Theoretical Background for the THD Cancellation

                    In this section, I have often quoted the Mutual Dynamic Characteristic (MDC) since it represents a key property to explain the non linear behaviour of active devices as vacuum tubes. The MDC can be easily approximated with a low order polinomial function whose coefficients are costant for a preset bias of the circuit taken into consideration. Here I show as you can reach a such algebraic form for the MDC starting, for semplicity, from triode’s equation.

          If Ib, Eb, Ec, represent the plate current, the plate voltage, and the grid voltage rispectively when no-signal excites the input and ip, ep, eg are the dinamic versions of same previous quantities, you can write:

ib= Ib+ip

eb=Eb+ep

ec=Ec+eg

where ib, eb, ec represent the plate current, plate voltage and grid voltage in the static+dynamic state, i.e. when an ac signal is superimposed to the tube bias. Glass Audio readers know the difficulties existing in writing a mathematical expression to analytically find the following relation:

ib=ib(eb,ec)=ib[(Eb+ep), (Ec+eg)]   (32)

because it’s the same difficulty that exist to generate an accurate SPICE model.

As first approximation  you can suppose:

Ib=Ib(Ec+Eb/m); ip=ip(eg+ep/m)     (33)

(i.e. the plate current is a function of the cumulative potential eg+ep/m)

therefore   (32) becomes:

ib=Ib+ip=ib[Ec+eg+(Eb+ep)/m]        (34)

and setting:

a=Ec+Eb/m e b=eg+ep/m

you can re-write (34) in the following simplified form:

ib=ib(a+b) (35).

Expression (35) is now expanded in Taylor’s Series:

f(a+b)= f(a)+f’(a)*b+1/2!*f’’(a)*b^2+….+

f(a+b)-f(a)=f’(a)*b+(1/2!)*f’’(a)*b^2+…+

Since:

f(a+b)-f(a)=ip

you have:

ip=f’(a)*b+(1/2!)*f’’(a)*a+b ^2+….+

where:

f’(a)=gm=m/rp,

f’’(a)=d(gm)/d(Ec),

f’’’(a)=d(f’’(a))/d(Ec), ...

Setting:

f’(a)=a1 and f’’(a)=a2

you obtain:

ip=a1*(eg+ep/m)+a2*(eg+ep/m)^2+…+     (36)

At least theoretically you can express ep as function ip:

ep=-Z*ip

where Z is the impedance “seen” from the tube plate in dynamic conditions. For a common-cathode amplifier this coincides merely with the dinamic load but in complex configurations  Z is more difficult to determine. Therefore you can re-write (36), after tedious calculi, in the following manner [1]:

ip=b1*eg+b2*eg^2+…+          (37

with bi=bi(ai,Zi) i=1,2,..,

Eq. (37) can be generalized to represent the MDC for a generic vacuum tubes amplifier stage.

A backward analysis shows that bi (i=1,2,…) coefficients result constant only for both a given bias and load. In real world this conditions are true only as first approximation (not excessive signal swing, weakly reactive loads etc.) and  their validity is limited: in this context calculi made by hands are time consuming and sometimes inaccurate.

          Eq. (37) for triode and pentode stages can be approximated as:

ip=b1*eg+b2*eg^2        (38)

ip=b1*eg+b2*eg^2+b3*eg^3 (39)

respectively.

[1] A.V. Eastman  Fundamentals of Vacuum Tubes Mc-Hill B.C., Chap. XII, New York, 1957