---
title: "Four-quadrant multiplier: concept and output"
subject: "Advanced Electronics"
chapter: 6
key: "semester_4/advanced_electronics:four-quadrant-multiplier-concept"
source: https://www.bctengineer.com/q/semester_4/advanced_electronics/four-quadrant-multiplier-concept
site: BCT Engineering
---

# Four-quadrant multiplier: concept and output

Advanced Electronics · Chapter 6: Log-Antilog Circuit Applications

- Asked in: 2070 Chaitra, 2+5 marks
- Video: [L25 | Analog Multipliers | Integrated Circuits (KEC501) | Hindi](https://www.youtube.com/watch?v=Fcqv5EIrTgU) — Dr. Vaibhav Jain
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/advanced_electronics/four-quadrant-multiplier-concept
- Whole chapter: https://www.bctengineer.com/md/semester_4/advanced_electronics/chapter-6

## Question

What do you understand by four-quadrant multiplier? Draw the circuit diagram and derive expression for its output voltage.

## Short answer

A four-quadrant multiplier correctly gives Vo = K·Vx·Vy for any sign of Vx and Vy, enabling AC signal processing. The Gilbert cell uses two cross-coupled differential pairs to obtain ΔIo = IEE·tanh(Vx/2VT)·tanh(Vy/2VT); for small signals Vo ≈ (IEE·RL/4VT²)·Vx·Vy, working in all four quadrants.

## Answer

#### 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**.

```text
            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)

```text
         +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).

```text
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:

```text
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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is a four-quadrant multiplier?
   A. One with four inputs
   B. One that works only for positive inputs
   C. One with four transistors only
   D. One that multiplies correctly when both inputs can be positive or negative
   Answer: D. It covers all four quadrants of the Vx-Vy plane. (Key point: Definition)

2. In quadrant III (Vx < 0, Vy < 0), what is the sign of Vo?
   A. Zero
   B. Undefined
   C. Positive
   D. Negative
   Answer: C. A negative times a negative is positive. (Key point: Sign of product)

3. What is the Gilbert cell output current difference?
   A. IEE exp(Vx Vy)
   B. IEE ln Vx ln Vy
   C. IEE tanh(Vx/2VT) tanh(Vy/2VT)
   D. IEE (Vx + Vy)
   Answer: C. The two differential stages multiply their tanh functions. (Key point: Derivation)

## Related in Advanced Electronics

- [Four-quadrant multiplier circuit and derivation](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Afour-quadrant-multiplier-derive) — Advanced Electronics, chapter 6
- [Four- vs single-quadrant multiplier](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Afour-vs-single-quadrant-multiplier) — Advanced Electronics, chapter 6
- [DIAC drawback; TRIAC four-quadrant operation](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Adiac-drawback-triac-quadrants) — Advanced Electronics, chapter 7

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