7 Gilbert Cell & Analog Multipliers (4p)

1Lecture 7. Gilbert cell & Analog MultipliersRecommended Text: Gray, P.R. & Meyer. R.G., Analysis and Design of Analog Integrated Circuits (3rdEdition), Wiley (1992) pp. 667-681

DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 2.

Introduction

Nonlinear operations on continuous-valued analog signals are often 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 techniques forperforming multiplication and division within a monolithic integrated circuit

In analog-signal processing the need often arises for a circuit that takes two analog inputs and produces an output proportional to their product.

Such circuits are termed analog multipliers. In the following sections we examine several analog multipliers that

depend on the exponential transfer function of bipolar transistors .


The Emitter-Coupled Pair

The emitter-coupled pair, was shown in to produce output currents that were related to the differential input voltage by :

02211 =+ ibebei VVVV( )111 /ln ScTbe IIVV =( )222 /ln ScTbe IIVV =

Q1 Q2

IEEVi1+

-Vi2

Ic2

+

-

Ic1

( )TbeSc VVII /exp 111 =( )TbeSc VVII /exp 222 =

=

=

T

id

T

bebe

S

S

c

c

VV

VVV

II

II expexp 21

2

1

2

1

212121 /)()( ccFcceeEE IIIIIII ++=+= )/exp()()/exp( 121 TidcEETidcc VVIIVVII ==

Q1 Q2

IEE

Vid

Ic2

+

-

Ic1


Notes

2DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 5.



)/exp(11 TidEE

c VVII += )/exp(12 Tid

EEc VV

II +=

Q1 Q2

IEEVi1+

-Vi2

Ic2

+

-

Ic1

)/exp()()/exp( 121 TidcEETidcc VVIIVVII ==

)/exp())/exp(1(1 TidEETidc VVIVVI =+

)/exp(1)/exp(1)/exp(

1Tid

EE

Tid

TidEEc VV

IVVVVII +=+=

Q1 Q2

IEE

Vid

Ic2

+

-

Ic1


Notes




)/exp(11 TidEE

c VVII += )/exp(12 Tid

EEc VV

II +=

)2/tanh(21 TidEEccc VVIIII ==

Q1 Q2

IEE

Vid

Ic2

+

-

Ic1

The differential output current that were related to the differential input voltage by :

)/exp(1)/exp(121 TidEE

Tid

EEccc VV

IVV

IIII +++==

Lets show that

or )2/tanh()exp(1

1)exp(1

1 xxx

=+++DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 8.

Notes


Gilbert cell

First show that

)()( 2222 -x/x/-x/x/-xx - eeee ee +=

-xx

-xx

ee - ee (x) +=tanh

( )( )( ) ( )

( )( )( )( ) )2/tanh(

1111

11

11

2/2/2/2/

2/2/2/2/

2/2/2/2/2/2/

xeeee

eeee

eeeeeeee

eeee

ee

xxxx

xxxx

xxxxxx

xx

xx

xx

xx

=+++=

=++=

=+++=++

)2/tanh(1

11

1 xee xx

=++

))((22 bababa +=

xxx

xxx

xx

eeeeee

ee

===

2/2/

2/2/

2/2/ 1

)2/tanh()/exp(1)/exp(1 TidEETid

EE

Tid

EEc VVIVV

IVV

II +=+++= Therefore


Notes



This relationship is plotted => and shows that the emitter-coupled pair by itself can be used as a primitive multiplier.

Assuming Taylor series expansion

For x


Two-quadrant Multiplier

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

With the addition of more circuitry, we can make IEE proportional to a second input signal.

Thus we have

The differential output current of the emitter-coupled pair can be calculated to give

)( )(2 onBEioEE VVKI

T

onBEiidoc V

VVVKI

2)(

)(2

Q1 Q2

IEE

Vid

-

Ic2

+

-

Ic1

Q3 Q4Vi2

R

+

-


Notes


Two-Quadrant restriction

Thus we have produced a circuit that functions as a multiplier under the assumption that Vid is small, and that Vi2 is greater than VBE(on).

The latter restriction means that the multiplier functions in only two quadrants of the Vid - Vi2 plane, and this type of circuit is termed a two-quadrant multiplier.

The restriction to two quadrants of operation is a severe one for many communications applications, and most practical multipliers allow four-quadrant operation.

The Gilbert multiplier cell, shown, is a modification of the emitter-coupled cell, which allows four-quadrant multiplication.


Gilbert multiplier cell

The Gilbert multiplier cell is the basis for most integrated-circuit balanced multiplier systems.

The series connection of an emitter-coupled pair with two cross-coupled, emitter-coupled pairs produces a particularly useful transfer characteristic,.

)/exp(1 11

3T

cc VV

II += )/exp(1 11

4T

cc VV

II +=

)/exp(1 12

5T

cc VV

II += )/exp(1 12

6T

cc VV

II +=

I3

Q1

Q5 Q6Q4Q3

Q2

I4I5 I6

I35 I46

IEE

I2I1

V1

V2

IO =I35 - I46


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 Ic6 get :

)/exp(1 21

T

EEc VV

II += )/exp(1 22 TEE

c VVII +=

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

c VVVVII ++=

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

c VVVVII ++=

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

c VVVVII ++=

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

c VVVVII ++=

I3

Q1

Q5 Q6Q4Q3

Q2

I4I5 I6

I35 I46

IEE

I2I1

V1

V2

IO =I35 -I46


Notes


Gilbert cell

The differential output current is then given by

Similar:

( ) ( ) ( )546364536453 cccccccccc IIIIIIIIIII =++==

( )( ) ( )( )( ) ( ) ( ) ( ) )2/tanh(11 11 11

1111

2////

////63

1221

2121

TVVEE

VVVVVVEE

VVVVEE

VVVVEE

cc

VVeI

eeeI

eeI

eeIII

TTTT

TTTT

+=

+++=

=++++=

)2/tanh(1

11

1 xee xx

=++

( )( ) ( )( ) ( ) )2/tanh(11111 2/////54 12121 TVVEEVVVV EEVVVV EEcc VVeIee Iee III TTTTT +=++++= DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 20.

Notes


Gilbert cell

The differential output current is then given by

Where

Finally The dc transfer characteristic is the product of the hyperbolic

tangent of the two input voltages. There are three main application of Gilbert cell depending of the

V1 an V2 range:

( ) ( )( ) ( ) )2/tanh()2/tanh()2/tanh(11

)2/tanh(1

)2/tanh(1

212//

2/2/

11

11

TTEETVVEE

VVEE

TVVEE

TVVEE

VVVVIVVeI

eI

VVeIVV

eII

TT

TT

=

++=

=++=

( ) )2/tanh(1 2/63 1 TVVEEcc VVeIII T+= ( ) )2/tanh(1 2/54 1 TVVEEcc VVeIII T+=( ) ( )5463 cccc IIIII =

)2/tanh()2/tanh( 21 TTEE VVVVII =

2/ )2/tanh( 2,12,1 TT VVVV DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 22.

Notes


1. Gilbert cell as Multiplier

Thus for small-amplitude signals, the circuit performs an analog multiplication.

Practically, the amplitudes of the input signals are often much larger than VT,

An alternate approach is to introduce a nonlinearity that predictors the input signals to compensate for the hyperbolic tangent transfer characteristic of the basic cell.

The required nonlinearity is an inverse hyperbolic tangent characteristic

xVVVV TT +=


Pre-warping circuit -inverse hyperbolic tangent

We assume for the time being that the circuitry within the box develops a differential output current that is linearly related to the input voltage V1. Thus

Here Io1 is the dc current that flows in each output lead if V1 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 - lnVKIVKIV

IVKIV

IVKIVV

o

oT

s

oT

s

oT

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

=

1

111tanh2o

T IVKVV

=

2

22

1

11 oo

EE IVK

IVKII


Notes


Analog Multiplier Approach

2121

2

2

1

13 1.0 VVVVI

KIKKIV

ooEEout ==


Notes


In this case the voltage drop across resistors must be included into analysis

In this case the collector currents are

and the output differential current is

This equation has no analytical solution in form Iod=f(Vid) but the effect of resistors may be understood as negative feedback desensitising the effect of the input voltage Vid.

Emitter Degeneration

Another way to increase the range of input voltage over which emitter couple works as linear amplifier additional resistors (RE) are used in series with emitters

EoBEBEidEEBEBEid RIVVVRIRIVVV +==++ 212121 0

Vid

Q1 Q2

IEE

RE

IC1 IC2

RE( )TEodidEE

c VRIVII

/)(exp11 +=

( ) 2/)(tanh21 TEodidEEcccod VRIVIIIII ===( )TEodid

EEc VRIV

II/)(exp12 +=


Notes


Emitter Degeneration

Assuming that (Vid-IodRE)> 2VT the output is:

Thus or Therefore the maximal input voltage is And the value of emitter resistor can be chosen as

( ) TEodidEETEodidEEod VRIVIVRIVII 2/)( 2/)(tanh =TidEETEEEod VVIVRII 2/)2/1( =+

EEE

idEE

TEEE

TidEEod RI

VIVRI

VVII += 2/12/

TTEEE

id

TEEE

TEEETEEEid

TEEE

TEEEid

TEEE

TidEEEidodEid

VVRI

VVRI

VRIVRIV

VRIVRIV

VRIVVIRVIRV

22/12/1

2/2/1

)2/1

2/1(2/12/


Similar result can be achieved assuming that (Vid-IodRE)

=+=

2/ 2/

2

1

idEEC

idEEC

KVIIKVII


Notes

10


Complete Analog Multiplier

( ) ( )[ ]

XX

XXodTcc

TcTccccc

IVKIVVII

VVIVVIIIIII2)2/tanh(

)2/tanh()2/tanh(

221

22215463

=====

idYEYY

idYYod VKRI

VII 2== I3

Q1

Q5 Q6Q4Q3

Q2

I4 I5 I6

I35 I46

IYY

I2I1

VX VY

IO =I35 - I46

REY REY

Q9 Q10

IXX

REX REX

Q7 Q8

I7 I8

I9 I10

RC RC

EXXX

X

EYYY

YYY

XX

XX

YY

YYYY RI

VRI

VII

VKI

VKII == 22


Notes


Analog Multiplier Design

Design four-quadrant analog multiplier with for input voltage range 10 V

The differential output voltage is:

For linear regime: For input voltage range 10V and IXX=IYY =1 mA

211.0 VVVout =

EXXX

X

EYYY

YYY RI

VRI

VII =

EXXX

X

EYYY

YYYCC RI

VRI

VIRIRV ==

EYYYYEXXXX RIVRIV and

k 10 and k 1010V/1mA/ ==== EYXXXEX RIVR

YXYX

C VVVVmARV 1.01010

1 ==

== kmAVRC 101/10

211.0 VVVout =


Notes

11


Gilbert cell Applications

and it woks as analogue multiplier

(2) If one of the inputs of a signal that is large compared to VT, this effectively multiplies the applied small signal by a squarewave, and acts as a balanced modulator.

(3) If both inputs are large compared to VT, and all six transistors in the circuit behave as nonsaturating switches. This is useful for the detection of phase 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 TTEE VVVVII =xVVVV TT +=

12


Spectra for balanced modulator

The spectrum has components located at frequencies m above and below each of the harmonics of c, but no component at the carrier frequency c or its 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


Notes


3. Phase Detector

If unmodulated signals of identical frequency are applied to the two inputs, the circuit behaves as a phase detector and produces an output whose dc component is proportional to the phase difference between the two inputs.


Notes

13


3. Phase Detector

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

where areas A1 and A2 are as indicated. Thus

[ ]2120 1)()(21 AAtdtVV ooaverage ==

=

= 12

CEECEECEEaverage RIRIRIV

If input signals are comparable to or smaller than VT, the circuit still acts as a phase detector.

However, the output voltage then depends both on the phase difference and on the amplitude of the two input waveforms


Notes


Four-quadrant multiplier AD534

Figure shows the complete multiplier AD534 based on Gilbert cell. Four-quadrant operation is achieved by using two transconductance

pairs with the bases driven in antiphase and the emitters driven by a second V-I converter. ))(( 212121 YYXXKZZ =

xxy

z

IRRRK =


Notes

14


AD534 Basic Configuration

The basic connection for four-quadrant multiplication is used in amplitude modulation, voltage-controlled amplification, and

instantaneous power measurements.

When one of the inputs is zero,the output should also be zero, regardless of the signal at the other input. In practice, a small fraction of the other input will feed through to the

output, causing an error. This can be minimized by applying an external voltage to the X2 or

Y2 input. This basic configuration has a number of useful variations.

For instance, tying the inputs together yields the squaring function. Deriving Z1 from Vo via a voltage divider allows for scale factors other than 1/10 . Applying a signal to the Z1 terminal will cause it to be summed to the output


Notes


AD534 Applications

))(( 212121 YYXXKZZ =)()10/1( oxz VVV =

xzo VVV /10=

)()10/1( ooz VVV =

zo VV = 10


Test

Show that ( ) 10/22 yxo VVV = ))(( 212121 YYXXKZZ =4k30k10

k 101

OO

VVZ =+=

2 and 11 YXX

VVYVX +==

41.0

221.0

221.0

4

22YXYXYXYXYX

XO VVVVVVVVVVVV =+=+

+=

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7 Gilbert Cell & Analog Multipliers (4p)

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