Four-quadrant multiplier: concept and output

Four-quadrant multiplier

A multiplier gives Vo = K·Vx·Vy. If both inputs can be positive or negative and the output takes the correct sign, the multiplier works in all four quadrants of the Vx–Vy plane and is called a four-quadrant multiplier.

            Vy
     II      |      I
  Vx<0,Vy>0  |  Vx>0,Vy>0
   Vo < 0    |   Vo > 0
 ------------+------------ Vx
  Vx<0,Vy<0  |  Vx>0,Vy<0
   Vo > 0    |   Vo < 0
     III     |      IV

A log-antilog multiplier works only in quadrant I (one-quadrant). A four-quadrant multiplier is needed for AC signals, e.g. in modulators, phase detectors and power measurement.

Circuit (Gilbert cell)

         +Vcc            +Vcc
          RL              RL
          |  Io1     Io2  |
          +--+--  ...  +--+----> Vo
         Q3  Q4       Q5  Q6    <- Vx (cross-coupled)
           \/           \/
           I1           I2
           Q1 ---------- Q2     <- Vy
                  |
                 IEE
                  |
                -VEE

Collectors of Q3, Q5 give Io1; collectors of Q4, Q6 give Io2.

Derivation

Differential pair with tail current I and input V: ΔI = I·tanh(V/2V_T).

Lower pair:   I1 − I2   = IEE·tanh(Vy/2V_T)
Upper pairs:  IC3 − IC4 = I1·tanh(Vx/2V_T)
              IC6 − IC5 = I2·tanh(Vx/2V_T)

ΔIo = (IC3 + IC5) − (IC4 + IC6)
    = (IC3 − IC4) − (IC6 − IC5)
    = (I1 − I2)·tanh(Vx/2V_T)
    = IEE·tanh(Vx/2V_T)·tanh(Vy/2V_T)

For small signals, tanh(u) ≈ u:

ΔIo ≈ IEE·Vx·Vy/(4V_T²)
Vo  = RL·ΔIo = (IEE·RL/4V_T²)·Vx·Vy = K·Vx·Vy

The product of two odd functions keeps the correct sign for every combination of input polarities, so the circuit is four-quadrant. IC versions (AD633, MC1495) give Vo = Vx·Vy/10.