Non Linear Behaviour of the SRPP

Fig. 1


The SRPP circuit (acronym of Shunt Regulated Push-Pull) is known since the 50’s [1], [2] but it has gained his fame as “basic block” for audio designs recently, at least in the west. Its use could be seen as an intrinsic characteristics of the renewed interest of vacuum tubes applied to the world of Hi-Fi reproduction. Oddly, this circuit is poorly used in the pro-audio sector and solid-state designers ignore it altough proposed by Milmann and others [3]. Altough imperfect in its operation (as any amplifier stage), the SRPP topology is able to mix good electrical performances with semplicity of building and interesting sonic results. Since many characteristics of this topology are not immediatly evident regarding to HCT point of view I will dedicate wide space to analysis.

 


 The schematic of a SRPP is in Fig. 1. The circuit operation is such for wich:

io= ip1-ip2   (1).

Ref. [4] contains a good explanation on the operation. The primary purpose of this configuration was in the driving of low impedance loads, tipically “long” cables terminated on low impedances  (a worry matter for the early television technique). The “tricks” consist in varying the values of R or RL in order to alter the amplitude of ip1 and ip2 and, for the (1), to minimize the THD. By substituing the Small Signals Equivalent Circuit (SSEC) to tubes VT1 and VT2 in Fig. 1 the above condition it’s obtained when:

R= 1/gm + 2*RL/m

(2)

RL= m*(R*gm – 1)/2*gm

 

with obvious meaning of symbols. Unfortunately a more deepened analysis that includes non linear effects reduces the precision of (2). The main observation is that both formulae furnish the solution to a non linear problem starting from a linear presupposition (the SSEC hypothesis). In fact the upper triode receives an input signal already corrupt in its harmonic content by the lower tube and therefore the minimun in the THD not necessarily is obtained by differentiating two currents with the same amplitude although out-of-phase; a similar mechanism happened in Cascade Stages. The problematic is better understood with a Spice simulation. If the circuit in Fig. 1 has the following charecteristics:

VT1=VT2=6SN7; Vin=4Vpp e Rcath=1k,    (3)

 

Fig. 2


in the condition of greather symmetry for ip1 and ip2, io presents a THD of 3.053% and R take a value very close to the first of the 2. The harmonic spectrum is decreasing with the frequency, Fig.2. In the Fig. 3 you have the best condition for the harmonic distortion, THD=0.5% and R=8K. You can see that ip1 and ip2 don’t result equal at all and the spectrum presents a 3rd harmonic with position of dominion respect to the others.

           A very important characteristic of SRPP derived directly from (1) but not easily comprehensible in a intuitve way consists in the ability to alter the phase relationship between ip1 and ip2 by varying R or RL. Fig. 4 shows the result of a Spice simulation of the circuit in Fig. 1 (with the istances 3) when R is varying between 4k and 10k by steps of 500W. Apart an obvious bias alteration,  ip1, the current  passing through the upper tube, is subjected to a phase inversion when R becomes elevated. Likewise, by varying RL between 5k and 100k by steps of 5k with R fixed you obtain the results of Fig. 5. For high load resistors ip1 and ip2 are in-phase, while the same becomes out-of-phase for low ones.

Fig. 3

       From here it emerges that  utmost potentialities of SRPP, as a shunt type push-pull, are obtained when, by varying R or RL, ip1 and ip2 are out-of-phase. This peculiarity of SRPP circuit was well understood from his inventor; unfortunetely now, in the majority of the cases, we use it like a kind of common cathode amplifier with an active load and a low output impedance.

       The examen of the precedent simulations light up a fairly “curious” aspect: from Fig. 3 and 4 a value for R or RL exists that nulls ip1 current (remember ip1 and ip2 are dynamic variations) since a phase inversion is concerned. In these conditions the upper triode doesn’t influence the elaboration of the information that virtually will follow the path underlined in Fig. 6. This behaviour give us a wide margin in the choice of the upper device  because, when ip1=0, it doesn’t compete for  sonic texture. For example you could  experiment  with pentodes or j-fet as upper device and Direct Heating Triodes as lower device.


 

Fig. 4
 

A better understanding of R’s or RL’s role can be obtained with a non linear analysis of Fig. 1. Since:

ip1= h1*eg1 + h2*eg12                   (4)

ip2 = g1*Vin + g2*Vin2                  (5)

eg1 = -Rip2                                           (6)

 

After a bit of algebraic calculi you obtain the following expression:

 

io=ip1-ip2= a*Vin4 + b*Vin3 + c*Vin2 +d*Vin  (7)

 

where:

a= g2^2 * h2 * R^2

b= 2* g1*g2*h2

c= g1^2*h2*R^2 – g2*(h1*R-1)

d= -(g1*h1*R+g1)

                                                   (8) 

 

and for the (6):

io= Co + C1*sinwt + C2*cos2wt + C3*sin3wt +C4*cos4wt     (9)

where:

Co= 3/8*g2^2*h2*R^2*Vp^4 + ˝*Vp^2*(g1^2*h2*R^2-g2*h1*R-g2);

C1= -Vp*(g1*h1*R + g1);

C2= -1/2*g2^2*h2*R^2*Vp^4 –1/2*Vp^2(g1^2*h2*R^2-g2*h1*R-g2);

C3= -1/2*g1*g2*h2*R^2*Vp^3;

C4= 1/8*g2^2*h2*R^2*Vp^4;

(10)

Fig. 5

The main problem consist in the correct determination of [gi] and [hi] coefficients. Such terms could be determined after a characterization of the Mutual Dynamic Characteristics (MDCs). You can follow an analytical or numerical way.

Fig. 6


Clearly, the latter (that include also the experimental approach) leads to  faster results since to use a “curve fitting” less informations are requested  for the tube characterization. Fig. 7 shows the DC loadlines of a 6SN7’s SRPP powered by 500Vcc source and Fig. 8 their MDCs rispectively evalued by PSpice. Please observe the difficulty to deduce MDCs from loadlines because you cannot extract from here the true phase relationship between ip1 and ip2  currents. A curve fitting based on the numerical values in Fig. 8 allow us to determine [gi] and [hi] coefficients and therefore to appraise partially the goodness of (10s), Table 1.

 

Fig. 7

Tab. 1

        Amplitude (Vp)

2nd Harmonic Calculated (V)

2nd Harmonic  Pspice Evaluated (V)

1

6.422e-7

6.210e-7

2

2.569e-6

2.507e-6

3

5.759e-6

5.647e-6

4

1.027e-5

1.002e-5

5

1.605e-5

1.561e-5

6

2.310e-5

2.238e-5

7

3.144e-5

3.029e-5

8

4.100e-5

3.927e-5

 

Fig. 8

 Unfortunately mathematical expressions don’t have a large utility to correctly determine R or RL values because a lot of physical effects are keep out from (10’s). However the C2 expression in the (10’s) permits to intuit that a value in R or RL exists in order to minimize this coefficient therefore you can entrust the “crunch numbers” task to the electronic simulator or to the benchmark to directly determine the resistive values.  I have easily characterized the SRPP behaviour with a set of simulations summarized in Figs 9-12 for  input signals of 0.2Vpp, 2Vpp, 4Vpp, 6Vpp rispectively. On the graphics the X-axis always represents R values while Y-axis can have  1st, 2nd, 3rd, 4th harmonics and THD of output current.

Fig. 9

Since the gain of a 6SN7’s SRPP is about 20dB you could think the first graphic set in Fig. 9 as referred to a line preamplifier, the sets of Figs 10 and 11 could represent the behaviour of gain stages in  power amplifiers and finally the set in Fig. 12 could be the behaviour of a driver for power amplifiers.

Fig. 10

Fig. 11

 

Fig. 12

All graphics always show a minimum in the THD. When this minimum is reached, an ulterior increment in R produces an abrupt THD surge. You can see from the graphs that the optimal value in the THD is given when an opportune (dise-)equilibrium  between  2nd, 3rd  and 4th harmonics exists. 2nd harmonic is the spectral component with a greater weight since its trend is similar to THD. The 2nd harmonic reduction is counterbalanced by a growth in the 3rd harmonic. The increase in the 3rd harmonic involves, when this is followed by a reduction in 2nd harmonic, a better simmetry of output signal respect to Y-axis. Besides all graphs show a left-shift, for the R values producing the minimum in the THD, when the input signals increase, Tab.2.

Tab.2

Input Signal

 R values for minimun in  THD

0.2Vpp

13kW

2Vpp

11kW

4Vpp

9kW

6Vpp

7kW

 The fundamental presents a tendency weakly increasing with R in Figg. 9-10, while in Figg. 11-12 it presents even a maximum. Since by varying the R or RL values you can alterate the harmonic spectra of the signals, this metodology could be used to change the timbric imprint of the produced sound.

 References:

(1) A. Peterson, D.B. Sinclair A Single Ended Push-Pull Audio Amplifier, Proc. IRE, Vol.40, pp. 7-11, Jan 1952
(2) Yeh Chai Analisys of a Single Ended Push-Pull Audio Amplifier, Proc. IRE, Vol. 11, pp.743-47, June 1953
(3) J. Millman and C.C. Halkias Integrated Electronics, McGraw-Hill, New York, Appendix C, Problems 10.22, 10.23, 1972
(4) J. Milmann and H. Taub Pulse, Digital and Switching Waveforms,  McGraw-Hill, New York, Par. 3.17, 1965

 

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