurement and calculation results agree approximately.
G
Jor" Vc' d(wt)
less up to the positive saturation than core A had to in th~ first half-cycle [Figure 2 (b)]. In the second half-cycle the values of the dependent, variable quantities are described by the same relations as in the first one, with the difference that : (a) cores A and B change places, (b) the initial flux value of core B is different from that of core A . In the second half-cycle, also, and in all succeeding ones-by an application of the equations of Table I-the values of the dependent, variable quantities may be determined. The effect of the control voltage Vc' is to bring about a steady change in the middle value of the output voltage, V L [see hatched areas Tl and T2 of Figure 2 (c)]. From Table 1 it may be demonstrated that the change in VL is in the first halfcycle after connecting Vc'
Assumptions We have made the following assumptions: (a) The dynamical hysteresis loop of the magnetic material
applied is square. (b) The rectifiers in the reverse direction are of infinite, in the forward direction of finite and constant resistance. (c) The load is a pure, ohmic one. (d) The control voltage, Vc, is a positive direct voltage. This
V Ll = D = D
13
1114
r~ [/I -
J~I
2lco(Ri
J:[[vo' -
+ R L )] d(wt) (1)
u
+ 2Ico(R; + R L )] dWf
1.
NAGY
effect of Uc' is to change UL by a constant 'speed' until it reaches its extreme value. By again making Uc' equal to zero, however, the change in UL is stopped. Consequently, with a rigid, negative feedback from the output signal, UL is changed until the voltage U,' does not equal Uc'. On the basis of the above, the steady-state operation of the s.m.a. fed back from the output signal (Figure 1, switch K open) is readily understandable. Since the current flowing on RL after the saturation may be several orders of magnitude higher than before, the feedback voltage, as compared with Uc', can be neglected in sections I and 11. A difference with respect to the s.m.a. without feedback may be discovered only in section Ill. This section must be divided into two further parts. In the first part the flux of the unsaturated core goes on changing in a negative direction (u,' > Uc'), whereas in the second part the flux change is already in the positive direction (u,' ~ Uc'). The direction of the change in the flux depends on the sign of the difference between the control voltage and the feedback voltage. The dependent, variable quantities in the two parts of section III are illustrated by the relations given in columns 4 and 5 of Table I. Figure 2 (f-h) were plotted with the aid of Table I. As compared with the ordinary s.m.a. 3 , considerable deviation appears in the change in time of the flux and of the control current. In steady-state conditions the change in
and in all succeeding ones, e.g. in the n
where D
=
RL
1 7TRi
+ RL
is the firing angle before connecting Uc', and r:t.n is the firing angle in the half-cycle n, calculated from the connection of Uc'. In general, we are interested only in the change with time of the output voltage (current) middle value. This may be illustrated by Figure 3 (a), or curve 1 of Figure 3 (b) (equation 2), neglecting the change of UL within the first half-cycle. (In the illustration, T denotes half of the cycle time.) Figure 3 (a) illustrates the change in UL where it would seem as if the total voltage change in the loading occurred suddenly at the beginning of the half-cycle. Curve 1 of Figure 3 (b) corresponds to a case in which it would seem as if the voltage change took place at a constant speed during the half-cycle. Of the two conceptions, the application of that leading to more simple relations in the case of feedback from the output signal is more advantageous. From our considerations it follows that the s.m.a.-should our assumptions be fulfilled-is an integrating element. The r:t.
Um
1+C05'"
.
(\
--1T- : sin r III
+---~-~--
(a)
: IW~G .r;,1TU~
I
iL
(c)
t
I I I
d(wt) :
I
I I
I
I ·r
le
(d) Yr_~L-
Figure 2
14
1115
_____
~~L-
______
uJt
~_
Table I Section I - - ---- -- ..
Section 11 - ~
Fulfilled inequalities
-
Vc' :> u
-- _.
Section III
0.;;; wt .;;; (31 here V' sin PI = --E..
I
Vc' .;;;
I I
I
11
(31 ';;; wt .;;;
I here
IX, .;;;
IXI
wt .;;;
VI
Ll
Current
iA
= =
N W
-firing angle
IXI
U
0
2Iro(Ri
11 -
It
Vc'd(wt) 1N1 (0
G
sin
f'
(i [11 1
+
2/"0( Ri
0
R1J
+
0
RIJ] d(wt)
Flux change LlB
Current
iB
2Iro
0
0
=
0
=
0
-11
+
Vc'
+
Ri
2Ieo(Ri
+
+
RL
Vc
RI
, RI
Ri RI
+
No
+ Ri + RL Ne 11 <
(3I1I
Pm .;;; wt .;;;
Vc'
TT
RL Vc Vrn
I
o
1
o
I
RI
+
,
I
+
r
RI)] d(wt)
Ri+RL Ji.
l eo
I I I I I
I
I r(iI ll(,
--
I uING• '"
Vc -
C
11
Rf
-
11
RINu/Ne R, + R i + RI,
RING/Ne) . d(ult) - I mNu + Ri + RL
f"
filII
(V' c
-11
RI
o
0 :
1
0
v'
,
I
C
-Vc'd(wt) wNu '"
I
=
PIlI = R f +
1
V'
R L)
If'
N fl 1 [-II+V/ (I) u + 2IeO(Ri 0
I Load current Control current ic'
wt .;;;
>
II
=
h,=
Nu -11 RE Nc
+
!
11
'B' core voltage liB
RI R i
here
V' c
Flux change
+
IX .;;;
TT
'A' core voltage =
RI
I
Um
11.4
Section 111 (b) ~ -
I
I Range of validity
Section III (a)
--~~
11
2/,0 -Ieo
Ri
I I
+
11
RL
RI + Ri I
1"0 -11'0
If feedback of the voltage reaches resistance RI in the expressions of section H, (R, + RJ must be written instead of Ri' leo = coercive current referred to the gate winding. Ri = sum of the resistances of the current source, the gate winding and the rectifier (in the forward direction).
1116
I
+ RE leo
RINu/Nc) d( wl ) + Ri + RI,
J. NAGY
flux of core A during a half-cycle is of the same magnitude, but in the opposite direction to that of core B. Substituting the corresponding values from Table I Uc' = (RfNclRLNc)UL = Ut'
the s.m.a. without feedback. The equations describing the static characteristics have been determined by using Figure 2 (g) and (h), Table 1, and equation 3. The final result is summarized in Table 2. The static characteristics plotted on the basis of the table are shown in Figure 4. The figures include also the static characteristics of the s.m.a. without feedback (dotted line). The deviation between the characteristics is obvious.
(3)
From the above physical considerations and from equation 3 it becomes apparent that the middle values of the control and feedback voltages are equal under steady-state conditions. The amplifier is of voltage-transforming character. From equation 3 the gain of the feedback is
Transfer Function-Equivalent Time Constant-Nyquist Diagram
(3a)
Af = R,NG /RI_Ne
For the dynamical examination of the regulating system, the transfer function of the elements appearing in the regulator is necessary. When the response is known [Figure 3 (a) and (b)], the transfer function of the s.m .a. without feedback may easily be determined and our results are given in the first line of Table 3. The last expression in the table refers to when time T with respect to the duration of the transient process is short and therefore- as in Figure 3 (b)-the response may be approached by line 2 of Figure 3 (b) . On the basis of these results, the transfer function of the s.m.a. with feedback-considering equation 3a- may be calculated in an elementary way (second line in Table 3). According to the elements with single time constant, the term
Static Characteristics The static characteristics of the s.m.a. with a negative feedback show an important deviation as compared with those of
R __ L U'
(a)
\
(4)
'R l '
characteristic for the operating speed of the s.m.a. with feedback, and being the so-called equivalent time constant, may be introduced. With all three Y*- for a step input- T,. equals, for unit final value, the area limited by the final value and the instantaneous values. While the response applying Y1 * may easily be determined, having a simple form [Figure 3 (c)] , the response based on Y 2 * is complicated. In the application of the three transfer functions, the following may be stated: with s.m.a. of 'fast' response Y1 * may be used, with that of 'slow' response Ya* is appropriate ; the use of Y 2 * is disadvantageous. In connection with s.m.a. of 'fast' and 'slow' response, the following must be considered. To characterize the speed of response, the time (T63) could also be introduced; this, calculated from the connection of the control voltage, is necessary for the middle value of the output voltage to reach 63 per cent of the final value. In Figure 5 the change in T63 relative to functions Uu(t) and that in Tr in function of the resultant gain AA, are plotted. Naturally, in a response of exponential form Tr = T63 . With small values of AA, this corresponds to our case, the transfer function being, with small loop gain, i.e. with slow response, Y3*' however, Tp and T63 at values AA, near to the unit differ considerably, and then the use of Y1 * is advantageous. Of the two time constants the choice of Tp is motivated, on the one hand, by the more simple way of determination, and on the other, by T63 being not interpretable (negative number) at values of AA, near to the unit. According to equation 4, increasing A, causes Te to decrease. By a suitable choice of A, it may be possible, by a step in Uc' in the first half-cycle, to let UL reach its constant value in the second half-cycle. The fastest speed of operation to be reached theoretically by the magnetic amplifier may be realized in this way. Then Te = T and (5)
I I
I
0
2",
"t
3,;
4,;
5'(
2 (b)
t-~
- - Uc ' 't
1:
2~
31:
4..:
5'[
Uu
1
4.0' AAf
(c)
o
or:
2'(
3'(
41:
The operation during a cycle is illustrated clearly in Figure 3 (c). The increase of the output voltage in the third and in all succeeding half-cycles is-because of equation 5-zero.
51:
Figure 3
16
1117
ANALYSIS OF A FAST-RESPONSE MAGNETIC AMPLIFIER
U'c
It tg~
--/-...-=--l·lt.,max
tg~
=Rc
_-====--,tL--4 U'C,L
=s
-le'
-Ic:o
l'cl
(a)
Ul,max
/ -U~
,/
/
/
/
- 1.
2 (c)
(b)
Ul
---:;;;--!-------------..._---/~-
leo
I
/ U'c U~,l
(d) Figure 4
Table 2
In the case of feedback, an approximation of Rc = 0 and I L,min = 0 was taken in the control range Ranlie ot'validity
Control current value
I
h ~1
Load current
Control VOltage Ve'
S(Jro
1/, [
< le'
+ 1/ )
--------
i
~i
N,. Rf N' S(/,o
,
+ 1/)
Vc'. 1 + R/(/e' - le', I)
--------
Ip i ~1
Input power
-1,,0';;; l e' ,;;; le', 1
(Vc',l -
1;' = 1
R/lc',l)lc'
+ Rc'Ic'2
1
Table 3 A = RL/CRt
+ R + R L) j
Accordinli to FiliI/re 3 Ca) (n
Transfer function of a s.m.a. without feedback Yn(p) =
I
---------------------------------1
=
I)
L[V",] RL e- PT L[V/] = Ri + RI. 1 - e-PT
A ccordinli to cl/I've I ot' F(fure 3 (b) (n = 2) L[ V L2 ] L[V/]
RI.
=
Ri +RL
e -pT
Pr
Transfer function of a s.m.a. with feedback
*
_
Yn(p)
e-PT
_
A --,---.,---;----::pT + AA,e jJT
Yn (p) - 1 + AY ( ) J "' P
17 2
1118
According to Cl/I've 2 of Filiure 3 Cb) (n = 3)
I. NAGY
Self-saturating Magnetic Amplifier as a d.c. Impedance Transformer The s.m.a. with feedback permits the development of an amplifier of high input impedance and, depending upon the manner of feedback, of high or low output impedance. A physical explanation for this can easily be given. The input impedance is increased because the control voltage must be balanced not only with the ohmic voltage drop but also with the feedback voltage; this is generally many times greater than the control voltage. The output impedance, in the case of feedback from RI or RL> is therefore high or low, respectively, because with constant Uc' the voltage reaching RI or RL is also constant. A constant current flows through the load in spite of RL being changed, consequently the amplifier is first a current generator, then a voltage generator and UL remains practically constant (Uc' = constant) even with a change between wide limits of the supply voltage. Table 4 presents simple relations for calculating the input and output impedances.
f :50 Hz
1)
T : 1 (log 0'37 63 2f log (l-AAf)
0'1
01
0'01
Figure 5
Increasing Figure of Merit
In Figure 6 are plotted the Nyquist diagrams of the s.m.a. with and without feedback.
All things considered, the quality of the s.m.a. connection is characterized by the figure of merit, J. J is a result of the power gain and of the equivalent time-constant ratio. By power gain the ratio Po,max/Pi,max is meant (the maximum output power is marked by Po,max, but the maximum input by
A A. f : 0·5
Pi,max), Real aXIs
0'6 0'5
.,
0'4
U 0'3 0'2 0'1 0
OD: c,1 (self-saturating magnetic amplifier without feedback)
0'4
Ri + RL Y3 (jttl't) RL
0'8
1'2
1'6
2'0
2'2
At Figure 7
Ri +RL RL
Ri + RL RL
'lf/15
'lf/15
Y,(jw't)
The value of J is higher for a s.m.a. with feedback than it is without feedback, and may be increased to a certain extent by increasing the gain of the feedback (Figure 7). A further, considerable increase in the figure of merit may be realized by certain circuit supplements. To understand and calculate the necessary complements, the following points must be considered. Figure 8 shows a block diagram of the s.m.a. with feedback; the voltage drop on the control circuit resistance is disregarded. In accordance with Table 3 and Figure 3 (c), it follows that, except for the first half-cycle following the change in Uc', the alteration in the output voltage middle value in any half-cycle as compared to that in the previous half-cycle is the product of the middle-value of the resultant control voltage supplied to the control circuit and of factor A. (This statement is valid also
Y2 (jW1:)
'If /15
Figure 6
18
1119
ANALYSIS OF A FAST-RESPONSE MAGNETIC AMPLIFIER
Table 4
Feedback
From R,
,
Input impedance
Zi =
oVe' -aI-e' ~
I
i
-------------------------Output impedance
~ Ri Re'lcO 1T
for the s.m.a. without feedback.) The middle value of the resultant control voltage in any half-cycle may be obtained by subtracting from Vc' not only Vt' but also Ic'Re' (Figure 8 (b». Knowing VI- and ID the value of le' can be determined from the equation of Table 2. (Between fe' and fL time delay is neglected.) The block diagram resulting is shown in Figure 8 (b),
U~
UL
+
At Vm
Figure 8 (c) indicates the way in which the intrinsic, negative feedback can be compensated. The s.m.a. must be provided with a bias - lcoNr; connected to a separate control winding as well as with a 'weak' positive feedback drawn from the output voltage, or current, and supplied to a third control winding. The control current will be--in the control range-zero when the gain of the positive feedback is Ai = 1/(SR1J. The circuit diagram of Figure 9 is based upon this information.
(a) R
( b)
Figure 9
Only the role of the constant voltage VI; is unexplained; its task is, where Vc = and VI;; = 0, whatever high value for current 12 may be chosen, le cannot be reduced to zero since the voltage Vt,min resulting from the minimum of IL will maintain a current (, of a certain value. To reduce to zero the value of I,. with the help of '2--also in the case of zero control voltage or near to it--the constant direct voltage VI; = Vt,mill is necessary. It must be emphasized that the circuit supplements described may easily be realized, so the dynamic behaviour--equivalent time constant--of the s.m.a. should not be subject to variations.
°
(c ) Figure 8
Appendix. Measurement Results
where, in addition to the extrinsic, negative feedback, an inherent, intrinsic, negative feedback also appears. Compensating this intrinsic, negative feedback the value of the output sign may be influenced, theoretically with zero control current, merely by changing the control voltage. This means that a S.m.a. of 'infinitely high' power amplification and input impedance may be realized.
To verify the correctness of our theoretical considerations and their suitability for numerical calculations, measurements have been carried out with a magnetic amplifier of 10 W, made of a magnetic material Permenorm 5,000 Z with toroidal core. During measurement the total output voltage was negatively fed back. 19
1120
r. NAGY
IL
fAJ u~
0·9
[vJ 2·25
08
15
0'3
075
0·2
05
0 ·1
0'25 I~
-2 ,8
o
-2
[mAJ
-2-8
o
-2
The dotted line refers to the case without feedb;!ck
Cb)
Ca)
u'c
[V] 275
2 '5 225
2 175 15
1' 25
0 ' 75
0·5 0·25
5
4
Cd) Figure 10
20
1121
3
2
o
ANALYSIS OF A FAST-RESPONSE MAGNETIC AMPLIFIER
measured h = hUe') are shown in Figure 11. The characteristics I, 2, and 3 measured in the order of enumeration by an ever-weakening positive feedback refer to case Uk = O. Therefore, in the environment of a zero control voltage le' is relatively great. For us characteristic 2 is of special interest: here, except for the zero control voltage and for its immediate
The characteristics
h = hUc'), Uc' = Uc'Uc'), Pi = PiUe')
= UL(Uc')
and UL
plotted with different values of RL are illustrated in Figure 10 (er Figure 4). It is obvious that calculation and measurement give qualitatively the same characteristics.
[1\] •- - - .
\/
:y:-
r' ! .
I·
\
\.
\ '.
\
i\ R.llkA ,
1 I
i i
5 \ 6-1 1'7
.\0'1 ,.
'., if 1/ , ..
f -2
j,.44kll.
'~
1/1243
/.
-N
"".{
k11kll. 02
..
\
A t curves 1.2.3 and 4 l2 ' 3'4 mA At curves 5.6 and 7 12' 2'3 mA
/
44k.fi--J
,,.~\/ .,
J
10'4
f
Supply voltage : 18 V Supply frequency : 50 Hz RL·, 20 ll.
/.
I
I
\;I
2·4
".
-1-6
• 1.2.3.
}.
4
;V!
/
°
-1,2 -08 -0·4
1c'
!
0·4
0·8
[mAl •
Figure 11 Table 5
:1[lJ I I RL
[.n] RI:
i L .mlia~
I;"I I
I
15
I-· ,~
;
-R l
!
reet.
!
I
Ig; ,5
i
5
~percentage error
MeaSUring Calculatio
!dlmension
I
:
[mm'] selenium
20
,
10 4 11 5
~mw] x
Iperc~n-
I,ma
50 x 50
0'564 .10 '
0'733 . 10'
I
23
1·7 45
I
43 x 43
0·564.10'
0679.10' i
17
1762
46
2 84
I
Z,
_ tage
252
I
2 305
308
1'095 .10'
I
13 7
i
149
1
tage
11 '49510' I
!
236
[.0.] Zo
5ec T.
sec-I
.
r, ~P
r5e c " ]J,'ca.. without feedbad )
Calculation Calculation C.lcu lati o
27
0·0 656
o 1204
19250
95
00812
o 0914
17100
6 77 0
I
I
I
1285
IPercen-
Measurin g k atcu,ation i error
Measur ing :calCulationi e rror
,
[fi]
. -- - . ~ -- -
4 3 x 43
0'442.10 3 0'463.10 3
1
3 26
1'706. 10 '
128
11'88'. 10'
I
- -- --
43 x 43 43 x 43
0'348.10 3
0368.10' 1 5 5
! 0,298 .10 3 , 0298.10 3
!
0
337 38
383 415
12
2 .10 '
1222510' 1 10 5
2-28510' ' 24110,1 • . I
84
00865 . . - - - -5 00941
00765 16350 - - - - _. --0·0699 14·900
5 930 ---- -
-
-
-
5·560 ~. -
-- ,
5015
vicinity, Ic' is approximately zero . (The value of l e' remained at ± 10 IlA.) When plotting curve 2, the value R = 22 kQ was applied (Figure 9) instead of R = SRLN1/N,. = 23'9 H2, which was necessary in principle. When introducing the voltage Uk = 2/,o RL = 0·134 V, the characteristic 4 was obtained. Here already the control current is approximately zero, even at low values of Uc' By changing the bias, the characteristics are shifted by themselves parallel to each other (see curves 5, 6, and 7).
For quantitative comparison some characteristic values of the connection have been calculated, on the one hand from the measurement results and on the other on the basis of the formulae derived in the study and summarized in Table 5. The difference between the results of calculations and of the measurements is partly attributable to the fact that the resistance of the selenium rectifier in the reverse direction is finite. This is verified by comparison of the measurement and calculation results obtained by rectifiers 50 x 50 mm2 and 43 x 43 mm 2-in the case of R L = 5Q-as well as by the fact that the calculated value of S is higher than that obtained by measurement (see Figure 4 (a». It is well known that with an increase in the current of the rectifier in the reverse direction the slope of the characteristic IL = I LU/) gets smaller. Measurements have been carried out on the same magnetic amplifier with the connection as in Figure 9. The characteristics
References 1
3
D. G. Fas t response with magnetic amplifiers. Trans. Amer. Inst. elect. Engrs 72, Pt. I (1953) 741 GEYGER, W. A. Magnetic amplifiers of the self-balancing potentiometer type. Trans. Amer. Inst. elect. En.l{rs 71, Pt. I (1952) 383 S TORM , H. F. Ma"fjnetic Amplifiers. 1955. New York; Wiley S CORGIE ,
Summary The object of our investigation has been the analysis of the selfsaturating magnetic amplifier. The basic connection of the amplifier IS derived from the d.c. input and output s.m.a. by its negative, rigid feedback from the output signal. In the interest of better comprehenSion, computability and design, a detailed analysis of the connection operation has been effected.
Firstly the s.m.a. with a negative feedback from the output signal is dealt with. It is shown that the change in time of the fluxes and of the control current deviates considerably from that of the s.m.a. without feedba ck. The equations of the characteristics (namely the load current, the control voltage and the input power as functions of the control 21
1122
1. NAGY
current) have ueen determined, having a most simple form. They reveal the basic difference between the static characteristics of the s.m.a. with and without feedback. The transient process of the amplifier is discussed and the possible expressions given for the transfer function, the response and the equivalent time constant. A relation is obtained for the operating conditions during a cycle. The Nyquist diagram is also plotted. The connection permits the realization of an amplifier of high input impedance and of low output impedance depending upon the manner of the feedback. In the course of examination, simple and relatively accurate relations have been found for the calculation of the input and output impedances. According to our considerations, the figure of merit is higher in
case of feedback than with the ordinary s.m.a" growing with the gain of the feedback. A most valuable conclusion may be drawn from the block diagram of the amplifier, namely, if completing the s.m.a. fed back by (a) a positive feedback taken from the output voltage, or current having a rate that can be read from the block diagram, (b) a constant d.c. bias, and (c) a small direct voltage compensating the VOltage drop on the feedback resistance caused by the minimum load current, it is possible theoretically to make the control current zero in the whole control range. To verify our quantitative results, measurements have been effected. They demonstrated the correctness of our results and their applicability for numerical calculations.
Sommaire On etudie les phenomenes transitoires dans I'amplificateur et !'on donne des expressions possibles de la transmittance, de la reponse et de la constante de temps equivalente. On obtient une relation reliant les parametres de fonctionnement au cours d ' un cycle. On trace egalement le diagramme de Nyquist. Selon les resultats obtenus, le facteur de merite est plus eleve avec contre-reaction que pour un amplificateur autosature ordinaire, ceci d'autant plus que le gain de la chalne inverse est plus grand .
L'objet de cette etude est un amplificateur magnetique autosature. Le montage de base de I'amplificateur derive de I'amplificateur magnetique autosature it entree et sortie continue, avec contre-reaction invariable du signal de sortie. Pour en faire mieux comprendre le fonctionnement, le calcul et la conception, on a effectue une etude detaillee du montage. On etudie d'abord I'amplificateur magnetique autosature avec contre-reaction du signal de sortie. On etablit que les variations des flux et du courant de commande different considerablement du cas OU il n'y a pas de contre-reaction.
Zusammenfassung fUr die Arbeitsweise wahrend einer Periode. Diese Angaben werden durch das Nyquist-Diagramm ergaznt. Die SchaItung macht den Aufbau eines Verstarkers moglich , der in Abhangigkeit von der Geg~nkopplung hohe Eingangsund niedrige Ausgangsimpedanz hat. Bei einem Magnetverstiirker des vorIiegenden Typs kann mit Hilfe von (a) einer Mitkopplung durch Ausgangsspannung oderstrom, deren Grad dem BIockschaItbild zu entnehmen ist, (b) einem konstanten Gleichstrom und (c) einer niedrigen Gleichspannung, die den durch den Mindeststrom i.iber dem Ruckopplungswiderstand verursachten Spannungsabfall kompensiert, der Steuerstrom im ganzen Regelbereich theoretisch zu Null gemacht werden. Die Ergebnissewurden durch Versuche bestatigt.
Die Grundschaltung eines MagnetversUirkers in Selbstsattigungsschaltung. Mit Gleichstromeingang und -ausgang kommt uber eine starre Gegenkopplung zustande. Beim Magnetverstarker mit Geg~nkopplung vom Ausgang her wird gezeigt, daf3 sich die zeitliche Anderung von Fluf3 und Steuerstrom von der des Verstarkers ohne Gegenkopplung erheblich unterscheidet. Die Gleichungen der wesentlichen Grof3en (d.h. Arbeitsstrom, Steuerspannung und SteuerIeistung als Funktionen des Steuerstromes) offenbaren den grundlegenden Unterschied im statischen Verhalten der Verstarker mit und ohne Gegenkopplung. .. Die Beziehungen fUr Ubertragungsfunktion , Frequenzgang und aquivalente Zeitkonstante werden angegeben . Es folgt eine Beziehung
DISCUSSIONS R. A.
LIPMAN,
(H. Storm and M. Rozenblat). The negative voltage feedback hardly changes the ampere-turns gain and, consequently, has no influence on the dynamic figure of merit of the amplifier. In the case of low time constants and 'high' frequencies in the control signal, it is necessary to take into consideration the discrete character of the transient response of the amplifier. In the case of a 100 per cent voltage feedback, such a magnetic amplifier represents a delay element of half a cycle of the supply frequency, i.e. it becomes fast acting. However, it is known that the dynamic figures of merit of ordinary and of fast acting magnetic amplifiers are nearly equal and they are practically equivalent, from the point of view of the dynamic as well as static characteristics. The statement of the author that the dynamic figure of merit of the amplifier is improved by negative feedback is not understood because, in the analysis of the circuit operation it is assumed that the side sections of the hysteresis loop have infinite slope and that the width of the hysteresis loop does not depend on the rate of change of the inductance, i.e. the influence of eddy currents and magnetic viscosity is not taken into consideration. However, as is shown in the book of M. Rozenblat, under such conditions an ordinary magnetic amplifier with self-saturation (without additional feed backs) will have an infinite ampere-turns gain and, consequently, it will have an infinite dynamic figure of merit. If there is an
U.S.S.R.
There are objections to one of the basic assumptions of the paper, namely, that the negative electrical voltage feedback improves the dynamic figure of merit of the amplifier. Usually, the dynamic figure of merit is understood to be the ratio of the power gain to the time constant. For magnetic amplifiers with a 'large' (compared to the half-cycle of the supply frequency) time constant, and also for magnetic amplifiers with any time constant, provided that 'Iow' control signal frequencies (Iow compared with the supply frequency) are considered, it is possible to disregard the discrete character of the transient response of the amplifier. Therefore, the magnetic amplifier can be considered as an inertia element of the first order with a time constant equalling
where K is the voltage gain, N e and N g are respectively the number of turns on the control and excitation windings. Under these conditions the negative feedback reduces to an equal extent the voltage gain and the time constant, thus the dynamic figure of merit remains unchanged. The correctness of this assumption also follows from the fact that the dynamic figure of merit is proportional to the ampere-turns gain. 22
1123
ANALYSIS OF A FAST-RESPONSE MAGNETIC AMPLIFIER
linear but slower. Since the time constant is proportional to the decrease At, the figure of merit will, indeed, increase with increasing At> see Figure 7 (in writing the above equation it was assumed that in using the static characteristics h = hUe') the control current lco/2 is located in the control range and also that all the assumptions made under point (d)* of the 'basic assumptions' of the paper are fulfilled). R. A. Lipman has in mind the third definition of the power amplification described in the book by Storm, namely, the product of the current gain and the voltage gain for small increments.
appropriate shift, such an amplifier will operate without taking any power from the control signal source. In reality, the high figures of merit for the experimental amplifier, described in the paper, were achieved by using external positive feedback. With this feedback it is possible theoretically to obtain an infinitely large ampere-turns gain and, consequently, also an infinite dynamic figure of merit. In practice, however, the degree of effectiveness of the positive feedback is limited by the instability of the amplifier characteristics during changes of ambient temperature, supply voltage and frequency, etc. I. NAGY in reply. I believe that basically the comments of R. A. Lipman result from a difference in the definition of the term 'power amplification'. In the paper this term is understood to be the ratio of the maximum output power (Po llIax) to the maximum input power (Pi max). If all the output voltages are transmitted by means of the feedback circuits, Po max will remain practically constant, whilst At (feedback gain) will increase. The value P,: lI1ax (the last row in Table 2) equals:
= 8"
I am sorry not to be able to agree with R. A. Lipman on the point that negative voltage feedback results in hardly any amplification with respect to the ampere-turns and, consequently, it does not influence the figure of merit of the amplifier. A detailed study of the Figures 4(a) and lO(a) of the paper shows clearly that the differential amplification with respect to the ampere-turns does change.
At
7f
Pimax
K =dh.dUL P dIe dUe
UnJco At
+1 +
Rj
* (Translator's note: The Russian text specifies d but, due to the difference between the sequence of the Cyrillic and the Latin letters, d in this case may well be e).
RL
This means that with increasing At the increase will not be
23
1124