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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 .
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 3.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 4.
Notes
2DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 5.
The Emitter-Coupled Pair
The emitter-coupled pair, was shown in to produce output currents that were related to the differential input voltage by :
)/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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 6.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 7.
The Emitter-Coupled Pair
The emitter-coupled pair, was shown in to produce output currents that were related to the differential input voltage by :
)/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
3DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 9.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 10.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 11.
The Emitter-Coupled Pair
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
4DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 13.
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
+
-
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 14.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 15.
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.
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 16.
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
5DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 17.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 18.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 19.
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
6DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 21.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 23.
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 +=
7DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 25.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 26.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 27.
Analog Multiplier Approach
2121
2
2
1
13 1.0 VVVVI
KIKKIV
ooEEout ==
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 28.
Notes
8DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 29.
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 +=
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 30.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 31.
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/
9DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 33.
Similar result can be achieved assuming that (Vid-IodRE)
=+=
2/ 2/
2
1
idEEC
idEEC
KVIIKVII
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 36.
Notes
10
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 37.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 38.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 39.
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 =
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 40.
Notes
11
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 41.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 45.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 46.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 47.
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.
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 48.
Notes
13
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 49.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 50.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 51.
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 =
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 52.
Notes
14
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 53.
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
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 54.
Notes
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 55.
AD534 Applications
))(( 212121 YYXXKZZ =)()10/1( oxz VVV =
xzo VVV /10=
)()10/1( ooz VVV =
zo VV = 10
DT021/4 Electronic Systems Lecture 7. Gilbert cell Applications 56.
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 =+=+
+=