Analog Multiplier s

7/29/2019 Analog Multiplier s

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Analog Multipliers

Recommended Text: Gray, P.R. & Meyer. R.G.,Analysis and Design of Analog Integrated Circuits (3rd

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


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Introduction

Nonlinear operations on continuous-valued analog signals areoften required in instrumentation, communication, and control-

system design. These operations include

rectification,

modulation,

demodulation,

frequency translation,

multiplication, and

division.

In this chapter we analyze the most commonly used techniquesfor performing multiplication and division within a monolithic

integrated circuit


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In analog-signal processing the need often arises for a circuitthat takes two analog inputs and produces an output proportional

to their product. Such circuits are termedanalog multipliers.

In the following sections we examine several analog multipliers

that depend on the exponential transfer function of bipolartransistors.


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The Emitter-Coupled Pairas a Simple Multiplier

The emitter-coupled pair, was shown in toproduce output currents that were relatedto the differential input voltage by :

This relationship is plotted =>and shows that the emitter-

coupled pair by itself can beused as a primitive multiplier.

)/exp(11

Tid

EEc

VVII+

=)/exp(1

2

Tid

EEc

VVII

+=

)2/tanh(21 TidEEccc

VVIIII ==

)2/(,1)2/(assumingor TidEEcTid VVIIVV =



The current IEE is actually the biascurrent for the emitter-coupled pair.

With the addition of more circuitry, we

can make IEE proportional to a secondinput signal.

Thus we have

The differential output current of theemitter-coupled pair can be calculated to

give

)( )(2 onBEioEE VVKI

T

onBEiido

c

V

VVVKI

2

)(

)(2



Two-Quadrant restriction

Thus we have produced a circuit that functions as amultiplier under the assumption that Vid is small, and that

Vi2 is greater thanVBE(on). The latter restriction means that the multiplier functions in

only two quadrants of the Vid - Vi2 plane, and this type ofcircuit is termed atwo-quadrantmultiplier.

The restriction to two quadrants of operation is a severe onefor many communications applications, and most practicalmultipliers allow four-quadrantoperation.

The Gilbert multiplier cell, shown, is a modification of theemitter-coupled cell, which allows four-quadrantmultiplication.



Gilbert multiplier cell

The Gilbert multiplier cell isthe basis for most integrated-circuit balanced multiplier

systems. The series connection of an

emitter-coupled pair with twocross-coupled, emitter-coupled pairs produces aparticularly useful transfercharacteristic,.

)/exp(11

13

T

cc

VV

II

+=

)/exp(1 1

14

T

cc

VV

II

+=

)/exp(1 1

25

T

cc

VVII

+=

)/exp(1 1

26

T

cc

VVII+

=



Gilbert cell - DC Analysis

The two currents Ic1 and Ic2 are related to V2

Substituting Ic1 and Ic2 in expressions for

Ic3 , Ic4, Ic5 and Ic6get :

)/exp(12

1

T

EEc

VV

II

+=

)/exp(12

2

T

EEc

VV

II

+=

[ ][ ])/exp(1)/exp(1 213

TT

EEc

VVVVII

++=

[ ][ ])/exp(1)/exp(1 214

TT

EEc

VVVV

II

++=

[ ][ ])/exp(1)/exp(1 215

TT

EEc

VVVV

II

++=

[ ][ ])/exp(1)/exp(1 216

TT

EEc

VVVVII

++=



Gilbert cell Applications

The differential output current is then given by

The dc transfer characteristic, then, is the product of the hyperbolic tangentof the two input voltages. The are three main application of Gilbert celldepending of the V1 an V2 range:

and it woks as multiplier

(2) If one of the inputs of a signal that is large compared to VT, thiseffectively multiplies the applied small signal by a square wave, and acts as amodulator.

(3) If both inputs are large compared to VT, and all six transistors in thecircuit behave as nonsaturating switches. This is useful for the detection ofphase differences between two amplitude-limited signals, as is required in

phase-locked loops, and is sometimes called the phase-detector mode.

( ) ( ) ( )

)2/tanh()2/tanh(21

546364536453

TTEE

cccccccccc

VVVVI

IIIIIIIIIII

=

==++==

:thenandIf(1) 21 TT VVVV


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P i i it


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Pre-warping circuit -inverse hyperbolic tangent

We assume for the time being that the circuitry within the box develops adifferential output current that is linearly related to the input voltage 7i. Thus

HereIo1 is the dc current that flows in each output lead ifV1 is equal to zero,and K1 is the transconductance of the voltage-to-current converter

11121111 and VKIIVKII oo =+=

The differential voltage developed across the

two diode-connected transistors is

Using the identity:

We get

And finally

+=

+=

111

111111111 lnln-lnVKI

VKIV

I

VKIV

I

VKIVV

o

oT

s

oT

s

oT

( ) /2x)-x)/(1(1lnxtanh-1

+=

=

1

111tanh2o

TI

VKVV

=2

22

1

11 oo

EEI

VK

I

VKII


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Complete Analog Multiplier

2121

2

2

1

13 1.0 VVVV

I

K

I

KKIV

oo

EEout ==

Gilb t ll


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Gilbert cell as aBalanced Modulator

In communications systems, the need frequently arises for the multiplicationof a continuously varying signal by a square wave.

This is easily accomplished with the multiplier circuit by applying a

sufficiently large signal directly to the cross-coupled pair.

tVtV mmm cos)( =

4/2sinwhere,cos)( 1

nn

AtnAtV nn cnc

==

=

[ ]

( ) ( )ttnttnVA

K

tntVAKtVtVKtV

ncn

ncmn

c

n

mmnmco

+=

===

=

=

coscos2

coscos)()()(

1

1

Spect a fo balanced


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Spectra for balancedmodulator

The spectrum has components located at frequencies m above and beloweach of the harmonics ofc, but no component at the carrier frequencyc orits harmonics. The spectrum of the input signals and the resulting output

signal is shown below. The lack of an output component at the carrier frequency is a very useful

property of balanced modulators. The signal is usually filtered following the

modulation process so that only the components near c. are retained

Gilbert cell as a


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Gilbert cell as aphase detector

If unmodulated signals of identical frequency coo are applied to the twoinputs, the circuit behaves as aphase detector and produces an output whosedc component is proportional to the phase difference between thetwo inputs.


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The output waveform that results is shown in Fig. 10.16c and consists of a dccomponent and a component at twice the incoming frequency. The dccomponent is given by:

where areasA1andA2are as indicated. Thus

[ ]212

0

1)()(21 AAtdtVV ooaverage

==

=

= 12

CEECEECEEaverage RIRIRIV

If input signals are comparable to or

smaller thanVT, the circuit still acts asa phase detector.

However, the output voltage thendepends both on the phase differenceandon theamplitudeof the two inputwaveforms


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Analog Multiplier s

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