ch02-analogmultiplier
TRANSCRIPT
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
Microelectronics
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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Microelectronics
Winter 2011 4
Introduction
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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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Electronics and Electrical Engineering Department
Microelectronics
Winter 2011 6
Simple Emitter Coupled Circuit
(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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Microelectronics
Winter 2011 7
Simple Emitter Coupled Circuit
(ECC)
Therefore:
T
id
V
V
C
C eI
I
2
1
And assuming high
IEEII CC 21
(1)
(2)
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Microelectronics
Winter 2011 8
Simple Emitter Coupled Circuit
(ECC)
T
id
V
VC
e
IEEI
1
1T
id
V
VC
e
IEEI
1
2
From (1) and (2):
&
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Electronics and Electrical Engineering Department
Microelectronics
Winter 2011 9
Simple Emitter Coupled Circuit
(ECC)
And also:)
2tanh(
T
idout
V
VIEEI
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Electronics and Electrical Engineering Department
Microelectronics
Winter 2011 10
Simple Emitter Coupled Circuit
(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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Microelectronics
Winter 2011 11
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:Tid VV
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Electronics and Electrical Engineering Department
Microelectronics
Winter 2011 12
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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Electronics and Electrical Engineering Department
Microelectronics
Winter 2011 13
Gilbert Cell As a 4-quadrant
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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Microelectronics
Winter 2011 14
Gilbert Cell As a 4-quadrant
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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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.
+
-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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Microelectronics
Winter 2011 16
Gilbert Cell As a 4-quadrant
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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Electronics and Electrical Engineering Department
Microelectronics
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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Electronics and Electrical Engineering Department
Microelectronics
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
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
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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Electronics and Electrical Engineering Department
Microelectronics
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.
Prof. Dr. Soliman Mahmoud & Dr. Ahmed Madian
Electronics and Electrical Engineering Department
Microelectronics
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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Electronics and Electrical Engineering Department
Microelectronics
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
Electronics and Electrical Engineering Department
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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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Electronics and Electrical Engineering Department
Microelectronics
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
MOS Multiplier: NMOS Differential
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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Electronics and Electrical Engineering Department
Microelectronics
Winter 2011 28
MOS Multiplier: NMOS Differential
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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Microelectronics
Winter 2011 29
MOS Multiplier: NMOS Differential
Pair
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Microelectronics
Winter 2011 30
MOS Multiplier: NMOS Differential
Pair
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MOS Multiplier: NMOS Differential
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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Electronics and Electrical Engineering Department
Microelectronics
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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Microelectronics
Winter 2011 33
BJT and MOS Analog Multiplier
END of Chapter 2
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Electronics and Electrical Engineering Department