Analog multiplier using an Emitter coupled Transistor pair

Analog multiplier using an Emitter coupled Transistor pair:

The output currents IC1 an d IC2 are related to the differential input voltage  V₁ by

where V is thermal voltage and the base currents have been neglected. Combining above eqn., difference between=the two output− currents as

The dc transfer characteristics of the emitter – coupled pair is shown in figure. It shows that the emitter coupled pair can be used as a simple multiplier using this configuration. When the differential input voltage  V₁ << V_T, we can appropriate as given by

∆I_C =I_EE ( V₁/2V_T)

The current I_EE is the bias current for the emitter – coupled pair. If the current I_EE is made proportional to a second input signal V₂, then

I_EE =K₀ (V₂ - V_BE)/2V_T

Substituting above eqn. , we get   ∆I_C  = K₀V₁ (V₂ - V_BE)/2V_T

This arrangement is shown in figure. It is a simple modulator circuit constructed using a differential amplifier. It can be used as a multiplier, provided  V₁ is small and much less than 50mV and V₂ is greater than V_BE (on). But, the multiplier circuit shown in figure has several limitations. The first limitation is that V₂ is offset by V_BE (on).

The second is that V₂ must always be positive which results in only a two-quadrant multiplier operation. The third limitation is that, the tanh (X) is approximately as X, where X =  V₁ /2VT. The first two limitations are overcome in the Gilbert cell.