Transfer function of a Gilbert cell multiplier
A Gilbert cell multiplier is a circuit that takes two inputs as differential currents and produces a differential current that is proportional to the product of the two inputs Let X and Y be arbitrary values between 1 and -1. Let I x and I y be arbitrary non-zero currents. The combined emitter current of the left differential pair, \(I_{le}=I_y\left(1+Y\right)\), and the combined emitter current of the right differential pair, \(I_{re}=I_y\left(1-Y\right)\). Let V 1 be the collector voltage of the leftmost transistor and V 2 be the collector voltage of the rightmost transistor. Let I s be the saturation current of the base-emitter junction. Let V t be the thermal voltage, about 0.025865 volts. Part 1 Consider the leftmost transistor. Note that its base and collector are connected together. Recall that the base current is \(I_s\left(\frac{e^{V_1/V_t}-1}{\beta}\right)\). Therefore the total current flowing through through the transistor is \(I_s\left(\frac{\beta+1}{\beta}\l