Design of four quadrant bipolar multiplier by Manraj Singh Gujral

1

Design of four-quadrant Bipolar Multiplier Manraj Singh Gujral, M.Sc.

School of Electronics & Computer Science, University of Southampton

SO17 1BJ, UK

[email protected]

Abstract— This document investigates the design of Gilbert Cell.

The process includes explanation of basic concepts like Long Tail

Pair and using it to implement the Analog multiplier circuit. The

software used for simulation is LT Spice IV version 4.08u with

preloaded cell library. The final objective is to design a

multiplier giving +/1 10 V output for +/- 5V input

Keywords— gilbert cell, multiplier, long tail pair (LTP), design

flow INTRODUCTION

In analog signal processing a need often arises for a circuit

that takes two inputs and produces an output proportional to

their products. Such circuits are now known as the multipliers.

A multiplier is a 2-input device which performs linear product

of two signals Vx and Vy generating an output of Vo=K.Vx.Vy.

From a block diagram perspective a Multiplier can be shown

as in figure 1.

Fig. 1 Block diagram of a Multiplier [1]

Multipliers, figure 2, can be categorized as

a. Single quadrant, when both v1and v2 are positive.

b. Two quadrant, when v1 is positive and v2 is bipolar.

c. Four quadrant, when both v1 and v2 are bipolar.

(c)

Fig. 2 Types of Multiplier (a) Single Quadrant, (b) Double Quadrant (c) Four

Quadrant

From a mathematical point of view a multiplier is a “four

quadrant” operator. But in physical hardware in electronics

implementing a first and second quadrant multiplier is

relatively much simpler than a four-quadrant multiplier. Also,

a lot of applications do not require a four quadrant multiplier

therefore the other types of multipliers still exist. There are

different types of multipliers being used in the industry. As an

example, AD539 is a two quadrant multiplier with two bipolar

inputs Vx1 and Vx2 and a common unipolar input Y. The

output is limited by lower bandwidth. Schematic of an AD539

is shown in figure 3

Fig. 3 AD539 Block Diagram [1]

Another basic type of multipliers is the logarithmic types.

They work on the principle that the multiplication of two

numbers is addition in a logarithmic form, and the output is

passed though an anti-log to convert back to the same units as

the input. Figure 4 explains the basic functionality of a

logarithmic multiplier. [1]

(a) (b)

2

Fig. 4 Block Diagram for Logarithmic Multiplier [1]

This is however a single quadrant multiplier with low

bandwidth. Therefore a much better approach, Gilbert Cell, is

used to explain the multiplier operation.

Before moving further we should understand the difference

between a modulator and a multiplier. A modulator is a circuit

which can have a number of inputs and one input can be used

to modify or control the signal flow from another input.

Fig. 5 Block Diagram for a Modulator

A multiplier is a two input device which is independent of

the input signals and performs a multiplication by a factor, k,

dependent on the multiplier parameter.

Fig. 6 Block Diagram for a Multiplier

Therefore, it can be said that a modulator is a special case

of a multiplier.

LONG TAIL PAIR [2]

An important concept used in any multiplier is the use of a

Long-Tail Pair, also known as the emitter-coupled pair circuit.

Fig. 7 Long Tail Pair or Common Emitter Pair

Figure 7 shows a basic construction of a Long Tail Pair.

The outputs current, Ic1 and Ic2 are related to the differential

voltage by:

�� = ��

�� (1)

�� = ��(

�� ) (2)

therefore, ∆�� = �� − ��

∆�� = ��( �

��

− �

��

) (3)

As,

11 +�� − 1

1 +�� = ��/��/� +��/� −

��/��/� +��/�

��/� − ��/��/� +��/� = tanh(�/2)

So, Eqn. (3) can be re-written as,

∆�� = ��tanh(!� �!�) (4)[2]

Assuming the Applied voltage, Vid<< VT (for small signal

analysis) we can write

∆�� = ��("#$/2"%) (5)

Therefore in order to control the Ic we can control the IEE,

making IEE the bias current for the Long Tail pair. Here we see

that If we can control the IEE externally, we can alter the

biasing of the LTP. So we add extra circuitry to make IEE

proportional to an external voltage source, as shown in figure

8

Fig. 8 Long Tail Pair with controlling voltage

�� ≅'(("#� − ")�((*)) (6)

So now, the differential output current at Eqn. (4) of the

LTP can be re-written as,

�+ ≅'(("#� − ")�((*))tanh(",-2".)

(7)[2]

3

Fig. 9 Long Tail Pair circuit realization

The output, Eqn. (7), is a multiplier assuming that Vid is a

small signal and Vi2> VBE. In circuit realization in figure 9,

V5 is sweep from 1 to 5. Instead of using a pure differential

voltage Vid, we use two voltages V4 and V3 w.r.t ground and

fix on of them to 1V and sweep the other for simulation.

Fig. 10 Long Tail DC sweep waveform

As seen from the wave in figure 10 the voltage (V5) has to

be positive (1V to 5V). As from the waveform we can see the

central point when input voltage is 1 V, i.e. when V1 = V2 the

differential voltage output is zero. Also the curve stays linear

for very small range of differential input voltage (v4-v3) in the

range of +/- 0.6V. And beyond that the output voltage is

saturated at fixed level.

Analysing a random output wave, say at V5=4, the output

differential voltage/6.1V and is constant for any change in

the input. In the liner range, when V3 in in the range 0.96V to

1.04V, we can assume the value of K is constant we can

therefore generate a multiplier with the equation as mentioned

in the Introduction

Vo=K.Vx.Vy

We can adjust the factor K by varying the tail current or the

resistances values R1 and R2. In a practical circuit, however,

it is prudent to have a voltage controlled reference that a

variable resistor which we will use later in this report.

Now we perform a large signal analysis on the same circuit

and a voltage sweep for both negative and positive values of

input voltage V5. From -5 to +5 in steps of 1. The circuit is

shown in figure 10

As we can observe from the waveform, figure 12, that this

particular multiplier is basically a two-quadrant multiplier. All

the output values for the multiplier remain at 0V for any

voltages of V5 < 0.

Fig. 11 LTP circuit with voltage sweep for V5 from -5 to +5

Fig. 12 Waveform analysis for voltage sweep for V5 from -5 to +5

The Voltage V5 has to be positive at all times since the IEE ,

seen in figure 8, has to flow in order for the LTP to function.

This causes further restrictions on its usage. Therefore we will

use this principle of a cross coupled Long Tailed Pair to

modify the current circuit and use it in all four quadrants.

ANALYSIS OF GILBERT MULTIPLIER

Barry Gilbert in 1967 had invented an Analog multiplier,

which is now known as the Gilbert Cell, or the Gilbert mixer

[3]. A basic structure of a Gilbert Cell is two Long Tailed

Pair connected in series with one half of other long tailed pair

as shown in the figure 13

4

Ic3 Ic4

v1

+

_

Q1 Q2

+

_v2

Ic1

Q5 Q6

Ic5 Ic6

Q3 Q4

Ic2

IEE

IL1 IL2

Fig. 13 The basic structure of the Gilbert Cell

To understand the basic operation, consider V2 as reference

and V1 as input. If V1 is a positive voltage and large enough

that transistor Q1 and Q4 are powered on and Q2 and Q3 are

therefore powered off. Therefore as is the case in a standard

differential amplifier,

(a) Q5 is connected to IL1 load line, and

(b) Q6 is connected to IL2 load line.

Whereas, if the input voltage is a negative voltage large

enough to power on Q2 & Q3, and hence, power off Q1 and

Q4 then we have

(c) Q5 is connected to IL2 load line , and

(d) Q6 is connected to IL1 load line.

Which is also the standard differential amplifier, but with

negative polarity. [4]

Theoretically it should work for both the negative and

positive values of the input and output giving us the four

quadrant operation. We will now look at the mathematical

derivation of the output and later simulate and see the results

to prove it works.

As in Eqn. (4), here we have two voltage controlling the

output namely v1 and v2.

In the figure 13 , we have

,�0 = ,��1 + �(�12

1�) , ,�4 = ,��

1 + �(121�)(8)

and,

,�6 = ,��1 + �(�12

1�), ,�7 = ,��

1 + �(�121�)

(9)

And the second LTP currents as,

,�� = ,��1 + �(�19

1�) , ,�� = ,��

1 + �(191�)(10)

Substituting Eqn. (10) in (8) & (9) we find that

,�0 = ,��;1 + �(�<� <%⁄ )>;1 + �(�<� <%⁄ )>,

,�4 = ,��;1 + �(<� <%⁄ )>;1 + �(�<� <%⁄ )>,

,�6 = ,��;1 + �(<� <%⁄ )>;1 + �(<� <%⁄ )>,

,�7 = ,��;1 + �(�<� <%⁄ )>;1 + �(<� <%⁄ )>

(11)

Assuming the Change in collector currents as

∆�� = �?� − �?� = (,�0 + ,�6) − (,�4 +,�7)= (,�0 − ,�7) + (,�6 + ,�4) and substituting the values of individual collector currents

from Eqn. (11) we get,

∆�� = �� @A 11 + ��<� <%⁄ − 1

1 + �<� <%⁄ BC @A 11 + ��<� <%⁄

− 11 + �<� <%⁄ BC

=D∆�� = ��tanh( <2�<�)tanh(

<9�<�) (12)[5]

This Eqn. (12) forms the equation of Gilbert Cell where the

output is the product of hyperbolic tangents of the input

voltages.

We use the same concept in out circuit realization as shown

in figure 14. Instead of the Current source IEE, we employ a

current mirror controlled by a voltage reference. Note that Q5

and Q6 now act as separate arms for the current to flow for V3

to be negative and positive respectively. Transistor Q5 , in

figure 14, is responsible for the flow of current labelled as Ic1

in figure 13 when the V3 is positive. Transistor Q6, in figure

14, is responsible for flow of current Ic2 when V3 becomes

negative. Actual biasing of the transistors needs to be

appropriate since a negative differential voltage effectively

means that one of the transistors, say Q5, is active during the

positive voltage cycle and therefore Q6 in negative voltage

cycle since it sees a positive potential w.r.t. Q5.

Fig. 14 The Gilbert Cell – two cross coupled LTP

5

Fig. 15 Two cross coupled LTP waveform, with Ic(Q7) = 2.35mA.

Observing from the waveforms in figure 15, we now have

operations in all the four quadrants, i.e., the negative and

positive values of the input voltages.

Please note that we have maintained the reference V6 fixed

at 1V and therefore the changeover occurs at 1V because V4-

V6 = 0 at 1 V.

In figure 14 & 15 the current flowing current mirror arm

into Transistor Q7 is

Ic(Q7) = 2.35mA . (13)

As per Eqn.(12), we see that the output voltage is basically

dependent on three factors,

∆�� ∝ �� . tanh( G�). tanh(G�) (14)

For the sake of validating our equation we will change the

reference voltage V5 to control IEE and observe the output

(a)

(b)

Fig. 16 Two cross coupled LTP waveform, with Ic(Q7) = 6.25 mA. (a) Circuit

with V4=4V, (b) Resulting waveform and Ic(Q7

From figure 16 we can observe the output has changed

from a range of +/- 2.4 V to +/- 6 V. Hence, our equation,

Eqn.(14) holds true.

This is just an operational schematic intended to simplify

and explain the Gilbert Cell operation. As we move to a more

practical aspect, we would need to provide an actual

differential voltage across the cross coupled LTP instead of

two voltage sources, V4 and V6.

The small signal analysis of the same circuit as shown in

figure 17

Fig. 17 Small Signal analysis of LTP waveform, with Ic(Q7) = 6.25 mA

V3 from -0.1V to 0.1V in steps of 0.01V

Small signal analysis shows that the waveform follows the

Tanh graph and therefore the liner range is very small.

We have now established the basic working of a Gilbert

circuit. From figure 17 it can be observed that it shows the

features of a multiplier circuit. With small signal input voltage

of we get a linear characteristic output. In order to move to a

more practical circuit we need to properly bias the inputs ,

increase the linearity of the output and have some

controllability. Therefore, we start by replacing the Voltage

source V4 with an inverse hyperbolic tangent differential

voltage source. Details of the effect of introducing voltage-to-

current converter are given in figure 21. As of now, we were

using V4 and V6, see figure 16 (a), to supply the input voltage.

Since V6 is always fixed at 1V the output was seen across the

1V mark which was acting as the reference point of the circuit,

figure 16(b). As we now move to practical multiplier like the

one shown in figure 18 which is a very basic from multiplier,

we need only VX1-VX2 as a differential voltage source.

6

Fig. 18 Low cost analog multiplier AD633, 8-Lead Plastic DIP (N) Package

Also, instead of a differential output, we could employ the

circuit for a single ended output using current mirrors.

Therefore, implementing a differential voltage input and using

emitter degeneration in place of Q7 & Q8 we get the

following characteristics as shown in figure 19.

(a)

(b)

Fig. 19 Gilbert Cell with differential Inputs and single ended output (a)

Circuit diagram, (b) Resulting output waveform

The output characteristics have been modified and the

observations are:

(a) Input voltage is now passed to the gilbert Cell as a

differential voltage.

(b) The output is taken across the Resistor R5 as a single

ended output. The circuit implementation is nothing

but two current mirrors connected to the ∆�� arm of

the previously used circuit, figure 16(a). The current

mirrors both try to maintain the same current in its

arm and the differential current is passed through the

resistor R5 resulting as the output.

(c) The output range of the voltage is now < +/- 2V for

and input voltage = +/-5 V.

(d) The Output voltage is now dependent on Resistors,

R2 & R5, the input and output resistors resp. as well

as R4 the emitter degeneration resistor.

(e) The Linear range of the circuit is drastically

improved from figure 12, the range is now valid for

the input voltage range of +/-5 V.

(f) Addition of a degeneration emitter resistor not only

improves the liner range, but also makes the system

more stable since the Ic is no longer dependent on the

temperature varying parameter of the BJT, β.

Consider a simple Common Emitter configuration:

Rb

Rc

Re

Vb

Ic

Vcc

Vb

Fig. 20 Basic CE Configuration of an npn transistor [4]

For dc Bias , the KVL loop can be written as:

"H = �HIH +"H + �I

Assuming �� = J�H , if Re=0 then the KVL equation

becomes,

"H / �HIH +"H = �� IHJ + "H

i.e., the Bias current Ic depends on the transistor β.

Whereas assuming �� = J�H & � / �� if ReK0 then

the KVL equation becomes,

"H / �HIH +"H +��I ="H +��I

Since IbRb << IcRe . Therefore Vb controls Ic without

depending on the transistor β making the biasing

more stable. [4]

Also, it can be shown that the Output gain of the

common emitter amplifier (small signal)becomes :

L< /"(MN"#* = −I�I

And this depends only on the external resistors and

independent of transistor β. [4]

We will try and figure out the output voltage relation to these

circuit elements by performing certain tests.

Ic3 Ic4

v1

+

_

Q1 Q2

+

_v2

Ic1

Q4 Q5

Ic5 Ic6

Q3 Q4

Ic2

IL1 IL2

Differential-to-single-ended converter

Differential voltage to current converter

Vout

Differential voltage-to-current converter

Vcc

Fig. 21 Basic Gilbert Cell Schematic [6]

In figure 19, all we have done is provided the Basic gilbert

cell with proper differential voltages and single ended output,

figure 20.

7

(a) The single ended output is nothing but two

current mirrors trying to balance each other and

the differential current flowing through the Load

resistor. Addition of a differential-to-single-

ended converter adds a parameter in the circuit

as

"(MN =∆�� . ' (15)[6]

Where K is dependent on the load resistance and the

voltage controlled current source in the differential-to-single

ended converter.

(b) Since differential voltage-to-current converter

with input V2 is responsible for the tail currents

in the cross coupled LTP, it has an important

role in output where

"(MN ∝ !9�OP (16)

The Ieo current in Eqn.(16) can be seen in our simulated

circuit, figure 19(a), as the degenerated emitter current source

as, I3 & I4

(c) Similarly, the differential voltage-to-current

converter with input V2 is the main input signal

which controls the LTP pair transistors.

Therefore addition of this converter has an effect

on the output as

"(MN ∝ !2�Q�RRPRSTRROUV

(17)

The Imirror current in Eqn.(16) can be seen in our simulated

circuit, figure 19(a), as the degenerated current source as, I1 &

I2. Practically the purpose of this differential voltage-to-

current converter is to introduce an inverse hyperbolic

Tangent voltage to cancel the effect of the Gilbert Cell’s

equation given in Eqn. (12).

v1

+

_

Differential voltage-to-current converter

∆V

I1 I2

Vcc

Fig. 22 Inverse hyperbolic tangent circuit [6]

The differential voltage source produces a current in

addition to the I1 and I2 shown in the figure 21. The currents

are hence given by,

�� = �( +'�"� , and �� = �( −'�"� (18)

Here Io is the current when V1= 0V, and K1 is the factor

introduced by the voltage-to-converter.

Therefore, the differential voltage ∆" is given by,

∆" � "% ln A�( � '�"��X B �"% ln A�( � '�"��X B

� "% ln Y�P�Z2!2�P�Z2!2[ (19)

Using, tanh�� ln Y��[

We have,

∆" � 2"% tanh�� YZ2!2�P [ (20)

Substituting Eqn. 20 in Eqn.12 we get ,

∆�� tanh A G�2G%B tanh AG�2G%B �D ∆�� YZ2!2�P [ YZ9!9�OP [ (21)[6]

This makes it linear to the Input voltage rather than

hyperbolic tangential.

Our specification is to meet a specification of +/- 10 V

Output for an Input range of +/-5V with flexibility of

controlling the tail current in the cross connected LTPs.

Using Eqn.15, 16 & 17 to vary the circuit parameters, such

as the input and output resistances; the tail current settings;

and the differential voltage V3 controlling the tail currents of

cross coupled LTP in figure 18 is modified to get the desired

output.

(a)

(b)

Fig. 23 Analysing the output by varying circuit parameter R2

(a) Replacing the resistance R2 from 5k to 10k (b) Resulting output

waveform V(output) vs. Input voltage, V4

Figure 22 shows the response of the circuit when we vary

the resistance R2 from its set value of the circuit (shown in

after Figure 25). The output swing is restricted to +/- 5V from

+/- 10V proving the output is inversely proportional to the

input degenerated resistor.

8

(a)

(b)


(a) Replacing the resistance R4 from 1.8 to 3.65k (b) Resulting output




after Figure 25). The output swing is restricted to +/- 5V from

+/- 10V proving the output is inversely proportional to the

Emitter degenerated resistor (validating Eqn. (16)).

(a)

(b)


(a) Replacing the resistance R5 from 5k to 10k (b) Resulting output




after Figure 25). The output swing is doubled to +/- 20V from

+/- 10V proving the output is directly proportional to the

differential-to-single-ended converter factor given by Eqn. (16)

(a)

(b)

Fig. 26 Gilbert Cell with adjusted circuit parameters. (a) Circuit diagram, (b)

Resulting output waveform V(output) vs. Input voltage, V4

The final required output is obtained in Figure 25, for input

voltage swing of +/-5V the output ranges from +/- 10V. the

circuit also has a voltage controlled Tail current controller , as

V3, in the circuit which can adjust the multiplication factor.

Gilbert Cell described so far has been BJT based, the same

model can be developed for a MOS based Cell as well.

9

Fig. 27 MOS based Gilbert Cell [7]

Here Iout = I1-I2

Where currents through the nMOSes are given by,

�\� � Z2� ]"̂ X� �"%_�, �\� � Z9

� ]"̂ X� �"%_�, �\0 � Z`

� ]"̂ X0 �"%_�, �\4 � Za� ]"̂ X4 �"%_� (22)

It is essential that the transistors are matched (common

centroid techniques) i.e.,

'� �'� �'0 �'4 �b*c(� de

If we assume the LTP functioning of the circuit

Fig. 28 MOS based Gilbert Cell [7]

The output current can be shown to be

�?%f(MN � �\� � �\� �g�XX'"� hi1 � !294�jj Z⁄ k

(23)

And similar to Eqn. (12) derivation the Gilbert Cell Output

current can be written as

�(MN � �� \� � �\�� \0 � �\4� (24)

�(MN ≅ Z√� �"�. "�� (25)[7]

One important factor to note here is that the Output is no

longer a the product of hyperbolic tangents of the input

voltages as was the case with bipolar based multiplier, rather

it is a linear and directly proportional to the inputs

LEARNING OUTCOMES & CONCLUSIONS

Gilbert Cell can be used for wide range of applications in

electronics & communication. While working on this report a

large number of its applications were found working on the

principle of this cell, such as:

1. Frequency modulator where the V1 can be a modulating

small signal and the Carrier large signal can be supplied in V2

to get modulated output.

2. Phase detector, where the product of both the inputs

(both large signals) gives the out of phase voltage at the

output.

Use of current mirrors, alone and in conjunction with the

Long Tail Pair has many practical uses and is a very important

tool in electronic circuit design today. The individual concepts

for LTP and Current mirrors are not covered in detail in this

report but their functionality is quite apparent and similar to

the Gilbert Cell, they can be used in variety of modules.

REFERENCES

[1] Analog Devices - MT-079 TUTORIAL Analog Multipliers.

[2] Gray, P.R. & Meyer. R.G. (Dublin Institute of Technology), Analog

Multipliers, Analysis and Design of Analog Integrated Circuits (3rd

Edition), Wiley (1992) pp. 667-681.

[3] “The Gears of Genius” , SSCS IEEE Solid-State Circuits Society News;

2007 Volume 12, Number 4

[4] Prof. Keith W. Whites, Electronics II-Wireless Communication

Electronics (EE 322), Lecture 27 & 19 ,Dept. of Electrical & Computer

engineering, South Dakota School of Mines & Technology

[5] Analog Circuits and Systems Design (ECE6414) - Multipliers and

PLLs , ECE , Georgia Tech University,

[6] Dr. Mohamed Abd El Ghany, Communication Systems (COMM 704).

Department of Electronics and Electrical Engineering, German

University in Cairo.

[7] Associate Prof. Dr. Soilam Mahmoud, Microelectronics Lecture 2 & 3,

winter 2006, Electronics and Electrical Department , German

University Cairo

BIBLIOGRAPHY

[1] Gunhee Han and Edgar S´anchez-Sinencio, CMOS Transconductance

Multipliers: A Tutorial, Fellow, IEEE, IEEE transactions on Circuits

and systems –II: Analog and digital Signal Processing, Vol.45, No.12,

December 1998.

[2] Barrie Gilbert, Current Mode, Voltage Mode, or Free Mode?, Analog

Integrated Circuits and Signal Processing, 38, 83–101, 2004, Kluwer

Academic Publishers. Manufactured in The Netherlands

[3] Datasheet, Rev. D, AD633, Low cost Analog Multiplier, Analog

Devices.

Documents

Design of four quadrant bipolar multiplier by Manraj Singh Gujral