Microelectronics
Chapter 2:
( Lecture 2 and 3 )
BJT and MOS Analog Multiplier
Winter 2011
1
Analog Multiplier
Objectives:
1.
2.
3.
4.
5.
6.
7.
Introduction
Revision of Simple Emitter Coupled Circuit .
Gilbert Cell As a 4-quadrant Multiplier
Complete Gilbert Multiplier without any restriction on
the I/P Signals.
The application of the Gilbert Cell to realize a Phase
Detector (PD)
Gilbert Cell as a DSB-SC Modulator
MOS Gilbert Cell
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
2
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
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
3
Introduction
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Electronics and Electrical Engineering Department
Microelectronics
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Simple Emitter Coupled Circuit
(ECC)
The Basic equation of the ECC is :
Vi1  VBE1  Vi 2  VBE 2
But in the active region, the collector
current of the npn transistor is given by:
IC  Ioe

VBE
VT
VBE  VT ln(
IC
)
Io
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
5
Simple Emitter Coupled Circuit
(ECC)
Therefore
I C1
IC 2
Vi1 V i 2 Vid  VT ln( )  VT ln(
)
I o1
Io2
For a matched transistors
I o1  I o 2
I C1
Vid  VT ln(
)
IC2
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
6
Simple Emitter Coupled Circuit
(ECC)
Therefore:
I C1
e
IC2
Vid
VT
And assuming high
(1)

I C1  I C 2  IEE
(2)
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Electronics and Electrical Engineering Department
Microelectronics
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Simple Emitter Coupled Circuit
(ECC)
From (1) and (2):
I C1 
IEE
1 e
V
 id
VT
&
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
IC 2 
IEE
1 e
Vid
VT
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Winter 2011
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Simple Emitter Coupled Circuit
(ECC)
And also:
I out
Vid
 IEE tanh(
)
2VT
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
9
Simple Emitter Coupled Circuit
(ECC)
Note That:
For
Vid  VT
The output current of the ECC is given by:
I out  K (Vid ) * ( IEE )
Therefore, the circuit can operates as a multiplier
with two inputs
Vid and
IEE
Where K is the constant of the proportionality
and is given by:
1
1
[
V
]
K 
2VT
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
10
Simple Emitter Coupled Circuit
(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:
V  V
id
T
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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Gilbert Cell As a 4-quadrant
Multiplier
The Gilbert Cell is a circuit used to overcome the disadvantages of the ECC.
Ic13
Q1
+
Ic24
Q3
Q2
Q4
Vx
+
Q5
Q6
Vy
IEE
-
The output differential current of the Gilbert Cell is defined as:
I out  ( I C13  I C 24 )  ( I C1  I C 3 )  ( I C 2  I C 4 )
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
12
Gilbert Cell As a 4-quadrant
Multiplier
The output current can be written as:
Ic13
Ic24
I out  ( I C1  I C 2 )  ( I C 3  I C 4 )
Q1
+
VX
 VX
Iout  I C 5 tanh
 I C 6 tanh(
)
2VT
2VT
Q2
Q3
Q4
Vx
+
VX
Iout  ( I C 5  I C 6 ) tanh(
)
2VT
VY
VX
Iout  IEE tanh(
) tanh(
)
2VT
2VT
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Q5
Q6
Vy
IEE
-
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Winter 2011
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Gilbert Cell As a 4-quadrant
Multiplier
Note that:
From the expression of the output current of the Gilbert Cell:
VY
VX
Iout  IEE tanh(
) tanh(
)
2VT
2VT
For
VX & VY  VT

I out  K (V X * VY )
Therefore Gilbert Cell operates as 4-quadrant multiplier because
can take positive and negative values, Where K 
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Vx &Vy
IEE
2
[
mA
/
V
]
2
(2VT )
Microelectronics
Winter 2011
14
Gilbert Cell As a 4-quadrant
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.
VCC
Vcc
Rc
Rc
- Vout +
Ic13
R
Ic13
Ic24
R
-Vout +
Ic24
Q1
+
Q2
Q3
Q4
Vx
+
Q5
Q6
Gilbert Cell
Vy
IEE
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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Gilbert Cell As a 4-quadrant
Multiplier with Output Voltage
Vout  (VCC  I C 24 RC )  (VCC  I C13 RC )
VCC
Rc
Rc
Vout  ( I C13  I C 24 ) RC
Vout  I EE RC tanh(
- Vout +
Ic13
VX
V
) tanh( Y )
2VT
2VT
Q1
+
Q2
Q3
Q4
Vx
+
Vout  K (V X * VY )
Ic24
For VX & VY  VT
Q5
Q6
Vy
IEE
Where
1
K  I EE RC /( 2VT ) [V ]
2
-
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.
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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Complete Gilbert Multiplier without
any restriction on the I/P Signals
Basic Idea
+
Vx
-
VX 1  2VT tanh 1 ( K X VX )
+
VX1
-
+
Vout
Gilbert Cell
+
Vy
-
Vout
VY 1  2VT tanh 1 ( KYVY )
+
Vy1
-
-
V X 1 2VT tanh 1 ( K X V X )
V Y 1 2VT tanh 1 ( K Y VY )
 I EE RC tanh(
) tanh(
)
2VT
2VT
Vout  K (V X * VY ) where K  I EE RC K X KY [V 1 ]
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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tanh-1 circuit
From the shown circuit:
VX 1  VBE 7  VBE 8
I1
I2
VX 1  VT ln( )  VT ln( )
Io7
I o8
Assuming Q7 and Q8 are matched
I1
VX 1  VT ln( )
I2
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
18
tanh-1 circuit
Assuming the voltage to differential current
converter produces two Output currents
given by:
I 1  I dc (1  K X V X )
&
I 2  I dc (1  K X V X )
Therefore, the output voltage of
the tanh-1 circuit is given by:
1  K X VX
VX 1  VT ln(
)
1  K X VX

VX 1  2VT tanh 1 ( K X VX )
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
19
Voltage to Differential Current
Converter
From the shown circuit, we have:
VX
I 1  I dc 
RX
VX
I1  I dc (1 
)
I dc RX
 I 1  I dc (1  K X V X )
KX 
and
Similarly
1
I dc R X
I 2  I dc (1  K X V X )
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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The application of the Gilbert Cell to
realize a Phase Detector (PD)
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.
+
Vx
-
Phase Detector
(PD)
+
Vy
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Electronics and Electrical Engineering Department
+
Vout
-
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Linearized PD
Condition on the inputs for proper
operations:
VCC
Rc
Rc
- Vout +
The amplitudes of VX & Vy  VT
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:
Ic13
Q1
+
Ic24
Q2
Q3
Q4
Vx
+
Q5
Q6
Vy
IEE
-
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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VCC
Rc
Rc
- Vout +
Ic13
Q1
+
Ic24
Q3
Q2
Q4
Vx
+
Q5
Q6
Vy
IEE
T1

V out   vout (t )dt  

0
1
[ I EE RC  I EE RC (T1   )]
T1
V out   I EE RC [2 / T1  1]
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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Gilbert Cell as a DSB-SC Modulator
Condition 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
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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MOS Multiplier: NMOS Differential
Pair
The output current of the NMOS Differential pair is given:
I out  I 1  I 2
+
Where the currents of M1 and M2
in the saturation region are given by:
K1
K1 2
2
I1 
(VGS 1  VT ) 
A
2
2
I2 
Vid
+
VGS1
I1
I2
M1
M2
+
V GS2
-
-
Iss
K2
K
(VGS 2  VT ) 2  2 B 2
2
2
And assuming M1 and M2 are matched (K1=K2=  n C ox
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
W
L )
Microelectronics
Winter 2011
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MOS Multiplier: NMOS Differential
Pair
Therefore:
I out
K 2
K
2
 ( A  B )  ( A  B)( A  B )
2
2
And
K 2
I1  I 2  ( A  B 2 )  I SS
2
Where
+
Vid
A  B  VGS 1  VGS 2  Vid
+
VGS1
I1
I2
M1
M2
+
V GS2
-
-
Iss
Note: ( A  B) 2  ( A  B) 2  2( A 2  B 2 )
I SS
( A  B)  4
 V 2 id
K
2
I out

I SS
V 2 id
( A  B)  4
( 1
)
K
4 I SS / K
V 2 id
 I SS K Vid ( 1 
)
4 I SS / K
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
27
MOS Multiplier: NMOS Differential
Pair
Note:
I out
V 2 id
 I1  I 2  I SS K Vid ( 1 
)
4 I SS / K
and
I1  I 2  I SS
Therefore:
I SS 1
V 2 id
I1 

I SS K Vid ( 1 
)
2 2
4 I SS / K
I SS 1
V 2 id
I2 

I SS K Vid ( 1 
)
2 2
4 I SS / K
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
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MOS Multiplier: NMOS Differential
Pair
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
29
MOS Multiplier: NMOS Differential
Pair
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
30
MOS Multiplier: NMOS Differential
Pair
Note: if
2 I SS
Vid 
K
The nonlinear term of output current can be neglected and the output
current is given by:
I out 
and
I SS K Vid
I SS 1
I1 

I SS K Vid
2 2
I SS 1
I2 

I SS K Vid
2 2
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
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MOS Gilbert Cell
I13
I24
The output current can be written as:
I out  I13  I 24  ( I1  I 3 )  ( I 2  I 4 )
I out  I 5 KVX  I 6 KVX
I out  K ( I 5  I 6 )VX
+
M2
M1
M3
M4
Vx
+
M5
M6
Vy
and

( I5  I6 ) 
I out
K
Vy
2
Iss
K

(VX *Vy )
2
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
32
BJT and MOS Analog Multiplier
END of Chapter 2
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
Winter 2011
33