Skip to main content

Four- vs single-quadrant multiplier

Short answer

A single-quadrant multiplier, usually log-antilog, accepts only positive inputs, while a four-quadrant Gilbert cell accepts any signs and gives Vo = K Vx Vy. Derivation uses IEE tanh(Vx/2VT) tanh(Vy/2VT), approximating to K Vx Vy with K = IEE RL/(4VT^2).

How four quadrant multiplier differs from single quadrant multiplier? Derive output voltage of four quadrant multiplier with necessary diagrams.

Difference between four-quadrant and single-quadrant multiplier

PointSingle-quadrantFour-quadrant
Allowed inputsVx > 0 and Vy > 0 onlyVx and Vy of any sign
Output signAlways one polarityFollows sign of Vx·Vy
Typical circuitLog–antilog multiplierGilbert cell (transconductance)
UseDC/positive signalsAC signals, modulators, phase detectors

A log amplifier cannot accept negative input (ln of a negative number is undefined), so a log-antilog multiplier is only single-quadrant. A four-quadrant multiplier works in all four quadrants of the Vx–Vy plane.

Output voltage of four-quadrant multiplier (Gilbert cell)

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

Q3, Q5 collectors join to give Io1; Q4, Q6 collectors join to give Io2.

For a differential pair: ΔI = I·tanh(V/2V_T).

  1. Lower pair:
I1 − I2 = IEE·tanh(Vy/2V_T)
  1. Upper pairs (both driven by Vx, outputs cross-coupled):
IC3 − IC4 = I1·tanh(Vx/2V_T)
IC6 − IC5 = I2·tanh(Vx/2V_T)
  1. Output:
ΔIo = (IC3 + IC5) − (IC4 + IC6)
    = (I1 − I2)·tanh(Vx/2V_T)
    = IEE·tanh(Vx/2V_T)·tanh(Vy/2V_T)
  1. For small inputs, tanh(u) ≈ u:
ΔIo ≈ IEE·Vx·Vy / (4V_T²)
Vo  = RL·ΔIo = K·Vx·Vy,  K = IEE·RL/(4V_T²)

Because tanh is odd, a change of sign of Vx or Vy changes the sign of Vo, so all four quadrants are covered. Practical ICs (AD633) linearise the cell and give Vo = Vx·Vy/10 V.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Advanced Electronics chapter 6 questions →