ch02-AnalogMultiplier

Winter 2011 1

Microelectronics

Chapter 2:

( Lecture 2 and 3 )

BJT and MOS Analog Multiplier

Microelectronics

Winter 2011 2

Analog Multiplier

Objectives:

1. Introduction

2. Revision of Simple Emitter Coupled Circuit .

3. Gilbert Cell As a 4-quadrant Multiplier

4. Complete Gilbert Multiplier without any restriction on the I/P Signals.

5. The application of the Gilbert Cell to realize a Phase Detector (PD)

6. Gilbert Cell as a DSB-SC Modulator

7. MOS Gilbert Cell

Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian

Electronics and Electrical Engineering Department

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Winter 2011 3

Introduction

One of the fundamental building blocks in analog circuit design is the analog multiplier.

Multipliers are particularly important in communication and signal processing circuits.

Applications of Multipliers

• Nonlinear analog signal processing

• Mixing

• Phase difference detection

• Modulation and demodulation

• Frequency translation



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Introduction



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Winter 2011 5

Simple Emitter Coupled Circuit

(ECC)

The Basic equation of the ECC is :

2211 BEiBEi VVVV

T

BE

V

V

oC eII

)ln(o

CTBE

I

IVV

But in the active region, the collector

current of the npn transistor is given by:



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Winter 2011 6


(ECC)

Therefore

)ln()ln(2

2

1

121

o

CT

o

CTidii

I

IV

I

IVVVV

For a matched transistors

21 oo II

)ln(2

1

C

C

TidI

IVV



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Winter 2011 7


(ECC)

Therefore:

T

id

V

V

C

C eI

I

2

1

And assuming high

IEEII CC 21

(1)

(2)



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Winter 2011 8


(ECC)

T

id

V

VC

e

IEEI

1

1T

id

V

VC

e

IEEI

1

2

From (1) and (2):

&



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Winter 2011 9


(ECC)

And also:)

2tanh(

T

idout

V

VIEEI



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Winter 2011 10


(ECC)

)(*)( IEEVKI idout

The output current of the ECC is given by:

Therefore, the circuit can operates as a multiplier

with two inputs idV IEEand

Note That:

For Tid VV

Where K is the constant of the proportionality

and is given by:

TVK

2

1 ][ 1V



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Winter 2011 11


(ECC)

The disadvantages of the ECC as an multiplier:

1. The circuits operates as 2- quadrant multiplier

( Vid may be positive or negative but IEE must be positive)

2. The two inputs of the multiplier are not the same type

( one of the inputs is voltage Vid and the another is current)

3. The circuits operates as 2- quadrant multiplier under the

restriction:Tid VV



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Gilbert Cell As a 4-quadrant

Multiplier

Q1 Q2 Q3 Q4

Q5 Q6

IEE

+

-

Vy

Ic13 Ic24

+

-Vx

The Gilbert Cell is a circuit used to overcome the disadvantages of the ECC.

The output differential current of the Gilbert Cell is defined as:

)()()( 42312413 CCCCCCout IIIIIII



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Winter 2011 13


MultiplierThe output current can be written as:

Q1 Q2 Q3 Q4

Q5 Q6

IEE

+

-

Vy

Ic13 Ic24

+

-Vx

)()( 4321 CCCCout IIIII

)2

tanh(2

tanh 65

T

XC

T

XC

V

VI

V

VIIout

)2

tanh()( 65

T

XCC

V

VIIIout

)2

tanh()2

tanh(T

X

T

Y

V

V

V

VIEEIout



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Winter 2011 14


Multiplier

For

Note that:

From the expression of the output current of the Gilbert Cell:

)2

tanh()2

tanh(T

X

T

Y

V

V

V

VIEEIout

TYX VVV & )*( YXout VVKI

Therefore Gilbert Cell operates as 4-quadrant multiplier because

VyVx&

can take positive and negative values,Where ]/[)2(

2

2VmA

V

IEEK

T



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Winter 2011 15


Multiplier with Output Voltage

Note that : The output of the Gilbert Multiplier is a current, and if it is required

to have an output voltage, a differential current to a voltage converter is

needed. This can be realized by using the two resistors connected to Vcc as

shown.

+

-Vx

Q1 Q2 Q3 Q4

Q5 Q6

IEE

+

-

Vy

Ic13 Ic24

Rc Rc

+- Vout

VCC

Gilbert Cell

R R

+-VoutIc13 Ic24

Vcc



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Winter 2011 16


Multiplier with Output Voltage)()( 1324 CCCCCCCCout RIVRIVV

CCCout RIIV )( 2413

)2

tanh()2

tanh(T

Y

T

XCEEout

V

V

V

VRIV

)*( YXout VVKV TYX VVV &For

+

-Vx

Q1 Q2 Q3 Q4

Q5 Q6

IEE

+

-

Vy

Ic13 Ic24

Rc Rc

+- Vout

VCC

Where ][)2/( 12 VVRIK TCEE

Note that: For large signals, the Gilbert cell is not operate correctly as a

multiplier, therefore we try to make the I/P signals pass through certain

circuits that eliminate the nonlinearity of the tanh function.



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Winter 2011 17

Complete Gilbert Multiplier without

any restriction on the I/P Signals

Basic Idea

+

-

Vout

+

-

Vx

+

-

Vy

Gilbert Cell

+

-

VX1

+

-

Vy1

)(tanh2 1

1 XXTX VKVV

)(tanh2 1

1 YYTY VKVV

)2

)(tanh2tanh()

2

)(tanh2tanh(

1

1

1

1

T

YYTY

T

XXTXCEEout

V

VKVV

V

VKVVRIV

)*( YXout VVKV ][ 1 VKKRIK YXCEEwhere



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Winter 2011 18

tanh-1 circuit

Assuming Q7 and Q8 are matched

871 BEBEX VVV

)ln()ln(8

2

7

11

o

T

o

TXI

IV

I

IVV

From the shown circuit:

)ln(2

11

I

IVV TX



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Winter 2011 19

tanh-1 circuitAssuming the voltage to differential current

converter produces two Output currents

given by:

)1(1 XXdc VKII

)1(2 XXdc VKII

&

Therefore, the output voltage of

the tanh-1 circuit is given by:

)1

1ln(1

XX

XXTX

VK

VKVV

)(tanh2 1

1 XXTX VKVV



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Winter 2011 20

Voltage to Differential Current

ConverterFrom the shown circuit, we have:

X

Xdc

R

VII 1

)1(1

Xdc

Xdc

RI

VII

)1(1 XXdc VKII

and Xdc

XRI

K1

Similarly )1(2 XXdc VKII



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Winter 2011 22

The application of the Gilbert Cell to

realize a Phase Detector (PD)

Phase Detector

(PD)

+

-

+

-

Vx

Vy

+

-

Vout

Phase Detector is a circuit provide output voltage proportional to the phase

difference between two input signals.

Types of PD:

Linearized phase detector: The output is linearly proportional to the

phase difference between the two input signal.

Sinusoidal phase detector: The output is proportional to the Sin or

Cosine the phase difference between the two input signal.



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Winter 2011 23

Linearized PD

+

-Vx

Q1 Q2 Q3 Q4

Q5 Q6

IEE

+

-

Vy

Ic13 Ic24

Rc Rc

+- Vout

VCCCondition on the inputs for proper

operations:

The amplitudes of TyX VVV &

Therefore, the transistors

operate as a switch (ON/OFF)

Example: Draw the output

waveform and calculate its

average value for the input

signal waveforms shown below:



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Winter 2011 24

+

-Vx

Q1 Q2 Q3 Q4

Q5 Q6

IEE

+

-

Vy

Ic13 Ic24

Rc Rc

+- Vout

VCC

)]([1

)( 1

10

1

TRIRIT

dttvV CEECEE

T

outout

]1/2[ 1

TRIV CEEout Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian


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Winter 2011 25

Gilbert Cell as a DSB-SC ModulatorCondition on the inputs for proper operations:

The Gilbert Cell can be used as a switching modulator under the condition,

One of the input signal( Carrier ) is >> VT and the another input( modulating

Signal) is <<VT



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Winter 2011 26

MOS Multiplier: NMOS Differential

Pair

+

-

Vid

M1 M2

I1 I2

Iss

+

-VGS1

+

-V GS2

The output current of the NMOS Differential pair is given:

Where the currents of M1 and M2

in the saturation region are given by:

21 III out

212

11

12

)(2

AK

VVK

I TGS

222

22

22

)(2

BK

VVK

I TGS

And assuming M1 and M2 are matched (K1=K2= L

WCoxn )



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Winter 2011 27


PairTherefore:

))((2

)(2

22 BABAK

BAK

I out

And

Note: )(2)()( 2222 BABABA

SSIBAK

II )(2

22

21

idSS VK

IBA 22 4)( )

/41(4)(

2

KI

V

K

IBA

SS

idSS

idGSGS VVVBA 21Where

)/4

1(2

KI

VVKII

SS

id

idSSout

+

-

Vid

M1 M2

I1 I2

Iss

+

-VGS1

+

-

V GS2



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Pair

)/4

1(2

1

2

2

1KI

VVKI

II

SS

id

idSSSS

)/4

1(2

1

2

2

2KI

VVKI

II

SS

id

idSSSS

Therefore:

)/4

1(2

21KI

VVKIIII

SS

id

idSSout

SSIII 21

Note:

and



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Winter 2011 29


Pair



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Winter 2011 30


Pair



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Winter 2011 31


Pair

idSSSS VKI

II

2

1

22

Note: if K

IV SS

id

2

The nonlinear term of output current can be neglected and the output

current is given by:

idSSout VKII

and idSSSS VKI

II

2

1

21



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Winter 2011 32

MOS Gilbert Cell

+

-

+

-

Vx

Vy

M5 M6

M1 M2 M3 M4

I13 I24

Iss

The output current can be written as:

)()( 42312413 IIIIIIIout

XXout VKIVKII 65

Xout VIIKI )( 65

yVK

II2

)( 65 and

)*(2

yXout VVK

I



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Winter 2011 33

BJT and MOS Analog Multiplier

END of Chapter 2



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