The Gilbert multiplier cell is a modification of the emitter coupled cell and this allows four – quadrant multiplication.

**Gilbert
Multiplier cell:**

The
Gilbert multiplier cell is a modification of the emitter coupled cell and this
allows four – quadrant multiplication. Therefore, it forms the basis of most of
the integrated circuit balanced Multipliers. Two cross- coupled emitter-
coupled pairs in series connection with an emitter coupled pair form the
structure of the Gilbert multiplier cell.

The
collector current of Q_{3} and Q_{4} are given by

Similarly,
the collector current of Q_{5} and Q_{6} are given by

collector current I_{C1} and I_{C2}
of transistors Q_{1} and Q_{2} can be expressed as

Substituting
the above equation in I_{C3} and I_{C4}, we get

Similarly
substituting I_{c2}= in I_{c5} and I_{C6} , we get,

The
differential output current I is given by

∆I
=I_{L1} - I_{L2}

=
(I_{C3} +I_{C5}) - ( I_{C4}+I_{C6})

=I_{C3}
- I_{C6})-(I_{C4} - I_{C5})

∆I=I_{EE}
tanh( V_{1}/2V_{T}) tanh(V_{2}/2V_{T})

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