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Mullard 20W tube amplifier
Feedback loop gain and compensation networks optimization
Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
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1. Objective
The objective of this paper is to show how network analysis programs can be used to optimize feedback loop
gain and associated compensation networks necessary achieving good damping and stability margins without
peaking, when feedback is applied on an amplifier. The amplifier used to illustrate this purpose is the Mullard
20W tube amplifier.
2. Circuit diagram
The circuit diagram used to setup the amplifier model is given on the figure 1. For more details on the Mullard
20W tube amplifier, refer to (1).
3 Amplifier model
The amplifier model has been setup using the ESACAP network analysis program and according to (2). Despite
other network analysis programs can be used, a copy of the model, as developed in ESACAP, is given in annex:
1, for help and information.
The amplifier is described in section $$DES, where sub-section $FUN: Limit(x,min,max); defines the limit
function, sub-section $FUN: pwrs(x,y);defines the power function, sub-section $CON: gives the input signal
parameters, the tube parameters, the core geometry, the OPT magnetic parameters and topology and sub-section
$NET: describes the complete amplifier network, according to the schematic diagram.
4 NYQUIST diagram
The feedback loop transfer function will be graphically represented in the NYQUIST plane shown on figure 2,
on which have been added:
The critical point A (-1, 0)
The circle R=1, centred on point A, which defines the area of positive feedback
The circle Q=2.3 dB, having limit points A and O, which defines the minimum stability margins
The vertical line x=-0.5 which defines the peaking conditions
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NYQUIST Diagram
3
260 °
240 °
280 °
300 °
220 °
2
320 °
X=-0,5
2,3 dB
.
200 °
340 °
1
Imaginary part
R=1
0
O
A
180 °
0°
-1
20 °
160 °
40 °
-2
60 °
140 °
120 °
100 °
80 °
-3
-5
-4
-3
-2
-1
0
1
2
3
4
5
Real part
Figure 1: NYQUIST diagram
5 Feedback loop transfer function optimization
To be acceptable, the feedback loop transfer function must be warped in such a way that it stays:
Outside the circle Q=2.3 dB, to achieve good stability and damping
On the right hand side of the vertical line x=-0.5, to avoid peaking at both ends of the frequency
bandwidth.
The above conditions define the strategy for the optimization of the feedback loop transfer function.
It is generally achieved using an optimum feedback loop gain and additional compensation networks.
5.1 Gain optimization
The feedback voltage is defined using a voltage divider across the output load
transfer function is, assuming that:
Rt  Rk
H ( p) 
RL , as shown on the figure 3. Its
RL
Rk
Rt  Rk
Where:
Rk
Resistance set to
Rt
Resistance to be determined to define the feedback loop gain
.1k in the Mullard 20W amplifier
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Rt
Vi
Vo
Rk
Figure 3: Voltage divider
A value of
Rt  9k has been chosen. With this value, the Routh’s criterion for the NYQUIST diagram of the
feedback loop, shown on the figure 4, is not fulfilled. As a result, the amplifier is unstable. However, this choice
is maintained because it provides an important feedback loop gain and we will show that the present amplifier
instability can be overcome properly using dedicated compensation networks.
NYQUIST Diagram
Feedback loop
3
260 °
240 °
MULLARD 20W
220 °
2
280 °
300 °
320 °
X=-0,5
2,3 dB
.
200 °
340 °
1
Imaginary part
R=1
0
A
180 °
O
0°
-1
20 °
160 °
40 °
-2
60 °
140 °
120 °
100 °
80 °
-3
-5
-4
-3
-2
-1
0
1
2
3
4
5
Real part
Figure 4: Feedback loop Nyquist diagram with a voltage divider
5.2 Differential compensation
The first compensation network is obtained using a by-pass capacitor across the resistor
Rt of the voltage
divider defined in section 5.1, as shown on the figure 5.
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Ct
Rt
Vi
Rk
Vo
Figure 5 Phase lead compensation network
The transfer function of the above compensation network is:
H ( p) 
Rk
1  Rt Ct p   1 1  a d p
Rt  Rk 
 a 1 d p
Rt Rk
1  R  R Ct p 
t
k


With:
a  1
Rt
Rt Rk
Ct
and  d 
Rk
Rt  Rk
We recognize (3) the transfer function of a differential or phase lead compensation giving the maximum phase
lead
m  Arc sin
a 1
1
for the angular frequency m 
.
a 1
d a
Applying these results to the Mullard 20W tube amplifier, we get, using the value defined for
Rt in section 5.1,
m  78deg
a  91
Ct must be adjusted to have m  r , where r is the angular frequency resonance of the feedback loop after
the phase lead compensation effect.
A value of Ct  250 p fulfils the above requirements as it is shown on the NYQUIST diagram of the
compensated feedback loop given on the figure 5.
The main effect of the phase lead compensation on the feedback loop transfer function is that we get stability for
the amplifier. However, if the amplifier is now stable, the damping is not sufficient because a part of the
feedback loop transfer function is entering into the circle Q=2.3 dB and if no peaking appears at the low end of
the frequency bandwidth, because the transfer function stays on the right hand side of the vertical line x=-0.5, it
is important and unacceptable at the high end of the frequency bandwidth, because a part of the transfer function
is on the left hand side of the vertical line x=-0.5 (4) and (5).
This situation is better shown on the diagrams on the figures 7 and 8 giving the feedback loop gain and phase
shift versus the frequency and on the figures 9 and 10 giving the closed loop gain and phase shift versus the
frequency.
As a result, to achieve a better optimization of the feedback loop, it is necessary to improve the present situation
by using an additional compensation network.
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NYQUIST Diagram
Feedback loop
3
260 °
240 °
MULLARD 20W
220 °
280 °
300 °
320 °
2
2,3 dB
.
200 °
340 °
1
Imaginary part
R=1
Phase lead compensation
0
A
180 °
O
Rt=9k
Ct=250p
-1
0°
20 °
160 °
40 °
X=-0,5
-2
60 °
140 °
120 °
100 °
80 °
-3
-5
-4
-3
-2
-1
0
1
2
3
4
5
Real part
Figure 6: Feedback loop Nyquist diagram with a phase lead compensation
Frequency response - Gain
Feedback loop
40,00
30,00
MULLARD 20W
20,00
10,00
Gain dB
.
0,00
-10,00
Phase lead compensation
-20,00
Rt=9k
Ct=250p
-30,00
-40,00
-50,00
-60,00
-70,00
1,00E-01
1,00E+00
1,00E+01
1,00E+02
Frequency
1,00E+03
1,00E+04
1,00E+05
Hz
Figure 7: Feedback loop gain with a phase lead compensation
Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
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Frequency response - Phase shift
Feedback loop
180,00
160,00
140,00
MULLARD 20W
120,00
.
100,00
80,00
Phase lead compensation
60,00
Rt=9k
Ct=250p
Phase shift deg
40,00
20,00
0,00
-20,00
-40,00
-60,00
-80,00
-100,00
-120,00
-140,00
-160,00
-180,00
1,00E-01
1,00E+00
1,00E+01
1,00E+02
1,00E+03
1,00E+04
1,00E+05
1,00E+06
Frequency Hz
Figure 8: Feedback loop phase shift with a phase lead compensation
Frequency response - Gain
Closed loop
5,00E+01
MULLARD 20W
4,00E+01
3,00E+01
Phase lead compensation
Rt=9k
Ct=250p
Gain dB
.
2,00E+01
1,00E+01
0,00E+00
-1,00E+01
-2,00E+01
-3,00E+01
1,00E-01
1,00E+00
1,00E+01
1,00E+02
1,00E+03
1,00E+04
1,00E+05
Frequency Hz
Figure 9: Closed loop gain with a phase lead compensation
Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
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Frequency response - Phase shift Closed loop
1,80E+02
1,60E+02
MULLARD 20W
1,40E+02
1,20E+02
1,00E+02
8,00E+01
Phase lead compensation
Rt=9k
Ct=250p
Phase shift Deg
6,00E+01
4,00E+01
2,00E+01
0,00E+00
-2,00E+01
-4,00E+01
-6,00E+01
-8,00E+01
-1,00E+02
-1,20E+02
-1,40E+02
-1,60E+02
-1,80E+02
1,00E-01
1,00E+00
1,00E+01
1,00E+02
1,00E+03
1,00E+04
1,00E+05
1,00E+06
Frequency Hz
Figure 10: Closed loop phase shift with a phase lead compensation
5.3 Integral compensation
The second compensation network to be used is obtained using a high frequency step circuit across the loading
plate resistance of the input stage of the amplifier, as shown on the figure 11.

Vi
Cu
Rp
Vo
Ru
Figure 11: Integral compensation network
Where:

Rp
Internal resistance of the input tube
Loading plate resistance
The transfer function of the above compensation network is:
H ( p) 
With:
Req 
 Rp
  Rp
1  Ru Cu p
1i p

1   R eq  Ru  Cu p 1  b i p
 i  Ru Cu
b  1
Req
Ru
Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
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We recognize (3) the transfer function of a gain reduction or integral compensation. To match this compensation
network, it is necessary to have
1
i
R
For the Mullard 20W tube amplifier, we have:
  2.5M
R p  0.1k
Cu and Ru must be adjusted to have  i
Req  96k
1
, where
r
r is the angular frequency resonance of the feedback
loop after the integral compensation effect.
Values of Cu  400 p and Ru  20k fulfil the above requirements, as it is shown on the NYQUIST diagram
of the compensated feedback loop given on the figure 12.
In conjunction with the differential compensation, the additional effect of the integral compensation on the
feedback loop transfer function is that it allows achieving the required damping and stability margins with
practically no peaking (4) and (5).
The transfer function has been warped enough to avoid entering into the circle Q=2.3 dB and the left hand side
of the vertical line x=-0.5.
This situation is again better shown on the figures 13 and 14 giving the feedback loop gain and phase shift versus
the frequency and on the figures 15 and 16 giving the closed loop gain and phase shift versus the frequency.
NYQUIST Diagram
Feedback loop
3,00E+00
MULLARD220
20W
°
260 °
240 °
280 °
300 °
2,00E+00
320 °
200 °
Q=2,3 dB
1,00E+00
Imaginary part
.
340 °
R=1
Compensation networks
0,00E+00
A
Phase lead compensation
R13=9k C9=250p0 °
Integral control
R3=20k C1=400p
O
180 °
-1,00E+00
20 °
X=-0,5
160 °
40 °
-2,00E+00
60 °
140 °
-3,00E+00
-5,00E+00
-4,00E+00
-3,00E+00
120 °
-2,00E+00
-1,00E+00
100 °
0,00E+00
80 °
1,00E+00
2,00E+00
3,00E+00
4,00E+00
Real part
Figure 12: Nyquist diagram with a phase lead compensation and an integral control
Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
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Frequency response - Gain
Feedback loop
4,00E+01
3,00E+01
MULLARD 20W
2,00E+01
1,00E+01
.
0,00E+00
Compensation networks
Gain dB
-1,00E+01
Phase lead compensation
R13=9k C9=250p
Integral control
R3=20k C1=400p
-2,00E+01
-3,00E+01
-4,00E+01
-5,00E+01
-6,00E+01
-7,00E+01
1,00E-01
1,00E+00
1,00E+01
1,00E+02
1,00E+03
1,00E+04
1,00E+05
1,00E+06
Frquency Hz
Figure 13: Feedback loop gain with a phase lead compensation and an integral control
Frequency response - Phase shift
Feedback loop
1,80E+02
1,60E+02
1,40E+02
MULLARD 20W
Compensation networks
1,20E+02
Phase lead compensation
R13=9k C9=250p
Integral control
R3=20k C1=400p
1,00E+02
.
8,00E+01
6,00E+01
Phase shift deg
4,00E+01
2,00E+01
0,00E+00
-2,00E+01
-4,00E+01
-6,00E+01
-8,00E+01
-1,00E+02
-1,20E+02
-1,40E+02
-1,60E+02
-1,80E+02
1,00E-01
1,00E+00
1,00E+01
1,00E+02
1,00E+03
1,00E+04
1,00E+05
Frequency Hz
Figure 14: Feedback loop phase shift with a phase lead compensation and an integral control
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Frequency response - Gain
Closed loop
5,00E+01
MULLARD 20W
4,00E+01
Gain dB
.
3,00E+01
2,00E+01
Compensation networks
Phase lead compensation
R13=9k C9=250p
Integral control
R3=20k C1=400p
1,00E+01
0,00E+00
-1,00E+01
-2,00E+01
-3,00E+01
1,00E-01
1,00E+00
1,00E+01
1,00E+02
1,00E+03
1,00E+04
1,00E+05
1,00E+06
Frequency Hz
Figure 15: Closed loop gain with phase a lead compensation and an integral control
Frequency response - Phase shift
Closed loop
180,00
160,00
MULLARD 20W
140,00
120,00
100,00
Compensation networks
Phase lead compensation
R13=9k C9=250p
Integral control
R3=20k C1=400p
Phase shift Deg
.
80,00
60,00
40,00
20,00
0,00
-20,00
-40,00
-60,00
-80,00
-100,00
-120,00
-140,00
-160,00
-180,00
1,00E-01
1,00E+00
1,00E+01
1,00E+02
1,00E+03
1,00E+04
1,00E+05
Frequency Hz
Figure 16: Closed loop phase shift with a phase lead compensation and an integral control
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6 Comments
With the above defined feedback loop gain and associated compensation networks, the objective which was to
warp the feedback loop transfer function in order that it stays outside the circle Q=2.3 dB and on the right hand
side of the vertical line x=-0,5 has been achieved. As a result, on the Mullard 20W tube amplifier, a feedback
loop gain of at least 20 db from 30 Hz to 6 kHz and 15 dB from 15 Hz to 15kHz is available. These results are
interesting if we consider that the amplifier shows no peaking at both ends of the frequency bandwidth, with
excellent damping and stability margins.
7 Conclusion
Applying a certain amount of feedback on an amplifier is a difficult exercise according to the predefined
optimization requirements. From that point of view, it is clear that using a network analysis programs is helpful,
as it has been shown in this paper, because it allows defining simply and surely, feedback loop gain and
associated compensation networks.
References
(1) Mullard Tube circuits for Audio amplifiers - Second reprint edition - Audio Amateur Publications, Inc.
(2) Pierre Touzelet “Accurate non linear models of valve amplifiers including output transformers” - AES
preprint 6830 - 120th AES convention Paris, France
(3) J.-Ch.Gille, P.Decaulne, M. Pelegrin “Théorie et calcul des asservissements linéaires” - DUNOD Paris
1967
(4) Norman H. Crowhurst “Understanding Hi-Fi circuits” - First reprint edition - Audio Amateur Press.
(5) R. Brault “Basse Fréquence et Haute Fidélité” - Second reprint edition - Librairie de la Radio. Paris
Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
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Annex: 1
ESACAP model of the Mullard 20 W tube amplifier
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# MULLARD 20W AMPLIFIER
$$DES
$FUN: limit(x,min,max);
limit=MIN(max,MAX(min,x));
END;
$FUN: pwrs(x,y);
IF(x.GT.0) THEN
pwrs=x**y;
ELSE
IF(x.LT.0) THEN
pwrs=-((-x)**y);
ELSE
pwrs=0;
ENDIF;
ENDIF;
END;
$CON:
# Sine input signal
A=.160;
F=1k;
per=1/F;
# Maximum amplitude
# Frequency
# EF86 Tube parameters
# Norman L.Koren's model
p11=201;
# Tube parameter
p21=48.4;
# Tube parameter
p31=.929;
# Tube parameter
p41=329;
# Tube parameter
p51=9.31;
# Tube paramater
p61=p41*.8693;
# Tube parameter
# ECC83 Tube parameters ( first triode)
# Norman L.Koren's model
t11=680;
# Tube parameter
t21=106;
# Tube parameter
t31=7910;
# Tube parameter
t41=1.06;
# Tube parameter
t51=437;
# Tube paramater
# ECC83 Tube parameters (second triode)
# Norman L.Koren's model
t12=680;
# Tube parameter
t22=106;
# Tube parameter
t32=7910;
# Tube parameter
t42=1.06;
# Tube parameter
t52=437;
# Tube paramater
# EL34 Power tube parameters (first pentode)
# Combined Norman L. Koren's and Menno van der Veen's models
p12=40.8;
# Tube parameter
p22=9.99;
# Tube parameter
p32=1.99;
# Tube parameter
p42=1970;
# Tube parameter
p52=3.70;
# Tube parameter
p62=p42*1.077;
# Tube parameter
# EL34 Power tube parameters (Second pentode)
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# Combined Norman L. Koren's and Menno van der Veen's models
p13=40.8;
# Tube parameter
p23=9.99;
# Tube parameter
p33=1.99;
# Tube parameter
p43=1970;
# Tube parameter
p53=3.70;
# Tube parameter
p63=p43*1.077;
# Tube parameter
# Core geometry
Ac=13.44E-4;
Lc=0.208;
Lc1=Lc/2;
Lc2=Lc/2;
Lc3=0.25;
Lg=3E-5;
Lambda=1.1;
# Cross section area
# Average magnetic path length
# Average magnetic path length for primary
# Average magnetic path length for secondary
# Air gap length for magnetic flux leak
# Air gap length
# Fringing coefficient
# OPT Topology
X=0.43;
# Screen feed back ratio
# OPT parameters
Np=2419;
Rp=58;
Ns=84;
Rs=0.26;
Cip=250p;
# Primary turns
# Primary resistance
# Secondary turns
# Secondary resistance
# Anode to anode stray capacitance
# Primary winding n°1
Npa1=Np/2;
Rpa1=Rp/2;
# Anode winding turns
# Anode winding resistance
# Screen winding n°1
Nps1=Npa1*ABS(X);
Rps1=Rpa1*ABS(X);
# Winding turns
# Winding resistance
# Primary winding n°2
Npa2=Np/2;
Rpa2=Rp/2;
# Winding turns
# Winding resistance
# Screen winding n°2
Nps2=Npa2*ABS(X);
Rps2=Rpa2*ABS(X);
# Winding turns
# Winding resistance
# Output Load
RL=8;
END;
# Load resistance
$NET:
# Voltage amplifier section
Ein(10,0)=A*SIN(2*PI*F*TIME); # Sine input Voltage
R1(10,0)=1M;
R2(10,20)=4.7k;
R3(70,60)=20k;
R4(30,40)=2.2k;
R5(40,0)=100;
R6(80,60)=100k;
R7(80,50)=390k;
R8(110,0)=82k;
R9(90,80)=270k;
R10(60,120)=1M;
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R17(100,90)=15k;
C1(70,80)=400p;
C2(50,30)=.05u;
C3(30,40)=50u;
C4(80,0)=8u;
C5(90,0)=8u;
C6(120,0)=.25u;
C12(100,0)=8u;
E1(100,0)=440;
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# DC supply
# EF86
%Up1=V(50,30)/p11*LOG(1+EXP(p11*(1/p21+V(20,30)/(V(50,30)+1e-12))));
%Vp1=limit(%Up1,0,1E6);
%Alpha1=.8693*pwrs(2/PI*ATAN(V(60,30)/(V(50,30)+1E-12)),1/p51);
Jap1(60,30)=pwrs(%Vp1,p31)/p61*%Alpha1;
Jsp1(50,30)=pwrs(%Vp1,p31)/p61*(1-%Alpha1);
Cakp1(60,30)=5.3p;
Cgkp1(20,30)=3.8p;
Cgap1(20,60)=.05p;
# Phase splitter section
R11(90,11)=180k;
R12(90,12)=180k;
R14(21,0)=470k;
R15(22,0)=470k;
R18(21,31)=2.2k;
R19(22,32)=2.2k;
R20(41,0)=470;
R21(42,0)=470;
C10(11,21)=.5u;
C11(12,22)=.5u;
C13(41,0)=50u;
C14(42,0)=50u;
# First 1/2 ECC83
%Xt1=V(11,110)/t11;
%Ut1=%Xt1*LOG(1+EXP(t11*(1/t21+V(60,110)/pwrs(pwrs(V(11,110),2)+t31,0.5))));
%Vt1=limit(%Ut1,0,1E6);
Jat1(11,110)=pwrs(%Vt1,t41)/t51;
Cagt1(11,60)=1.7p;
Cgkt1(60,110)=1.6p;
Cakt1(11,110)=0.46p;
# Second 1/2 ECC83
%Xt2=V(12,110)/t12;
%Ut2=%Xt2*LOG(1+EXP(t12*(1/t22+V(120,110)/pwrs(pwrs(V(12,110),2)+t32,0.5))));
%Vt2=limit(%Ut2,0,1E6);
Jat2(12,110)=pwrs(%Vt2,t42)/t52;
Cagt2(12,120)=1.7p;
Cgkt2(120,110)=1.6p;
Cakt2(12,110)=0.46p;
# Power section
R24(71,51)=1k;
R25(72,52)=1k;
# First power tube (EL34)
%Up2=V(51,41)/p12*LOG(1+EXP(p12*(1/p22+V(31,41)/(V(51,41)+1e-12))));
%Vp2=limit(%Up2,0,1E6);
%Alpha2=pwrs(2/PI*ATAN(V(51,41)/(V(61,41)+1E-12)),1/p52);
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Jap2(61,41)=pwrs(%Vp2,p32)/p62*%Alpha2;
Jsp2(51,41)=pwrs(%Vp2,p32)/p62*(1-%Alpha2);
Cakp2(61,41)=8.4p;
Cgkp2(31,41)=15.4p;
Cgap2(31,61)=1.1p;
Cskp2(51,41)=0.5p;
# Second power tube (EL34)
%Up3=V(52,42)/p13*LOG(1+EXP(p13*(1/p23+V(32,42)/(V(52,42)+1e-12))));
%Vp3=limit(%Up3,0,1E6);
%Alpha3=pwrs(2/PI*ATAN(V(52,42)/(V(62,42)+1E-12)),1/p53);
Jap3(62,42)=pwrs(%Vp3,p33)/p63*%Alpha3;
Jsp3(52,42)=pwrs(%Vp3,p33)/p63*(1-%Alpha3);
Cakp3(62,42)=8.4p;
Cgkp3(32,42)=15.4p;
Cgap3(32,62)=1.1p;
Cskp3(52,42)=0.5p;
# OPT magnetic core
%EP=-(Npa1*I(R28)+Nps1*I(R26))
+(Npa2*I(R29)+Nps2*I(R27));
%ES=Ns*I(R30);
%Hc1=%Hc1+%Hc1*Lc1+%Hc3*Lc3-%EP;
%Hc2=%Hc2+%Hc2*Lc2-%Hc3*Lc3+Lg*%Bc2/(µ0*Lambda)-%ES;
%Hc3=1/µ0*(%Bc1-%Bc2);
%Bc1=10000*µ0*%Hc1;
%Bc2=10000*µ0*%Hc2;
%PHIc1=%Bc1*Ac;
%PHIc2=%Bc2*Ac;
%D1=%PHIc1';
%D2=%PHIc2';
# Primary circuit n° 01
Epa1(100,81)=-Npa1*%D1;
R28(81,61)=Rpa1;
Copt(61,62)=Cip;
Eps1(100,91)=-Nps1*%D1;
R26(91,71)=Rps1;
# Primary circuit n° 02
Epa2(100,82)=Npa2*%D1;
R29(82,62)=Rpa2;
Eps2(100,92)=Nps2*%D1;
R27(92,72)=Rps2;
# Secondary circuit
Es(140,0)=-Ns*%D2;
R30(140,130)=Rs;
R31(130,0)=RL;
#R32(130,150)=9k;
#R33(150,0)=100;
#C15(130,150)=250p;
# NFB
Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
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R13(130,40)=9k;
C9(130,40)=250p;
END;
$$STOP
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