# NYQUIST diagram - DOC

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```					Pierre Touzelet                                                                        Page                1
Issue               3
Date         24/08/08

Mullard 20W tube amplifier

Feedback loop gain and compensation networks optimization

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                        Page                2
Issue               3
Date         24/08/08

Document change record
01             25/02/08    All                  First release of the document
02             27/02/08    All                  Update of the document
03             24/08/08    All                  Update of the document

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                           Page               3
Issue              3
Date        24/08/08

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,
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

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                                                                                    Page               4
Issue              3
Date        24/08/08

NYQUIST Diagram

3
240 °             260 °            280 °
220 °                                                              300 °

2                                                                                                               320 °
X=-0,5
2,3 dB
200 °
.

1                                                                                                                       340 °
R=1
Imaginary part

A                     O                                          0°
0         180 °

-1                                                                                                                      20 °

160 °

-2                                                                                                             40 °

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 RL , as shown on the figure 3. Its
transfer function is, assuming that: Rt  Rk  RL

Rk
H ( p) 
Rt  Rk
Where:
Rk                           Resistance set to         .1k in the Mullard 20W amplifier
Rt                           Resistance to be determined to define the feedback loop gain

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                                                                                   Page               5
Issue              3
Date        24/08/08

Rt

Vi                         Rk               Vo

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
240 °             260 °             280 °
MULLARD 20W °
220                                                                         300 °

2                                                                                                              320 °
2,3 dB                             X=-0,5
200 °
.

1                                                                                                                      340 °
R=1
Imaginary part

0                                                        A                      O                                           0°
180 °

-1                                                                                                                     20 °

160 °

-2                                                                                                            40 °

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.

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                               Page               6
Issue              3
Date        24/08/08

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       Rt Rk        a 1 d p
1  R  R Ct p 
      t     k     
With:
Rt            Rt Rk
a  1      and  d          Ct
Rk           Rt  Rk
We recognize (3) the transfer function of a differential or phase lead compensation giving the maximum phase
a 1                                  1
lead   m  Arc sin        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,
a  91                      m  78deg
Ct must be adjusted to have m  r , where  r is the angular frequency resonance of the feedback loop after
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.

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                                                                                            Page                 7
Issue                3
Date          24/08/08

NYQUIST Diagram
Feedback loop

3
240 °                260 °            280 °
MULLARD 20W °
220                                                                                300 °

2                                                                                                                     320 °
2,3 dB
200 °

1                                                                                                                             340 °
.

R=1

Imaginary part

0                                                               A                      O                               Rt=9k         0°
180 °
Ct=250p

-1                                                                                                                             20 °

160 °

-2                                                                        X=-0,5                                      40 °

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

0,00
.

Gain dB

Rt=9k
-20,00                                                                                  Ct=250p

-30,00

-40,00

-50,00

-60,00

-70,00
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 7: Feedback loop gain with a phase lead compensation

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                                                             Page           8
Issue          3
Date    24/08/08

Frequency response - Phase shift
Feedback loop

180,00
160,00
140,00
MULLARD 20W
120,00
100,00
80,00
.

60,00                                                                   Rt=9k
Ct=250p
40,00
Phase shift deg

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

2,00E+01                                                                      Rt=9k
Ct=250p
.
Gain dB

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 9: Closed loop gain with a phase lead compensation

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                                                               Page             9
Issue            3
Date      24/08/08

Frequency response - Phase shift Closed loop

1,80E+02
1,60E+02
1,40E+02          MULLARD 20W
1,20E+02
1,00E+02
8,00E+01
6,00E+01                                                                       Rt=9k
Ct=250p
4,00E+01
Phase shift Deg

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.

           Cu

Vi                Rp                     Vo
Ru

Figure 11: Integral compensation network

Where:
                        Internal resistance of the input tube
The transfer function of the above compensation network is:
1  Ru Cu p        1i p
H ( p)                          
1   R eq  Ru  Cu p 1  b i p
With:
 Rp                                                               Req
Req                             i  Ru Cu                      b  1
  Rp                                                               Ru

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                            Page              10
Issue              3
Date        24/08/08
We recognize (3) the transfer function of a gain reduction or integral compensation. To match this compensation
1
network, it is necessary to have                       i 
R
For the Mullard 20W tube amplifier, we have:
  2.5M                                R p  0.1k                             Req  96k
1
Cu and Ru must be adjusted to have  i                                         , where      r is the angular frequency resonance of the feedback
r
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
240 °               260 °     280 °
20W
MULLARD220 °                                                                300 °

2,00E+00                                                                                                   320 °
Q=2,3 dB
200 °

1,00E+00                                                                                                            340 °
.

R=1

Compensation networks
Imaginary part

0,00E+00    180 °                                                                                      R13=9k C9=250p ° 0
Integral control
R3=20k C1=400p

-1,00E+00                                                                                                          20 °
X=-0,5

160 °
40 °
-2,00E+00
60 °
80 °
140 °              120 °             100 °

-3,00E+00
-5,00E+00      -4,00E+00   -3,00E+00   -2,00E+00    -1,00E+00     0,00E+00       1,00E+00       2,00E+00     3,00E+00   4,00E+00   5,00E+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
Pierre Touzelet                                                                                                             Page            11
Issue            3
Date      24/08/08

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

R13=9k C9=250p
-2,00E+01                                                      Integral control
R3=20k C1=400p

-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
MULLARD 20W
1,40E+02
Compensation networks
1,00E+02                                                                     R13=9k C9=250p
Integral control
8,00E+01
R3=20k C1=400p
6,00E+01
.

4,00E+01
Phase shift deg

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 14: Feedback loop phase shift with a phase lead compensation and an integral control

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                                                       Page            12
Issue            3
Date      24/08/08

Frequency response - Gain
Closed loop

5,00E+01

MULLARD 20W
4,00E+01

3,00E+01

2,00E+01                                                               Compensation networks
.

R13=9k C9=250p
Gain dB

1,00E+01                                                               Integral control
R3=20k C1=400p

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
140,00
MULLARD 20W
120,00
100,00                                                              Compensation networks
R13=9k C9=250p
.

60,00
Integral control
40,00                                                              R3=20k C1=400p
Phase shift Deg

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 16: Closed loop phase shift with a phase lead compensation and an integral control

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                             Page               13
Issue               3
Date         24/08/08
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
Pierre Touzelet                                                                        Page               14
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Date         24/08/08

Annex: 1
ESACAP model of the Mullard 20 W tube amplifier

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                        Page               15
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Date         24/08/08

# 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;                # Maximum amplitude
F=1k;                  # Frequency
per=1/F;

# 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)

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                        Page               16
Issue               3
Date         24/08/08
# 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;            # Cross section area
Lc=0.208;               # Average magnetic path length
Lc1=Lc/2;               # Average magnetic path length for primary
Lc2=Lc/2;               # Average magnetic path length for secondary
Lc3=0.25;               # Air gap length for magnetic flux leak
Lg=3E-5;                # Air gap length
Lambda=1.1;             # Fringing coefficient

# OPT Topology
X=0.43;                 # Screen feed back ratio

# OPT parameters
Np=2419;                # Primary turns
Rp=58;                  # Primary resistance
Ns=84;                  # Secondary turns
Rs=0.26;                # Secondary resistance
Cip=250p;               # Anode to anode stray capacitance

# Primary winding n°1
Npa1=Np/2;              # Anode winding turns
Rpa1=Rp/2;              # Anode winding resistance

# Screen winding n°1
Nps1=Npa1*ABS(X);       # Winding turns
Rps1=Rpa1*ABS(X);       # Winding resistance

# Primary winding n°2
Npa2=Np/2;              # Winding turns
Rpa2=Rp/2;              # Winding resistance

# Screen winding n°2
Nps2=Npa2*ABS(X);       # Winding turns
Rps2=Rpa2*ABS(X);       # Winding resistance

END;

\$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;

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                         Page               17
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Date         24/08/08
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;             # 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);

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization
Pierre Touzelet                                                                        Page               18
Issue               3
Date         24/08/08
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
Pierre Touzelet                                                                        Page               19
Issue               3
Date         24/08/08
R13(130,40)=9k;
C9(130,40)=250p;
END;
\$\$STOP

Mullard 20 W tube amplifier – feedback loop gain and compensation networks optimization

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