Help me solve a modulation exercise

In the circuit below, supose that the valves have a growth ratio between the plate current and grid voltage, as follow:

ip = gm . egk + β . (egk)2


( a ) calculate v ( t ) in function of the parameters.
( b ) calculate v ( t ) to the following cases:

b.l - va(t) = a(t), vb(t) = cos ωo t
b.2 - va(t) = cos ωo t vb(t) = a(t)
b.3 - va(t) = a(t) cos ωo t vb(t) = cos ωo t

Top valve input is Va - Vb

Bottom valve input is Va + Vb

Do a double substitution, first for the combination of Va and Vb then put this sum in the first transfer equation.

Output is gm times plate load resistance.

Sum the outputs noting that they are out of phase.

Your result should be that the output contains terms of Vb, combinations of Va and Vb, but no pure Va terms.

This is a standard single balanced mixer circuit from 50 years ago. It was used as a receiver and transmitter mixer. It was also used to generate voice SSB signals with a filter following it. Dual triodes were commonly used to get a close match between the gm terms of the two halves.

I think the intention of the exercise is figure that b1 is for upper side band, b2 is for lower side band and b3 is for demoudulation of upper side band.

How is calculate v(t) in function of the parameters? Someone can show it pass by pass?

What means each parameter in the equation:


ip = gm . egk + β . (egk)2


tks

the meaning of parameters are very simple
ip = the plate current of the tubes
beta= the modulation index
egk = may be the voltage

How does calculate v(t) in function of the parameters? Someone can show it pass by pass?


please, i need it urgent

thank you

Anybody can help, please?