1 Chapter1 Introduction 1.1 Background Due to a large amount of papers in the past 40 years before 1965. There are at least 5 methodologies for symbolic analysis [1]. It can be characterized as following. 1. The tree enumeration method 2. The signal flow graph method 3. The state variable eigenvalue method The state variable eigenvalue method discusses about how will you derive system of differential equation of KCL and Ohm’s law as a matrix form in time domain. After that use Laplace’s formula of differential equation to replace with the order of the system which transform the equation from time domain into frequency domain. Subsequently, the unknown of any order of the differential equation can be solve with inverse matrix. 4. The iterative method 5. The nodal analysis eigenvalue method. The methodologies present in this thesis may be different from nodal analysis eigenvalue method. It starting with the theory similar with Gaussian elimination but it is written in symbolic form. Subsequently, eliminate one nodal variable per equation until there no equation left in the matrix of the current matrix which can be written as nodal matrix multiplied by admittance matrix. Admittance matrix can be written in terms of small signal parameters such as drain to source conductance, parasitic capacitances, passive capacitance, passive inductance, etc. Nodal matrix is the listed of all node variables which are defined in the circuit. Usually, the left side of the equations which is current matrix which is zero, if someone do not want to derive input impedance. Then, from KCL, summation of the current flowing into the node is equal with current flowing out of the node. But it should be written with the same side so that someone can group node voltage with only one side of the equal sign, so the other side of the equal sign must be zero. Typical example can be written as following. 11 21 31 41 1 12 22 32 42 2 13 23 33 43 3 14 24 34 44 4 0 0 0 0 a a a a V a a a a V a a a a V a a a a V = (1) 11 12 13 14 21 22 23 24 44 , , , , , , , , ...., a a a a a a a a a are called coefficient of the nodal voltage. It can also be seen as admittance matrix which have 16 coefficients for four node problems.
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Transcript
1
Chapter1 Introduction
1.1 Background Due to a large amount of papers in the past 40 years before 1965. There
are at least 5 methodologies for symbolic analysis [1]. It can be characterized as following.
1. The tree enumeration method 2. The signal flow graph method 3. The state variable eigenvalue method
The state variable eigenvalue method discusses about how will you derive system of differential equation of KCL and Ohm’s law as a matrix form in time domain. After that use Laplace’s formula of differential equation to replace with the order of the system which transform the equation from time domain into frequency domain. Subsequently, the unknown of any order of the differential equation can be solve with inverse matrix.
4. The iterative method 5. The nodal analysis eigenvalue method.
The methodologies present in this thesis may be different from nodal
analysis eigenvalue method. It starting with the theory similar with Gaussian elimination but it is written in symbolic form. Subsequently, eliminate one nodal variable per equation until there no equation left in the matrix of the current matrix which can be written as nodal matrix multiplied by admittance matrix. Admittance matrix can be written in terms of small signal parameters such as drain to source conductance, parasitic capacitances, passive capacitance, passive inductance, etc.
Nodal matrix is the listed of all node variables which are defined in the circuit. Usually, the left side of the equations which is current matrix which is zero, if someone do not want to derive input impedance. Then, from KCL, summation of the current flowing into the node is equal with current flowing out of the node. But it should be written with the same side so that someone can group node voltage with only one side of the equal sign, so the other side of the equal sign must be zero. Typical example can be written as following.
11 21 31 41 1
12 22 32 42 2
13 23 33 43 3
14 24 34 44 4
0000
a a a a Va a a a Va a a a Va a a a V
=
(1)
11 12 13 14 21 22 23 24 44, , , , , , , ,....,a a a a a a a a a are called coefficient of the nodal voltage. It can also be seen as admittance matrix which have 16 coefficients for four node problems.
2
1.2 Thesis Motivation Thesis motivation is created by reading recent advance of electronic circuit in
Journal of Solid state circuits and Transactions on Circuit and Systems, IET Circuit and Devices, electronic letters compared with the references papers therein. Subsequently, it try to determine something different in the methodology of analysis of transfer function of electronic circuit. Usually, novel problem of circuit design methodology start with circuit analysis. By substituting small signal high frequency equivalent circuit of MOSFET into transistor circuit schematic. One can determine closed form transfer function easily by back substitution of nodal voltage as a function of other nodal voltage to eliminate one nodal voltage per equation.
The first motivation is when problem is more and more difficult, because the problem have more than 3 nodes. It might be interesting to derive something called map or route of the solution of back substitution or symbolic Gaussian elimination. Why does it useful? Because it is more systematic, so that the circuit designer do not duplicate back substitute the nodal voltage into other equation iteratively. Some of the electronic circuit analysis problem might have some nodal voltage which have no column duplicate with the same column, so it might be useless to substitute without eliminate one nodal voltage per equation.
The second motivation is to create novel artwork by modification of the old electronic circuit artwork with the hope that the specifications of the circuit looks better that the old circuit such as distributed amplifier, wideband amplifier with the circuit technique called inductive coupling. The process of create novel artwork is to mixed something called passive circuit such as transmission line, passive capacitor, passive resistor, passive inductor with general type of amplifier schematic such as cascade amplifier, folded cascade amplifier, regulated cascade amplifier.
The last motivation is to discuss operation of the presented electronic circuit as detail as possible by imagination and comparative study with the old paper journal which have something related with the presentation such as class of the CMOS oscillator, phase noise analysis which is still in discussion today.
3
1.3 Thesis Contribution My thesis contribution usually originate from artwork. Usually, it is drawn in
Cadence design system. Subsequently, it is redrawn in Microsoft Visio which is the most popular software in drawing electronic circuit schematic.
My first contribution is a modified regulated cascade bandpass amplifier and oscillator which is described in chapter2. The analysis and design methodology and analysis step is described in details in chapter2.
My second contribution is modified simple cross coupled oscillator with current source which is described in chapter3. The analysis and design methodology and analysis step is described in details in chapter3.
My third contribution is two stage operational amplifier with inductive compensation circuit. Analysis of the macro model of the proposed two stage amplifier. Design algorithm of the two stage amplifier with inductive compensation circuit. Equivalent output noise voltage of the presents circuit is described in chapter4.
My fourth contribution is power spectrum of simple cross coupled oscillator by impedance parameter analysis which is described in chapter5.
My fifth contribution is analysis methodology of the circuit which has more than three nodes. Usually, it is difficult to solve circuit which have more than three nodes. But this thesis presents analysis algorithm which is based on symbolic Gaussian elimination which is ideal systematic step. It is not software but it is written derivation report. Currently, the author present how to solve nine node problems which has approximately 47 pages of solution. But without direct electronic circuit analysis method by Kirchhoff’s current law and Ohm’s law and by grouping of nodal voltages in the circuit. The report is useless except to solve for the ratio of the real number instead of complex number as a function of frequency after substitute small signal parameters into the matrix. Another report which should be solved in the future is 12 nodes problem which is the proposed two stage CMOS complementary distributed amplifier.
4
Chapter2 Modified Regulated Cascode Bandpass Amplifier and Oscillator
2.1 Introduction of the oscillator
Usually, CMOS oscillator composed of second order resonance circuit. One of the most famous circuit is simple cross couple oscillator which have two, three, four or five transistors. The circuit can act as bandpass amplifier and oscillator at the same time when the solution of two pole positions as a function of current consumption can be conjugate imaginary pole. It is called natural frequencies.
The proposed oscillator can be drawn by accidentally modified the regulated cascode bandpass amplifier. It is well known that regulated cascode amplifier composed of three transistors. But the proposed modified version is different as following. By connecting gate of input transistor with the cascode transistor. So that gate souce voltage of both transistor has approximately similar value, eventhough it has some error between drain source voltage drop of both two transistors. The proposed figure and its small signal equivalent circuit can be drawn below.
1M
2M3M
BRLR
AR
CR
LCLL
inVCR
ARBR
LR
LL
LC
1dsg
1dsg
1 1m gsg V
1 1m gsg V
3 3m gsg V
2gsC
2gdC1dbC
1gdC
1gsC
3gsC
3gdC
3dsg3dbC
outV
outV
Fig.2.1 Modified Regulated cascade bandpass amplifier and oscillator
Fortunately, after analyzed this circuit, it can be found that this circuit can oscillate as sinusoidal signal at terahertz frequency. The solution can be rewritten here for convenience without derivation in details.
5
2.1.1 Periodic steady state (PSS) of modified regulated cascade BPF and oscillator
Periodic steady state means that special dc operating point which could not be moved as a function of time because it is dc offset of the oscillator circuit. In contrast with dc operating point meaning because dc operating point is voltage is constant as a function of time.
Class of this type of oscillator should be class B instead of class C or class D because it has dc voltage head room for negative signal 2Vds of input transistor and cascade transistor [1]. Its dc offset can also be tuned by adaptive resistor biasing RC and Ra. It should guess that negative signal is practical only if someone use negative power supply.
2.2 The Analysis algorithm of implementation in MATLAB of the proposed circuit
2.2.1 Algorithm of Polynomial Multiplication
First Step Multiply polynomial in the two brackets from the highest order of the first bracket to the highest order of the second brackets
1 2 1 21 2 0 1 2 0... ...n n n n n n
n n n n n na s a s a s a b s b s b s b− − − −− − − − + + + + + + + +
(2.1)
Second Step Reduce order to the next lower order or shift the multiplier term of the first bracket to the right one order, then multiply with the highest order of the second bracket
Third Step repeat step second, until the last term of the first bracket
Fourth Step repeat the first step, but reduce order of the second bracket to the next lower order in the polynomial.
Fifth Step repeat step four, until the last term of the second bracket
2.2.2 Algorithm of Grouping of coefficient from polynomial multiplication
First Step Coefficients in front of s parameter are small signal parameters of interest
Second Step Define the name of the new coefficients which are not duplicate with any group of the small signal parameters in the circuit, the name can be English alphabet or Greece alphabet
Third Step Subscript of the name of the new coefficient can have at least one number from 1 to 9. Its meaning of the first subscript is the order of the polynomial
Fourth Step 2nd number of the name of the new coefficient can have at least one number from 1 to 9. Its meaning of the second subscript is the name of the new coefficient which is not duplicated with other name which you created.
6
The design algorithm which implement in MATLAB has step as following
1. Assign all current value in the circuit 2. Assign physical constant of the CMOS process as following
The typical value is 0.5 micron from textbook of Sedra and Smith [2] can be referred to Appendix A
99.5 10 oxide thicknessoxT m−= × =
(1) ( )8 2460 10 / sec mobility of NMOSUon cm V carrier= × × =
(2) ( )8 2115 10 / sec mobility of PMOSUop cm V carrier= × × =
(3) 113.45 10 /oxide F mε −= ×
(4) 15
2Oxide Capacitance =3.63 10oxFCmµ
−= ×
(5) min 0.5 minimum gate length of processL mµ= =
(6) 0.7 threhold voltage of NMOStonV V= =
(7) 0.8 threhold voltage of PMOStopV V= − =
(8) 1/20.5 [V ] body effect parameter of NMOS threshold voltagegamman γ= = =
(9) 1/20.45 [V ] body effect parameter of PMOS threhsold voltagegammap γ= = =
(10) 0.8 [ ] 2 surface inversion potential of NMOSFphin V φ= = =
(11) 0.75 [ ] 2 surface inversion potential of PMOSFphip V φ= = =
(12) ox
ox
kn Uon Ckp Uop C
= ×= ×
(13) 60.08 10 lateral diffusion into the channel from source to drain diffusion regions of NMOSLovn m−= × =
(14)
60.09 10 lateral diffusion into the channel from the source to drain diffusion regions of PMOSLovp m−= × =
(15)
7
min
min
22
effN
effP
L L LovnL L Lovp
= − ×
= − ×
(16) 1 2 30, 1, 0sbn sb sbV V V= = =
(17) ( )( )( )( )( )( )
1 1
2 2
3 3
2 2
2 2
2 2
thn ton n f sbn f
thn ton n f sbn f
thn ton n f sbn f
V V V
V V V
V V V
γ φ φ
γ φ φ
γ φ φ
= + + −
= + + −
= + + −
(18)
11 / 1
MJdb
db aV
C CJ ADPB
= × +
(2.1)
( ) 11 / 1
MJSWdb
db bV
C CJSW PDPB
= × +
(2.2) 2
3 3gd gda C C=
(2.3)
( ) ( ) ( )22 2 2 2 2 3 2 3 2 3 2 3 2mb m ds gd db gs gd gd gd m gd ds ma g g g C C C C C C g C g g = − − − + + + + +
(2.4)
( ) ( )( )
2 2 2 2 3 2 3 21
2 3 2 2 2 2 2 3
mb m ds m db gd gd gd
gd m m mb m ds gd ds
g g g g C C C Ca
C g g g g g C g
− − + + + = + − − −
(2.5)
( )0 2 2 2 2 31
mb m ds m dsB
a g g g g gR
= − − +
(2.6)
( )( )3 2 2 3 2 3 2L gd db L db gs gd gdb L C C C C C C C= + + + + +
(2.7)
8
( )
( )
2 2 3
2
3 2 3 2 2 2 2
1
1
L gd db L dsB
L db gs gd gd ds L m gdL
L C C C gR
bL C C C C g L g C
R
+ + +
= + + + + + +
(2.8)
1 2 31 1
L ds dsL B
b L g gR R
= + +
(2.9)
( )0 3 3 2 3 21
ds db gs gd gdB
b g C C C CR
= + + + + +
(2.10)
2.3 Silicon Inductor Design Consideration
From [3], it can be concluded that there are at least 4 types of geometry which can be implemented on substrate to form inductance. They are square, hexagonal, octagonal and circular. It can be seen from reference that the circular shape have the highest quality factor, the second in quality factor is octagonal, the third in quality factor is hexagonal and the last is square. So the circuit designer can design silicon inductor according to many shapes but it is a little bit different less than 30 percent from square and circular shape. Thus, you should choose circuit shape because it has maximum quality factor.
Quality factor of silicon inductor can have at least two definition. From circuit theory point of view, it can be seen from equivalent circuit which can be extracted from experimental results. Quality factor of this view can be seen as imaginary part of input impedance of equivalent circuit divided by real part of equivalent circuit.
Second definition of quality factor can be described as a peak magnetic energy multiply by 2π divided by energy loss in one oscillation cycle.
It can discuss about three methodologies to design silicon inductor with equation. The first methodology is modified Wheeler formula
2
1 021
avgMW
n dL K
Kµ
ρ
= +
(2.3.1) 7
0 4 10 / permeability of free spaceH mµ π −= × =
1 2, layout dependent constantK K =
total turn of silicon inductorn =
( ) ( )1 fill factor= ; 0.1 0.9
nw n sl
ρ ρ+ −
= < <
2in out
avgd dd +
=
For square silicon inductor, if someone want to design 1 nanohenry with modified Wheeler how can he approximate , avgdρ
( )( ) ( )( )
( ) ( )
42 139 7
1 02
4 4
4
6
300 10 8821.59 101 10 2.34 4 101 1 2.75 1 2.75
1 2.75 8821.59 10 8821.59 10 1 2.75 0.9 2.475
3.475 3.938821.59 10
1 2.75 8821.59 10 8821.59 10
avgMW
nn d nL KK
n n
n
n n
µ πρ ρ ρ
ρ
ρ
− −− −
− −
−
−
× × = = × = × = + + +
+ = × → × − = =
= =×
+ = × → ×( ) ( )6
4
1 2.75 0.1 0.275
1.275 1.448821.59 10
n
−
−
− = =
= =×
(2.3.2)
10
( ) ( ) ( )( )
( )
6 6
5 55
3.93 14 10 2.93 4 101 0.9=
5.502 10 1.172 107.415 10
0.9
nw n sl l
l
ρ− −
− −−
× + ×+ −= =
× + ×= = ×
(2.3.3)
The second methodology is based on current sheet approximation, these method is based on many concepts such as geometric mean distance (GMD), arithmetic mean distance (AMD) and arithmetic mean square distance (AMSD). The closed formed formula can be written as following.
21 22
3 4ln2
avgGMD
n d c cL c cµ
ρ ρρ
= + +
(2.3.4)
For square silicon inductor, if someone want to design 1 nanohenry with GMD. It can be shown as a typical example below
( ) ( )( )
[ ]
7 2 62 9
13 2 9
42
4 10 300 10 1.27 2.07ln 0.18 0.13 1 102
if 0.9
2393.89 10 0.8329 0.162 0.1053 10
10 3.7968 1.9485 22633.7577
GMD
GMD
nL
L n
n n
πρ ρ
ρ
ρ
− −−
− −
× × = + + = × =
= × + + =
= = → = ≈
(2.3.5)
The third methodology is data fitted monomial expression, it has five physical variables in this model, and five fitting parameters, it can be rewritten here below
3 51 2 4mono out avgL d w d n sα αα α αβ=
(2.3.6)
For square silicon inductor, if someone want to design 1 nanohenry with this formula, it can be shown as a typical example below
2.4 Transmission Line Inductor design based on continue fraction expansion
Transmission line inductor design can be design with well known lossy transmission line which is hyperbolic tangent function of characteristic impedance and length of the transmission line. This equation can be rewritten as following
(2.4.1)
For ideal short circuit termination, then 0LZ = , as a result equation (2.4.1) can be rewritten as following
(2.4.2)
(2.4.3)
(2.4.4)
(2.4.5)
12
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
3 5 3 5
0 02 4 2 4
2 ... ...3! 5! ! 3! 5! !
2 1 ... 1 ...2 4! ! 2! 4! !
n odd n odd
in n even n even
l l l l l ll l
n n R j LZ Z Zl l l l l l
n n
γ γ γ γ γ γγ γ
ω γγγ γ γ γ γ γ
= =
= =
+ + + + + + + +
+ = = = + + + + + + + +
( )( ) ( ) ( ) ( )
( ) ( ) ( )
2 4
2 4
1 ...3! 5! !
1 ...2! 4! !
n even
n even
l l ln
ll l l
n
γ γ γ
γ γ γ
=
=
+ + + + + + + +
(2.4.6)
(2.4.7)
( )( ) ( ) ( ) ( )
( ) ( ) ( )
2 4
2 4
1 ...3! 5! !
1 ...2! 4! !
n even
in n even
l l ln
Z Rl j Lll l l
n
γ γ γ
ωγ γ γ
=
=
+ + + + = + + + + +
13
Chapter3 Modified Simple Cross coupled oscillator with current source
3.1 Introduction to simple cross coupled oscillator
Simple cross coupled oscillator appeared in literature after 1990. It is very popular type of oscillator inside phase locked loop system. Its design equation is well known to the engineering communities since 1998 [1].
3.2 Analysis of the simple CMOS cross couple oscillator
The analysis and design philosophy of simple CMOS cross couple oscillator have two philosophies since paper of Nhat Nguyen [?]. The first methodology is based on negative resistance concept. By deriving input impedance of CMOS cross couple oscillator we can determine symbolic formula of input resistance and input reactance of the circuit as a function of input frequency. Without crystal oscillator in phase locked loop block diagram, input frequency is not existed.
(b)Transfer function of simple cross coupled Oscillator
Gain stage transfer function can be derived as following
( )( )
gd m
gd ds
sC g sLVA
V Ls C C L s g L
R
−= =
+ + + +
2 2 22
1 2 22 2 2 2 2
2
1
(3.2.10)
16
Feedback stage transfer function can be derived as following
( )( )
gd m
gd ds
sC g sLV
V Ls C C L s g L
R
β−
= =
+ + + +
1 1 11
2 2 11 1 1 1 1
1
1
(3.2.11)
From feedback model concept, the ideal transfer function should be written as following
( )( )
( )( )
( )( )
gd m
gd ds
in
gd m gd m
gd ds gd ds
sC g sL
Ls C C L s g L
RV A
V A
sC g sL sC g sL
L Ls C C L s g L s C C L s g L
R R
β
−
+ + + +
= =+
− − + + + + + + + + +
2 2 2
2 22 2 2 2 2
22
1 1 1 2 2 2
2 21 21 1 1 1 1 2 2 2 2 2
1 2
1
1
1
1 1
(3.2.12)
17
3.3 Analysis of the modified simple cross couple oscillator
This schematic is different from simple cross coupled oscillator because there are additional two resistors which connected between RLC resonance circuit and drain terminal of the simple cross coupled oscillator. There are also have NMOS current source connected between source terminals of both two input transistors. Its current can be tuned by adapt voltage reference externally to tune oscillating frequency of its modified cross coupled oscillator.
1L
2L
1R
2R
1C2C
1M
3M2M
DDV
3R4R
2L
1R
1L
1C 2R2C
3R4R
3gsC2gsC
2gdC
1gdC
3gdC
1dsg
2dsg
3dsg2 2m gsg V 3 3m gsg V
inV
inI
2 2mb bsg V 3 3mb bsg V
Fig.3 (a) modified simple cross couple oscillator (b) its equivalent circuit and its input
impedance source is connected to input of the transistor
18
3.3 Phase noise discussion of the CMOS oscillator
Phase noise can be understood by considering power spectrum. There should have no phase noise for oscillator when the frequency of oscillation is at center frequency. Phase noise usually defined by measure power spectral density of output mean square noise divided by power of carrier signal at phase offset from center frequency. Usually, it can be assume that it has amplitude distortion as a result of self modulation of amplitude due to signal feedback from drain terminal to gate terminal as a typical case of simple cross coupled oscillator. Another case can be seen in simulation results in chapter2 of modified regulated cascode oscillator.
Second reasonable prove is based on flicker noise up conversion due to amplification and modulation of low frequency flicker noise. Which should be prove with mathematics in the ref [1].
Third reasonable prove is based on percentage error of power supply which make current flow into the circuit as constant as possible otherwise the center frequency or frequency of oscillation is fluctuating up and down randomly. The conclusion here is phase noise can be written as a function of power supply fluctuation.
19
Chapter4 Two stage operational amplifier with inductive compensation circuit
4.1 Introduction to two stage operational amplifier (op-amp)
Two stage CMOS operational amplifier is one of the most famous circuit in operational amplifier. Its existence is before 1982. It can be use as buffer circuit, switched capacitor filters, op-amp Wien Bridge Oscillator, second order continuous time filter, etc. It has connection of at least seven transistors in the circuit. Usually, it use compensation circuit which composed of series capacitor and resistor. Resistor in compensation circuit can be implemented with mosfet in triode region. But the author have idea to replace the compensation circuit with passive inductor with the hope to extending open loop bandwidth of the two stage CMOS op-amp. Figure4.1 is drawn to shown two stage op-amp with capacitive compensation circuit
1M 2M
3M 4M
5M
6M
7M
LC
inV +inV −
outV
DDV
SSV
inV1m ing V
1outR
2outR2 1m outg V
1outC
2outC
1outVprobeZ
outV
CC
CC
( )a
( )b Fig. 4.1 Two stage operational amplifier with capacitive compensation circuit
(a) Transistor diagram (b) ideal macro model
The figure below two stage op-amp in fig. 4.1 is ideal macro model of two stage op-amp with capacitive compensation circuit.
20
4.2 Analysis of the macro model of two stage op-amp with inductive compensation circuit
1M 2M
3M 4M
5M
6M
7M
LC
inV +inV −
CLoutV
DDV
SSV
inV1m ing V
1outR
2outR
CL
2 1m outg V
1outC
2outC
1outVprobeZ
outV
( )a
( )b Fig 4.2 Two stage operational amplifier with inductive compensation circuit
(a) Transistor diagram (b) ideal macro model
The closed form formula of two stage op-amp with inductive compensation circuit
was derived as following formula
( )
2 21 1 2 1 1
4 3 1 1 11 1 1 2 1 1 1 2
2 1 1
2 1 1 11 1 1 2 1 2
1 2
1 1m C m m C
probe probeout
in C C CC C C C
out in out
C C CC C C m
probe out out
s g L g s g LZ ZV
V L L Ls L C L C s L C L Cr Z r
L L Ls L C L C L gZ r r
− − + − = −
+ + +
+ + + + −
1 1 1
1 2
2C C C
probe out out
L L LsZ r r
+ + + +
(4.1)
As can be seen from fig. 4.2 (b), there are two voltage controlled voltage source
To represent two stage op-amp. Two output conductances to represent output conductance of first stage amplifier and second stage amplifiers. Two output capacitances to represent output capacitances of the first stage and second stage amplifier. Output capacitances can be seen as the lump of parasitic of the output node of the first stage and second stages. Such as 1 4 6 4db gs gdC C C C= + + is output capacitances of the first stage amplifier and 2 6 7db L dbC C C C= + +
21
From simulation results, two-stage op-amp with inductor coupling compensation circuit. It can be seen that the magnitude response have bandpass response. It can be seen as below.
Fig4.2 Magnitude and phase response when C1 is 5 pF.
From fig.4.2, it can be seen that center frequency is designed to be 3.0GHz at voltage gain equal to 0.486 dB for capacitive load equal to 5pF. Drain current consumption at the first stage is 2 microamperes. Drain current consumption at the second stage is 5 microampere. -3db frequency on the left side of center frequency is 2.82 GHz at -2.48dB. -3dB frequency on the right side of center frequency is 3.36 GHz at -2.48 dB. Consequently, quality factor is calculated to be approximately 6.0
Fig. 4.3 Magnitude and phase response when C2 is 15 pF
From fig.4.3, it can be seen that center frequency is designed to be 1.8 GHz at voltage gain equal to 0.003 dB for capacitive load equal to 15pF. Drain current consumption at the first stage is 2 microamperes. Drain current consumption at the second stage is 5 microampere. -3db frequency on the left side of center frequency is 1.71 GHz at -3.12dB. -3dB frequency on the right side of center frequency is 1.93 GHz at -3.06 dB. Consequently, quality factor is calculated to be approximately 6.0
Chapter5 CMOS Distributed Amplifier Analysis and Design based on Complementary Regulated Cascode amplifier
5.1 Introduction The first paper in distributed amplifier was published since 1948 [1] in the
proceeding of the I.R.E. The connection between traveling wave tubes (TWT) is called section which is coupled by inductor at the grid terminal which is shown in fig 5.1
Another connection of traveling wave tubes is at the plate terminal which is also coupled by inductor. It is called stage when the plate terminal of traveling wave tube is coupled by series capacitor and inductor.
inVgC gC gC gC gC
gC
gL gL gL gL
pLpLpL pLpCpC pC pC pC
B + B +4
4
output
3
3
1 2
21 Fig 5.1 Basic distributed amplifier based on TWT
Complementary regulated cascode amplifier (CRGC) was proposed by B.
J. Hosticka since 1979 [2]. Since the time it composed of at least 8 transistors. Its experimental result used CMOS array MC14007B. It consume current 1 mA. Its DC gain is 2300 times of the input signal and its 3dB frequency is 5.5 kHz.
The author have idea to used this amplifier architecture because it is high voltage gain architecture. Its circuit is redrawn below. It is different from original idea of [2] because drain node of the NMOS and PMOS regulated transistor which is the cascaded stage of the input transistor is connected with current mirror.
(b) Small signal High Frequency Equivalent circuit of (a)
KCL at node input
(5.2.5)
Grouping coefficients (small signal parameters) which has the same node voltage
(5.2.6)
KCL at node V1
(5.2.7)
Grouping coefficients (small signal parameters) which has the same node voltage
(5.2.8)
( ) ( ) ( )( ) ( )1 1 2 2 2 1 1 2 2
1 1 2 2 2
2 3 1 1 2
in gd m m gs x x out ds
x ds ds m mb
x gs db gd gs
V sC g V g sC V g s C V g
g g g g gC C C C C
− + + = + +
= + + +
= + + +
( ) ( ) ( ) ( ) ( )
( )1 1 2 1 2 2 2 1 2 1 1 2
1 1 1 3 1
0in gd gs m mb out ds
m in ds gs db
V V sC V V sC g V V g V V V g
g V V g s C C
− + − + − + − + −
= + + +
( ) ( ) ( )1 3 6 1 1
1 6 6 1 1
in in x gd gd
x gs gd gs gd
I V s C V sC V sC
C C C C C
= − − = + + +
( ) ( ) ( ) ( )6 3 6 1 1 10in in gs in gd in gs in gdI V sC V V sC V sC V V sC+ − = − + + −
27
( ) ( )( )( ) ( )
6 6 3 4 5 5 6 6 6 4 4 4
4 4 4 5 4
in gd m gs gd gs db gd ds ds m mb
m gs gd out ds
V sC g V s C C C C C g g g g
V g s C C V g
+ = + + + + + + − −
+ − + −
( ) ( ) ( )( ) ( )6 6 3 4 5 4 4 4 5 4
4 4 5 5 6 6
5 6 4 4 4
in gd m x x m gs gd out ds
x gs gd gs db gd
x ds ds m mb
V sC g V sC g V g s C C V g
C C C C C C
g g g g g
+ = + + − + −
= + + + +
= + − −
( ) ( ) ( )
( )( ) ( )
4 5 5 3 8 4 3 4 4 5
4 8 4 8 8 5 4 42
0
1
ds m m gs gd
ds gs db db out gdB
V g g V g V V V s C C
V g V s C C C V V sCR
− + + + − +
= + + + + + −
KCL at node Vout
(5.2.9)
Grouping coefficients (small signal parameters) which has the same node voltage
( ) ( )
( ) ( ) ( )( )4 4 4 3 4 4 4
1 2 2 2 2 2 2 2 4 2 4 4 2
gd m ds m mb
m mb ds m gd out ds ds db db gd gd
V sC g V g g g
V g g g V g sC V g g s C C C C
+ + − −
= − + + + − + + + + + +
(5.2.10)
( ) ( )( ) ( ) ( )( )
4 4 4 3 2
1 3 2 2 2 4 3
3 2 4 4 2
2 4 4 4
3 2 2 2
4 2 4
gd m x
x m gd out x x
x db db gd gd
x ds m mb
x m mb ds
x ds ds
V sC g V g
V g V g sC V g s C
C C C C C
g g g gg g g gg g g
+ +
= − + − + +
= + + +
= − −
= + +
= +
(5.2.11)
KCL at node V3
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( )
3 6 6 3 6
3 4 4 5 4 4 3 4 3 3 4 3 5 6
0 0
0
in gd m in ds
gs gd m mb out ds gs db
V V sC g V V g
V V s C C g V V g V V V g V s C C
− + − + −
= − + + − + − + − + +
(5.2.12)
Grouping coefficients (small signal parameters) which has the same node voltage
(5.2.13)
(5.2.14)
KCL at node V4
(5.2.15)
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )
4 4 4 4 3 4 3 3 4 2 2
2 2 1 2 1 1 2 2 4
0
0out gd m mb out ds out gd
m mb out ds out db db
V V sC g V V g V V V g V V sC
g V V g V V V g V s C C
− + − + − + − + −
= − + − + − + +
28
( )( ) ( )
( ) ( ) ( )
7 2 2 7 7 3 2 7 3 1 2 31
2 2 2 1 2 3
1
0
m gs db db ds m dsB
out gd gs gd
g V V s C C C V g g V V gR
V V sC V V s C C
+ + + + + + +
+ − + − + =
( )
( )( ) ( )
2 7 7 3 7 7 3 2 2 31
1 3 2 3 2
1m ds ds gs db db gd gs gd
B
m gs gd out gd
V g g g s C C C C C CR
V g s C C V sC
+ + + + + + + + +
= − + + +
( )( ) ( )1 3 2 3 22
7 6
6 7 7 3 2 2 3
7 7 7 31
1
m gs gd out gd
x x
x gs db db gd gs gd
x m ds dsB
V g s C C V sCV
g sCC C C C C C C
g g g gR
− + + +=
+
= + + + + +
= + + +
( )( ) ( )[ ]
3 8 4 5 44
6 5
5 8 8 5 4 5 4
6 8 5 81
m gs gd out gd
x x
x gs db db gs gd gd
x ds ds mB
V g s C C V sCV
g sCC C C C C C C
g g g gR
+ + +=
+
= + + + + +
= + + −
( )( )8 4 5
16 5
m gs gd
x x
g s C CH s
g sC
+ +=
+
Grouping coefficients (small signal parameters) which has the same node voltage
( )( )( )
( )8 5 823 8 4 5 4 4
8 8 5 4 5 4
1ds ds m
Bm gs gd out gd
gs db db gs gd gd
g g gRV g s C C V V sCs C C C C C C
+ + − + + = −
+ + + + + + (5.2.16)
( )( ) [ ] ( )3 8 4 5 4 6 5 4m gs gd x x out gdV g s C C V g sC V sC+ + = + −
(5.2.17)
(5.2.18)
KCL at node V2
(5.2.19)
Grouping coefficients (small signal parameters) which has the same node voltage
(5.2.20)
(5.2.21)
Intermediate transfer function can be define to make the path to finish derivation shorter.
(5.2.22)
29
( ) 42
6 5
gd
x x
sCH s
g sC=
+
(5.2.23)
( )( )3 2 3
37 6
m gs gd
x x
g s C CH s
g sC
− + +=
+
(5.2.24)
( ) 24
7 6
gd
x x
sCH s
g sC=
+
(5.2.25)
( ) ( )( )5 1 2 3 2 2x x m gsH s g sC H s g sC= + − +
(5.2.26)
( )( ) ( )3 2 3
5 1 2 2 27 6
m gs gdx x m gs
x x
g s C CH s g sC g sC
g sC
− + + = + − + +
(5.2.26b)
( )( )( ) ( )( )( )
( )1 2 7 6 3 2 3 2 2
57 6
x x x x m gs gd m gs
x x
g sC g sC g s C C g sCH s
g sC
+ + − − + + +=
+
(5.2.26c)
( )
( ) ( )
( )( ) ( )( )( )
21 7 2 7 6 1 2 6
23 2 2 3 2 2 3 2 3 2
57 6
x x x x x x x x
m m gs gd m gs m gs gd gs
x x
g g s C g C g s C C
g g s C C g C g s C C CH s
g sC
+ + +
− − + + − + +=
+
(5.2.26d)
( ) ( )( ) ( )
( ) ( )( )
211 11 11
57 6
11 2 3 2 2 6
11 2 7 6 1 2 3 2 2 3
11 1 7 3 2
x x
gs gd gs x x
x x x x gs gd m gs m
x x m m
s a sb cH sg sC
a C C C C C
b C g C g C C g C g
c g g g g
+ +=
+
= + −
= + − + −
= +
(5.2.26e)
30
( ) ( )( )6 2 4 2 2ds m gsH s g H s g sC= − +
(5.2.27)
( ) ( )26 2 2 2
7 6
gdds m gs
x x
sCH s g g sC
g sC
= − + +
(5.2.27b)
( )2
2 2 2 26 2
7 6
gd gs gd mds
x x
s C C sC gH s g
g sC
+ = − +
(5.2.27c)
( )( )2 2
2 2 6 2 2 2 2 7 21 11 016
7 6 7 6
21 2 2 11 6 2 2 2 01 2 7, ,
gd gs x ds gd m ds x y y y
x x x x
y gd gs y x ds gd m y ds x
s C C s C g C g g g s C sC gH s
g sC g sCC C C C C g C g g g g
− + − + − + += =
+ +
= = − =
(5.2.27d)
( ) ( )( )7 1 4 4 2gd m xH s H s sC g g= + −
(5.2.28)
( ) ( )( )8 4 3 2 4 4x x gd mH s g sC H s sC g= + − +
(5.2.29)
( ) ( ) ( )( )9 4 5 1 4 4 5x x m gs gdH s sC g H s g s C C= + + − +
(5.2.30)
( )( ) ( )( )( )
( )1 1 3 2 2 3
105
gd m m gd xsC g H s g sC gH s
H s
− − −=
(5.2.31)
( ) ( ) ( )( )( ) ( )( )( )
( )6 3 2 2 3
11 8 4 2 25
m gd xm gd
H s H s g sC gH s H s H s g sC
H s
− −= − − −
(5.2.32)
( ) ( ) ( ) ( )( )( ) ( )( )
712 11 2 4 4 5 4
9m gs gd ds
H sH s H s H s g s C C g
H s
= + − + −
(5.2.33)
31
( )( ) ( )
( ) ( )6 6 713 10
9
gd msC g H sH s H s
H s
+= −
(5.2.34)
( ) ( ) ( )
2 2 2 26 6 6 1 1 1
14 19 5
gd gd m gd gd mx
s C sC g s C sC gH s sC
H s H s
+ − = − −
(5.2.35)
( ) ( )( )
( )( )
( ) ( )( )( ) ( )2 4 4 5 41 613
15 612 5 9
m gs gd dsgdgd
H s g s C C gsC H sH sH s sC
H s H s H s
− + − = −
(5.2.36)
( ) ( )14 15
1inin
in
VZI H s H s
= =+
(5.2.37)
After finished closed form derivation of the proposed input impedance equation. It can be seen that equation (5.2.37) is still not in polynomial form. Thus, it can be substituted from top down to bottom of the procedure of derivation as following.
( ) ( )( )7 1 4 4 2gd m xH s H s sC g g= + −
(5.2.28)
Substitute equation (5.2.22) into equation (5.2.28) as following
( )( ) ( )8 4 5
7 4 4 26 5
m gs gdgd m x
x x
g s C CH s sC g g
g sC
+ + = + −
+
(5.2.38)
( )2
22 12 027
6 5
y y y
x x
s C sC gH s
g sC
+ +=
+
(5.2.39)
( )( )
22 4 5 4
12 4 5 4 4 8 5 2
02 8 4 2 6
,y gs gd gd
y gs gd m gd m x x
y m m x x
C C C C
C C C g C g C g
g g g g g
= +
= + + −
= −
(5.2.40)
Substitute equation (5.2.23) into (5.2.29), we got
32
( )2
23 13 038
5 6
y y y
x x
s C sC gH s
sC g+ +
=+
(5.2.41)
23 3 5
13 3 6 6 4 4 4
03 4 6
y x x
y x x x x m gd
y x x
C C C
C C g C g g C
g g g
=
= + −
=
(5.2.42)
Substitute (5.2.22) into (5.2.30)
( )2
24 14 049
6 5
y y y
x x
s C sC gH s
g sC+ +
=+
(5.2.46)
( )( )( )
224 4 5 4 5
14 4 6 5 5 4 5 4 8
04 5 6 8 4
y x x gs gd
y x x x x gs gd m m
y x x m m
C C C C C
C C g C g C C g g
g g g g g
= − +
= + + + −
= +
(5.2.47)
Substitute ( )3H s from equation (5.2.24) and ( )5H s from equation (5.2.26e) into equation (5.2.31), we got
( )( )( )
( )
21 1 25 15 057 6
10 26 711 11 11
gd m y y yx x
x x
sC g s C sC gg sCH ssC gs a sb c
− + + += ++ +
(5.2.50)
( )( )( )
( )
25 2 3 2
15 2 3 2 3 2 6 3
05 3 2 3 7
y gs gd gd
y gd m gs gd m x x
y m m x x
C C C C
C C g C C g C g
g g g g g
= − +
= + + −
= − +
(5.2.51)
From equation, it can be seen that there are terms in numerator and denominator which can be cancelled, after that you can multiplied the two brackets of polynomial.
( )3 2
36 26 16 0610 2
11 11 11
y y y ys C s C sC gH s
s a sb c
+ + + = + +
33
(5.2.52)
36 1 25
26 1 15 1 25
16 1 05 1 15
06 1 05
y gd y
y gd y m y
y gd y m y
y m y
C C C
C C C g C
C C g g C
g g g
=
= −
= −
= −
(5.2.53)
From equation (5.2.32), it can be seen that there are five polynomials which are called intermediate transfer function. Manipulate groups of polynomial in the bracket so that it can be written in polynomial form before multiply with other brackets.
( ) ( ) ( )( )( ) ( )( )( )
( )6 3 2 2 3
11 8 4 2 25
m gd xm gd
H s H s g sC gH s H s H s g sC
H s
− −= − − −
(5.2.32)
( ) ( ) ( )( ) ( )( ) ( )6
11 8 4 2 2 165
m gdH s
H s H s H s g sC H sH s
= − − −
(5.2.54)
( ) ( )( )( )2
23 13 0316 3 2 2 3
7 6m gd x
x x
s d sd dH s H s g sC gg sC+ +
= = − −+
(5.2.55)
( )( )
23 2 3 2
13 2 3 2 2 3
03 3 2
gs gd gd
gs gd m gd m
m m
d C C C
d C C g C g
d g g
= − +
= + +
= −
(5.2.56)
Next step, ( )( )
6
5
H sH s
can be defined as following
( ) ( )( )
2 221 11 01 21 11 016 7 6
17 2 25 7 611 11 11 11 11 11
y y y y y yx x
x x
s C sC g s C sC gH s g sCH sH s g sCs a sb c s a sb c
− + + − + + + = = = ++ + + +
(5.2.57)
After that, ( ) ( )17 16H s H s can be defined as following
34
( ) ( ) ( )2 2
21 11 01 23 13 0318 17 16 2
7 611 11 11
y y y
x x
s C sC g s d sd dH s H s H sg sCs a sb c
− + + + + = = ++ +
(5.2.58)
( )4 3 2
44 34 24 14 0418 3 2
35 25 15 05
s d s d s d sd dH ss d s d sd d
+ + + += + + +
(5.2.59)
Coefficients of equation (5.2.59) can be defined as following
44 21 23
34 21 13 11 23
24 21 03 11 13 01 23
14 11 03 01 13
04 01 03
35 11 6
25 11 6 11 7
15 11 7 11 6
05 11 7
y
y y
y y y
y y
y
x
x x
x x
x
d C d
d C d C d
d C d C d g d
d C d g d
d g d
d a Cd b C a gd b g c Cd c g
= −
= − +
= − + +
= +
=
=
= +
= +
=
(5.2.60)
Equation (5.2.54) can be rewritten as following
( ) ( ) ( )( ) ( )11 8 4 2 2 18m gdH s H s H s g sC H s= − − −
(5.2.61)
( ) ( )( )2 2
2 219 4 2 2
6 7
gd gd mm gd
x x
s C sC gH s H s g sC
sC g− +
= − =+
(5.2.62)
Substitute equation (5.2.41), (5.2.62) and (5.2.59) respectively into equation (5.2.61)
( )
( )
( )( )( )
6 5 4 3 261 51 41 31 21 11 01
6 5 4 3 262 52 42 32 22 12
6 5 4 3 263 53 43 33 23 13 03
11 3 25 6 6 7 35 25 15 05x x x x
s f s f s f s f s f sf f
s f s f s f s f s f sf
s f s f s f s f s f sf fH s
sC g sC g s d s d sd d
+ + + + + +
− − + + + + + − + + + + + + =
+ + + + +
(5.2.63)
35
Coefficients of equation (5.2.63) can be defined as following
( )( ) ( ) ( )
( ) ( )( )
61 35 23 6
51 35 23 7 13 6 25 23 6
41 35 6 03 13 7 25 23 7 13 6 15 23 6
31 35 03 7 25 6 03 13 7 15 23 7 13 6 05 23 6
21 25 03 7 15 6 03 13 7 05
y x
y x y x y x
x y y x y x y x y x
y x x y y x y x y x y x
y x x y y x
f d C C
f d C g C C d C C
f d C g C g d C g C C d C C
f d g g d C g C g d C g C C d C C
f d g g d C g C g d C
=
= + +
= + + + +
= + + + + +
= + + + ( )( )
23 7 13 6
11 15 03 7 05 6 03 13 7
05 05 03 7
y x y x
y x x y y x
y x
g C C
f d g g d C g C g
f d g g
+
= + +
=
(5.2.64)
( )( )( )( )
262 5 35
2 252 35 2 2 5 6 25 5
2 242 35 2 2 6 25 2 2 5 6 15 5
2 232 25 2 2 6 15 2 2 5 6 05 5
222 15 2 2 6 05 2 2 5 6
12 05 2
gd x
gd m x gd x gd x
gd m x gd m x gd x gd x
gd m x gd m x gd x gd x
gd m x gd m x gd x
gd
f C C d
f d C g C C g d C C
f d C g g d C g C C g d C C
f d C g g d C g C C g d C C
f d C g g d C g C C g
f d C
=
= − −
= + − −
= + − −
= + −
= 2 6m xg g
(5.2.65)
( )( )( )( )( )
63 5 6 44
53 5 6 34 5 7 6 6 44
43 44 6 7 34 5 7 6 6 24 5 6
33 34 6 7 24 5 7 6 6 14 5 6
23 24 6 7 14 5 7 6 6 04 5 6
13 14 6 7 04 5 7 6 6
03 04
x x
x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x
f C C df C C d C g C g d
f d g g d C g C g d C C
f d g g d C g C g d C C
f d g g d C g C g d C C
f d g g d C g C gf d
=
= + +
= + + +
= + + +
= + + +
= + +
= 6 7x xg g
(5.2.66)
From equation (5.2.63), Coefficients which have the same order can be grouped as folllowing
( )
( ) ( ) ( )( ) ( ) ( ) ( )
( )( )( )
6 5 461 62 63 51 52 53 41 42 43
3 231 32 33 21 22 23 11 12 13 01 03
11 3 25 6 6 7 35 25 15 05x x x x
s f f f s f f f s f f f
s f f f s f f f s f f f f fH s
sC g sC g s d s d sd d
+ − + − − + − − + − − + − − + − − + − =
+ + + + +
(5.2.67)
36
( )( )
( )( )( )6 5 4 3 2
64 54 44 34 24 14 0411 3 2
5 6 6 7 35 25 15 05x x x x
s f s f s f s f s f sf fH s
sC g sC g s d s d sd d
+ + + + + +=
+ + + + +
(5.2.68)
Coefficients of numerator of equation (5.2.68) can be defined as following
64 61 62 63
54 51 52 53
44 41 42 43
34 31 32 33
24 21 22 23
14 11 12 13
04 01 03
f f f ff f f ff f f ff f f ff f f ff f f ff f f
= + −
= − −
= − −
= − −
= − −
= − −
= −
(5.2.69)
Multiply three brackets of denominator polynomial in (5.2.68), we will get
( )( )
( )6 5 4 3 2
64 54 44 34 24 14 0411 5 4 3 2
55 45 35 25 15 05
s f s f s f s f s f sf fH s
s f s f s f s f sf f
+ + + + + +=
+ + + + +
(5.2.70)
Coefficients of denominator of equation (5.2.70) can be defined as following
( )( )( )( )
55 5 6 35
45 5 7 6 6 35 25 5 6
35 35 6 7 5 7 6 6 25 15 5 6
25 25 6 7 5 7 6 6 15 05 5 6
15 15 6 7 5 7 6 6 05
05 05 6 7
x x
x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x
x x
f C C df C g C g d d C C
f d g g C g C g d d C C
f d g g C g C g d d C C
f d g g C g C g df d g g
=
= + +
= + + +
= + + +
= + +
=
(5.2.71)
37
Equation (5.2.33) can be rewritten as following
( ) ( ) ( ) ( )( )( ) ( )( )
712 11 2 4 4 5 4
9m gs gd ds
H sH s H s H s g s C C g
H s
= + − + −
(5.2.33)
From equation (5.2.33), it can be seen that there are four polynomials which are called intermediate transfer function. Manipulate groups of polynomial in the bracket so that it can be written in polynomial form before multiply with other brackets
( ) ( )( )
222 12 027
19 29 24 14 04
y y y
y y y
s C sC gH sH s
H s s C sC g+ +
= =+ +
(5.2.72)
( )( ) ( )
( ) ( )( )2
4 4 5 4 420 2 4 4 5
5 6
gd gs gd gd mm gs gd
x x
s C C C s C gH s H s g s C C
sC g
− + += = − +
+
(5.2.73)
( )( ) ( )
( ) ( )( )( )4 42
4 4 5 4 64 5
21 2 4 4 5 45 6
gd mgd gs gd ds x
ds xm gs gd ds
x x
C gs C C C s g g
g CH s H s g s C C g
sC g
− + + −
− = = − + −+
(5.2.74)
( ) ( ) ( )4 3 2
41 31 21 11 0122 21 19 3 2
32 22 12 02
s g s g s g sg gH s H s H ss g s g sg g+ + + +
= =+ + +
(5.2.75)
( )( ) ( )
( ) ( )( )
41 22 4 4 5
31 22 4 4 4 5 12 4 4 5
21 22 4 6 12 4 4 4 5 02 4 4 5
11 12 4 6 02 4 4 4 5
01 4 6 02
y gd gs gd
y gd m ds x y gd gs gd
y ds x y gd m ds x y gd gs gd
y ds x y gd m ds x
ds x y
g C C C C
g C C g g C C C C C
g C g g C C g g C g C C C
g C g g g C g g C
g g g g
= − +
= − − + = − − − − + = − + −
= −
(5.2.76)
38
32 24 5
22 24 6 14 5
12 14 6 04 5
02 04 6
y x
y x y x
y x y x
y x
g C C
g C g C C
g C g g C
g g g
=
= +
= +
=
(5.2.77)
Equation (5.2.33) can be rewritten as following
( ) ( ) ( ) ( )
6 5 4 3 4 3 264 54 44 34 41 31 21
224 14 04 11 01
12 11 22 5 4 3 3 255 45 35 32 22 12 02
225 15 05
s f s f s f s f s g s g s gs f sf f sg g
H s H s H ss f s f s f s g s g sg g
s f sf f
+ + + + + + + + + + = + = + + + + + + + + +
(5.2.78)
( )
( )6 5 4 3
64 54 44 34 3 232 22 12 022
24 14 04
5 4 34 3 255 45 3541 31 21
211 01 25 15 05
12 5 4 355 45 35 3 2
32 22225 15 05
s f s f s f s fs g s g sg g
s f sf f
s f s f s fs g s g s gsg g s f sf f
H ss f s f s f
s g s g sgs f sf f
+ + + + + + + + +
+ ++ + + + + + + + =
+ + + + + + +
( )12 02g+
(5.2.79)
( )( )( )( )9 8 7 6 5 4 3 2
93 83 73 63 53 43 33 23 13 0312 5 4 3 2 3 2
55 45 35 25 15 05 32 22 12 02
s g s g s g s g s g s g s g s g sg gH s
s f s f s f s f sf f s g s g sg g
+ + + + + + + + +=
+ + + + + + + +
(5.2.80)
Coefficients of denominator of equation (5.2.80) can be defined as following
5.2.3 Derivation of Output Impedance of the MCRGC amplifier
1M
2M
3M
4M
5M
6M
inV
outV
inV
1m ing V
( )2 2 1mg V V−
1dsg
( )4 4 3mg V V−
( )4 30mbg V−
4dsg
( )6 0m ing V −
6dsg
outV
1V
3V
( )a
( )b
7M
8M
1BR
2BR
5 6gs dbC C+
3 1gs dbC C+
2 4db dbC C+
3, 2, 7D G D
1V
3V
2V2V2V
4V4V
4V
8 4mg V
4V4V
2V
8 8 5gs db dbC C C+ +
7 2mg V
7 7 3gs db dbC C C+ +0inI =
2dsg
( )2 10mbg V−
DDV
1 7/ /B dsR g
2 8/ /B dsR g
4 5gs gdC C+
2 3gs gdC C+2gdC
4gdC
1gdC
1gsC
5 3mg V5dsg
3 1mg V3dsg
6gsC 6gdC
outI
Fig 5.5 (a) Modified Regulated Cascode Amplifier
(c) Its small signal equivalent circuit for output impedance derivation
KCL at input node, current flow out of node 3 branches and current flow into node 1 branch
( ) ( ) ( ) ( )3 6 1 1 1 60in gd in gd in gs in gsV V sC V V sC V sC V sC− + − + = −
(5.2.124)
( ) ( ) ( )6 1 1 6 3 6 1 1 0in gd gd gs gs gd gdV s C C C C V sC V sC + + + + − =
(5.2.125)
( ) ( ) ( )1 3 6 1 1
1 6 1 1 6
0in x gd gd
x gd gd gs gs
V s C V sC V sC
C C C C C
+ − = = + + +
(5.2.126)
KCL at 3V , current flow out of node 5 branches and current flow into node 3 branches
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
3 6 6 3 6
3 4 4 5 4 4 3 4 3 3 4 3 5 6
0 0
0in gd m in ds
gs gd m mb out ds gs db
V V sC g V V g
V V s C C g V V g V V V g V s C C
− + − + −
= − + + − + − + − + +
(5.2.127)
( )( )
( ) ( )4 5 5 6 66 6 3 4 4 4 5 4
6 4 4 4
gs gd gs db gdin gd m m gs gd out ds
ds ds m mb
s C C C C CV sC g V V g s C C V g
g g g g
+ + + + + = + − + − + + − −
(5.2.128)
56
[ ] ( ) ( )
( )
6 6 3 2 2 4 4 4 5 4
2 4 5 5 6 6
2 6 4 4 4
in gd m x x m gs gd out ds
x gs gd gs db gd
x ds ds m mb
V sC g V sC g V g s C C V g
C C C C C C
g g g g g
+ = + + − + − = + + + +
= + − −
(5.2.129)
KCL at node outV , current flow into node 6 branches and current flow out of node 4 branches
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )
4 4 4 4 3 4 3 3 4 2 2
2 2 1 2 1 1 2 2 4
0
0
out gd m mb out ds out gd out
m mb out ds out gd db
V V sC g V V g V V V g V V sC i
g V V g V V V g V s C C
− + − + − + − + − +
= − + − + − + +
(5.2.130)
[ ][ ] ( )
4 4 4 3 4 4 4 2 2 2
1 2 2 2 4 2 2 4 2 4
gd m ds m mb out m gd
m mb ds out gd gd db db ds ds
V sC g V g g g i V g sC
V g g g V s C C C C g g
+ + − − + = − − + + + + + + + +
(5.2.131)
[ ] [ ] [ ]4 4 4 3 4 4 4 2 2 2 1 2 2 2 3 3
3 4 2 2 4
3 2 4
gd m ds m mb out m gd m mb ds out x x
x gd gd db db
x ds ds
V sC g V g g g i V g sC V g g g V sC g
C C C C C
g g g
+ + − − + = − − + + + + = + + +
= +
(5.2.132)
KCL at node 1V , current flow into node 5 branches, current flow out of node 3 branches
( ) ( ) ( ) ( ) ( )
( ) ( )2 1 2 3 2 2 1 2 1 1 2
1 1 1 1 1 3 1
0gs gd m mb out ds
in gd m in ds gs db
V V s C C g V V g V V V g
V V sC g V V g s C C
− + + − + − + −
+ − = + + +
(5.2.133)
( ) ( )
( )
2 3 1 3 12 2 3 2 1
1 2 2 2
2 1 1
gs gd gd gs dbgs gd m
ds m mb ds
out ds in m gd
s C C C C CV s C C g V
g g g g
V g V g sC
+ + + + + + − − − − − + = −
(5.2.134)
( ) [ ] ( )2 2 3 2 1 4 4 2 1 1
4 2 3 1 3 1
4 1 2 2 2
gs gd m x x out ds in m gd
x gs gd gd gs db
x ds m mb ds
V s C C g V sC g V g V g sC
C C C C C C
g g g g g
+ + − + + = − = + + + +
= − − − −
(5.2.135)
57
KCL at node 2V , current flow out of node 7 branches
( )
( ) ( ) ( ) ( )
7 2 2 7 3 7 3 1 2 71
2 1 2 3 2 13 2 2
1
0
m gs db db m dsB
gs gd ds out gd
g V V s C C C g V V gR
V V s C C V g V V sC
+ + + + + + + − + + + − =
(5.2.136)
( ) ( )7 7 3
12 1 3 2 3 2
7 3 7
2 2 3
1
0m ds ds
Bm gs gd out gd
gs db db
gd gs gd
g g gR
V V g s C C V sCC C Cs
C C C
+ + + + − + − = + + + + + +
(5.2.137)
( ) ( ) ( )2 5 5 1 3 2 3 2
5 7 3 7 2 2 3
5 7 7 31
0
1
x x m gs gd out gd
x gs db db gd gs gd
x m ds dsB
V g s C V g s C C V sC
C C C C C C C
g g g gR
+ + − + − = = + + + + +
= + + +
(5.2.138)
KCL at node 4V , current flow into node 4 branches, current flow out of node 3 branches
( ) ( ) ( )
( ) ( )
8 4 5 3 4 5 3 4 4 5
4 8 4 8 5 8 4 42
0
1
m m ds gs gd
ds gs db db out gdB
g V g V V g V V s C C
V g V s C C C V V sCR
+ + − + − + = + + + + + −
(5.2.139)
( ) ( ) ( )3 5 4 5 4
8 5 84 8 5 8
4 5 42
1
m gs gd out gd
gs db dbds ds m
gs gd gdB
V g s C C V sC
C C CV g g g s
C C CR
+ + + + +
= + + − + + + +
(5.2.140)
( ) ( ) ( ) [ ]3 5 4 5 4 4 6 6
6 8 5 82
6 8 5 8 4 5 4
1m gs gd out gd x x
x ds ds mB
x gs db db gs gd gd
V g s C C V sC V g sC
g g g gR
C C C C C C C
+ + + = +
= + + −
= + + + + +
(5.2.141)
58
From equation (5.2.126)
1 61 3
1 1
gd gdin
x x
sC sCV V V
sC sC
= −
(5.2.126b)
From equation (5.2.129)
[ ] ( ) ( )4 4 4 53 2 2 4
6 6 6 6 6 6
m gs gdx x out dsin
gd m gd m gd m
V g s C CV sC g V gV
sC g sC g sC g
− ++ = + −+ + +
(5.2.129b)
Let us define intermediate transfer function to reduce the time to finished the closed form formula as following
( ) ( ) ( )
( ) [ ]
( )( )
( ) ( )
3 4 4 3 5
2 24
6 6
4 4 53
6 6
45
6 6
in out
x x
gd m
m gs gd
gd m
ds
gd m
V V H s V H s V H s
sC gH s
sC g
g s C CH s
sC g
gH s
sC g
= + −
+=
+
− + =+
=+
(5.2.129c)
From equation (5.2.135)
( ) ( ) [ ]2 2 3 2 1 4 4 2
1 1 1 1 1 1
gs gd m x x out dsin
m gd m gd m gd
V s C C g V s C g V gV
g sC g sC g sC
+ + + = − +− − −
(5.2.135b)
From equation (5.2.141)
( ) ( )3 5 4 5 44
6 6 6 6
m gs gd out gd
x x x x
V g s C C V sCV
sC g sC g
+ + = ++ +
(5.2.141b)
Let us define intermediate transfer function to reduce the time to finished the closed form formula as following
59
( ) ( )
( )( )
( )( )
4 3 1 2
5 4 51
6 6
42
6 6
out
m gs gd
x x
gd
x x
V V H s V H s
g s C CH s
sC g
sCH s
sC g
= +
+ + =+
=+
(5.2.141c)
Substitute equation (5.2.141c) into (5.2.129c)
( ) ( ) ( ) ( ) ( )3 4 3 1 2 3 5in out outV V H s V H s V H s H s V H s= + + −
(5.2.129c)
After grouping the coefficients which have the same node voltage, we get
( ) ( ) ( ) ( ) ( ) ( )3 4 1 3 2 3 5in outV V H s H s H s V H s H s H s= + + −
(5.2.129d)
From equation (5.2.132), it can be rewritten here
( ) ( ) ( ) ( ) ( )( )( ) [ ]( )( ) [ ]( ) [ ]
4 6 3 7 2 8 1 9 10
6 4 4
7 4 4 4
8 2 2
9 2 2 2
10 3 3
out out
gd m
ds m mb
m gd
m mb ds
x x
V H s V H s i V H s V H s V H s
H s sC g
H s g g g
H s g sC
H s g g g
H s sC g
+ + = − +
= + = − −
= − = + +
= +
(5.2.132b)
Substitute equation (5.2.141c) into equation (5.2.132c), we get
( ) ( ) ( ) ( ) ( ) ( ) ( )3 1 2 6 3 7 2 8 1 9 10out out outV H s V H s H s V H s i V H s V H s V H s+ + + = − +
(5.2.132c)
After grouping the coefficients which have the same node voltage, we get
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 7 1 6 2 8 1 9 10 2 6out outV H s H s H s i V H s V H s V H s H s H s+ + = − + −
(5.2.132d)
60
Substitute equation (5.2.129d) into (5.2.126b)
( ) ( ) ( ) ( ) ( ) ( ) 1 63 4 1 3 2 3 5 1 3
1 1
gd gdout
x x
sC sCV H s H s H s V H s H s H s V V
sC sC
+ + − = −
(5.2.126c)
( ) ( ) ( ) ( ) ( ) ( )6
4 1 32 3 51
1 31 1
1 1
gd
xout
gd gd
x x
CH s H s H s
H s H s H sCV V V
C CC C
+ + − = +
(5.2.126d)
( ) ( )
( )( ) ( ) ( )
( ) ( ) ( ) ( )
1 3 11 12
64 1 3
111
1
1
2 3 512
1
1
out
gd
x
gd
x
gd
x
V V H s V H s
CH s H s H s
CH s
CC
H s H s H sH s
CC
= +
+ +
=
− =
(5.2.126e)
Substitute equation (5.2.126e) into (5.2.132d), we get
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )
3 7 1 6 2 8 3 11 12 9
10 2 6
out out
out
V H s H s H s i V H s V H s V H s H s
V H s H s H s
+ + = − + + −
(5.2.132e)
After grouping the coefficients which have the same node voltage, we get
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 7 1 6 11 9 2 8 10 2 6 12 9out outV H s H s H s H s H s i V H s V H s H s H s H s H s+ + + = + − − (5.2.132f)
Let us define intermediate transfer function to reduce the time to finished the closed form formula as following
Substitute equation (5.2.135i) into equation (5.2.132g)
( )( )
( )( ) ( ) ( ) ( )15 19
2 13 2 8 1418 18
out out outH s H s
V V H s i V H s V H sH s H s
+ + = +
(5.2.132h)
( ) ( )( ) ( ) ( ) ( )
( ) ( )15 13 19 132 8 14
18 180out out
H s H s H s H sV H s V H s i
H s H s
− + − + =
(5.2.132i)
Substitute equation (5.2.116e) into equation (5.2.138)
( ) ( ) ( ) ( ) ( )2 5 5 3 11 12 3 2 3 2 0x x out m gs gd out gdV g s C V H s V H s g s C C V sC + + + − + − =
(5.2.138b)
After grouping the coefficients which have the same node voltage, we get
( ) ( ) ( )( )( ) ( )( )
( )12 3 2 3
2 5 5 3 11 3 2 32
0m gs gd
x x m gs gd outgd
H s g s C CV g s C V H s g s C C V
sC
− + + + − + + = −
(5.2.138c)
Let us define intermediate transfer function to reduce the time to finished the closed form formula as following
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )2 5 5 3 11 3 2 3 20
20 12 3 2 3 2
0x x m gs gd out
m gs gd gd
V g s C V H s g s C C V H s
H s H s g s C C sC
+ + − + + =
= − + −
(5.2.138d)
63
Substitute equation (5.2.135i) into equation (5.2.138d), we get
( ) ( )( )
( )( ) ( ) ( )( ) ( )15 19
2 5 5 2 11 3 2 3 2018 18
0x x out m gs gd outH s H s
V g s C V V H s g s C C V H sH s H s
+ + + − + + =
(5.2.138e)
Let us define intermediate transfer function to reduce the time to finished the closed form formula as following
( )( )( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( )5 5
192 20 11 3 2 315
1811 3 2 318
0x x
out m gs gdm gs gd
g s CH s
V V H s H s g s C CH sH sH s g s C C
H s
+
+ + − + = + − +
(5.2.138f)
Let us define intermediate transfer function to reduce the time to finished the closed form formula as following
( ) ( )
( ) ( ) ( )( ) ( ) ( )( )
( ) ( ) ( )( ) ( ) ( )( )
2 21 22
1521 5 5 11 3 2 3
18
1922 20 11 3 2 3
18
0out
x x m gs gd
m gs gd
V H s V H s
H sH s g s C H s g s C C
H s
H sH s H s H s g s C C
H s
+ =
= + + − +
= + − +
(5.2.138g)
From equation (5.2.138g), we can write
( )( )
222
21out
H sV V
H s
= −
(5.2.138h)
Substitute equation (5.2.138h) into equation (5.2.132i)
( )( )
( ) ( )( ) ( ) ( ) ( )
( ) ( )22 15 13 19 138 14
21 18 180out out out
H s H s H s H s H sV H s V H s i
H s H s H s
− − + − + =
(5.2.138i)
64
After grouping the coefficients which have the same node voltage, we get
( )( )
( ) ( )( ) ( )
( ) ( )( ) ( )
22 15 138
21 18
19 1314
18
out out
H s H s H sH s
H s H sV i
H s H sH s
H s
− = − −
(5.2.138j)
( )( )
( ) ( )( ) ( ) ( ) ( )
( ) ( )22 15 13 19 138 14
21 18 18
1outout
out
VZi H s H s H s H s H s
H s H sH s H s H s
= =
− − −
(5.2.138k)
Substitute every function inside equation (5.2.126e)
( )
[ ] ( ) ( )
( )( )
( )( )
( ) [ ] ( )( )
5 4 5 4 4 5 62 2
6 6 6 6 6 6 1
111
1
5 4 5 41 2
6 6 6 6
4 4 52 24 3
6 6
,
,
m gs gd m gs gd gdx x
gd m x x gd m x
gd
x
m gs gd gd
x x x x
m gs gdx x
gd m g
g s C C g s C C CsC gsC g sC g sC g C
H sCC
g s C C sCH s H s
sC g sC g
g s C CsC gH s H s
sC g sC
+ + − ++ + + + + + =
+ + = =+ +
− ++ = =+
( ) ( )
( )
( ) ( ) ( )
45
6 6 6 6
4 4 54 4
6 6 6 6 6 6
121
1
, ds
d m gd m
m gs gdgd ds
x x gd m gd m
gd
x
gH s
g sC g
g s C CsC gsC g sC g sC g
H sCC
=+ +
− + − + + + =
(5.2.126f)
65
Multiply both numerator and denominator polynomial with ( )( )6 6 6 6gd m x xsC g sC g+ +
( )
( ) ( )
( )
( ) ( )
222 12 02
11 221 11 01
622 2 6 4 5 6 4 5 6 6 6
1
12 2 6 6 2 4 5 6 6 5 6 4
66 4 5 6 6 6 6
1
02 2 6 5 6 4
gdx x gs gd gd gs gd x x gd
x
x x x x gs gd m gd m x m
gdx gs gd x m x gd
x
x x m m m x
s a sa aH s
s a sa aC
a C C C C C C C C C CC
a C g C g C C g C g C g
Cg C C C g g C
C
a g g g g g g
+ +=
+ +
= + + − + +
= + + + + +
− + + +
= + +
( )
66 6 6
1
121 6 6
1
111 6 6 6 6
1
101 6 6
1
gdx m
x
gdgd x
x
gdgd x x m
x
gdm x
x
Cg g
C
Ca C C
C
Ca C g C g
C
Ca g g
C
+
=
= +
=
(5.2.142)
( )( ) ( ) ( )
( ) ( )
( )
( )( )
24 5 4 4 4 4 6 4 6
121 1 12
6 6 6 6 6 6 6 61 1 1
225 15 05
12 226 16 06
25 4 5 4
15 4 4 4 6
05 4
gs gd gd gd m ds x ds x
gd gd gdx gd x m gd x x m
x x x
gs gd gd
gd m ds x
ds
s C C C s C g g C g gH s
C C Cs C C s C g C g g g
C C C
s a sa aH s
s a sa a
a C C C
a C g g C
a g g
− + + − − =
+ + +
− + −=
+ +
= +
= −
= ( )
( )
( )
6
126 6 6
1
116 6 6 6 6
1
106 6 6
1
x
gdx gd
x
gdx m gd x
x
gdx m
x
Ca C C
C
Ca C g C g
C
Ca g g
C
=
= +
=
(5.2.143)
66
( ) ( ) ( ) ( ) ( ) ( )
( ) [ ]( )
[ ]
( )( )
( )
( )
13 7 1 6 11 9
25 4 5 22 12 0213 4 4 4 4 4 2 2 22
6 6 21 11 01
5 4 51
6 6
6 4 4
222 12 02
11 2
m gs gdds m mb gd m m mb ds
x x
m gs gd
x x
gd m
H s H s H s H s H s H s
g s C C s a sa aH s g g g sC g g g g
sC g s a sa a
g s C CH s
sC g
H s sC g
s a sa aH s
s a
= + +
+ + + + = − − + + + + + + + + + + =
+
= +
+ +=
( ) [ ]21 11 01
9 2 2 2m mb ds
sa aH s g g g
+ +
= + +
(5.2.144)
( ) [ ]( )
[ ]
( )
[ ][ ]
( )
25 4 5 22 12 0213 4 4 4 4 4 2 2 22
6 6 21 11 01
24 4 4 6 6 21 11 01
25 4 5 4 4 21 11 01
13
m gs gdds m mb gd m m mb ds
x x
ds m mb x x
m gs gd gd m
g s C C s a sa aH s g g g sC g g g g
sC g s a sa a
g g g sC g s a sa a
g s C C sC g s a sa a
H s
+ + + + = − − + + + + + + + +
− − + + + + + + + + +
=[ ][ ]
[ ]
222 12 02 2 2 2 6 6
26 6 21 11 01
m mb ds x x
x x
s a sa a g g g sC g
sC g s a sa a
+ + + + + +
+ + +
(5.2.145)
67
( )
[ ] ( ) ( )( )
( )( )
( )( )( )
( )( )
3 26 21 6 21 6 11
4 4 46 01 6 11 6 01
4 5 4 114 34 5 4 21
4 5 4 4 5 21
4 5 4 01
24 5 4 4 5 11
21 5 4
13
x x xds m mb
x x x
gs gd gdgs gd gd
gs gd m gd m
gs gd gd
gs gd m gd m
m m
s C a s g a C ag g g
s C a g a g a
C C C as C C C a s
C C g C g a
C C C a
s C C g C g a
a g g
H s
+ +− −
+ + + + + + + + +
+ +
+ + + +
+
=
( )( )
( ) ( ) ( )
( ) ( ) ( )
4 5 4 4 5 01
5 4 11
5 4 013 2
22 6 22 6 12 6 12 6 02 6 02 6
3 26 21 21 6 6 11 6 01 6 11
gs gd m gd m
m m
m m
x x x x x x
x x x x x
C C g C g as
g g a
g g a
s a C s a g a C s a g a C a g
s C a s a g C a s C a g a
+ + + + + + + + + + +
+ + + + 6 01xg a+
(5.2.146)
From equation (5.2.132g), it can be rewritten after substitute 5 functions here
( )( ) ( ) ( )
( )( )[ ] ( ) ( )( ) ( )
( )
4 3 243 33 23 13 03
13 3 26 21 21 6 6 11 6 01 6 11 6 01
43 4 5 4 21
33 6 21 4 4 4 4 5 4 11 4 5 4 4 5 21 22 6
23 6 21 6 11 4
x x x x x x
gs gd gd
x ds m mb gs gd gd gs gd m gd m x
x x ds m
s a s a s a sa aH s
s C a s a g C a s C a g a g a
a C C C a
a C a g g g C C C a C C g C g a a C
a g a C a g g
+ + + + =+ + + + +
= +
= − − + + + + + +
= + −[ ] ( )( )( ) ( )
( )[ ] ( )( ) ( )
[ ]
4 4 4 5 4 01
4 5 4 4 5 11 21 5 4 22 6 12 6
4 5 4 4 5 0113 6 01 6 11 4 4 4 12 6 02 6
5 4 11
03 6 01 4 4 4 5 4 01 02 6
mb gs gd gd
gs gd m gd m m m x x
gs gd m gd mx x ds m mb x x
m m
x ds m mb m m x
g C C C a
C C g C g a a g g a g a C
C C g C g aa C a g a g g g a g a C
g g a
a g a g g g g g a a g
− + +
+ + + + + +
+ + = + − − + + + +
= − − + +
(5.2.147)
From equation (5.2.132g), it can be rewritten after substitute 5 functions here
68
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( )( )( )( )( )( )( )
14 6 2 6 12 9
24 25 15 05
14 5 3 4 4 2 2 226 6 26 16 06
25 3 6 6 26 16 06
24 4 26 16 06
225 15 05 2
14
gdx x gd m m mb ds
x x
x x x x
gd gd m
m
H s H s H s H s H s H s
sC s a sa aH s sC g sC g g g g
sC g s a sa a
sC g sC g s a sa a
sC sC g s a sa a
s a sa a g gH s
= − −
− + += + − + − + + + + +
+ + + +
− + + +
− − + + +=
( )( )
( )( )2 2 6 6
26 6 26 16 06
mb ds x x
x x
g sC g
sC g s a sa a
+ + + + +
(5.2.148)
( )
( )( )( )
( ) ( )( )
( )
( )( )( )
25 3 6 6 26 16 06
5 6 065 6 164 3 2
5 6 26 26 3 63 6 26
5 6 3 6 16
5 6 3 6 16 3 6 16 3 6 06
24 4 4 26 16 06
24
14
x x x x
x xx x
x x x xx x
x x x x
x x x x x x x x
gd gd m
gd
sC g sC g s a sa a
C C aC C a
s C C a s s a g gg C a
C g g C a
s C g g C a g g a g g a
sC sC g s a sa a
s C
H s
+ + + +
+ + + + + + + + +
− + + +
−
=
( )( )
( )( )( )
( )( )( )
2 24 4 26 16 06
4 2 3 24 26 4 16 4 4 26
2 24 06 4 4 16 4 4 06
225 15 05 2 2 2 6 6
325 6 2 2 2
225 6 15 6 2 2
gd m
gd gd gd m
gd gd m gd m
m mb ds x x
x m mb ds
x x m mb
sC g s a sa a
s C a s C a C g a
s C a C g a s C g a
s a sa a g g g sC g
s a C g g g
s a g a C g g g
+ + +
+ + − + + +
− − + + + + +
− + +
+ − + + +−
( )( )( )
( )( )( )
2
15 6 05 6 2 2 2
05 6 2 2 2
26 6 26 16 06
ds
x x m mb ds
x m mb ds
x x
s a g a C g g g
a g g g g
sC g s a sa a
+ + + + + + +
+ + +
(5.2.149)
69
( )
( )
( )
( )( )
( )
( )( )
4 25 6 26 4 26
5 6 16
3 6 2632
4 16 4 4 26
25 6 2 2 2
5 6 06
26 3 62
5 6 3 6 16
24 06 4 4 16
25 6 15 6 2 2 2
14
x x gd
x x
x x
gd gd m
x m mb ds
x x
x x
x x x x
gd gd m
x x m mb ds
s C C a C a
C C ag C a
sC a C g a
a C g g g
C C aa g g
s C g g C a
C a C g a
a g a C g g g
H s
−
+
+ − − + + + ++ +
− + − − + + +
=
( )
( )( )( )
( )
5 6 3 6 16
3 6 16
4 4 06
15 6 05 6 2 2 2
3 6 06 05 6 2 2 2
3 26 26 6 16 6
x x x x
x x
gd m
x x m mb ds
x x x m mb ds
x x x
C g g C ag g a
sC g a
a g a C g g g
g g a a g g g g
s C a s C a g a
+ + + − + + +
+ − + +
+ +( ) ( )26 6 06 6 16 6 06x x xs C a g a g a+ + +
(5.2.150)
( )
( )( )
( )( )
( )
4 3 247 37 27 17 07
14 3 238 28 18 08
247 5 6 26 4 26
5 6 16 3 6 2637 2
4 16 4 4 26 25 6 2 2 2
227 5 6 06 26 3 6 5 6 3 6 16 4 06
x x gd
x x x x
gd gd m x m mb ds
x x x x x x x x gd g
s a s a s a sa aH s
s a s a sa a
a C C a C a
C C a g C aa
C a C g a a C g g g
a C C a a g g C g g C a C a C
+ + + + =+ + +
= −
+ =− − + + +
= + + + − + ( )( )( )
( )( )( )
( )( )( )
4 4 16 25 6 15 6 2 2 2
5 6 3 6 16 3 6 1617
4 4 06 15 6 05 6 2 2 2
07 3 6 06 05 6 2 2 2
38 6 26
28 6 16 6 26
18 6 06 6 16
08
d m x x m mb ds
x x x x x x
gd m x x m mb ds
x x x m mb ds
x
x x
x x
x
g a a g a C g g g
C g g C a g g aa
C g a a g a C g g g
a g g a a g g g g
a C a
a C a g a
a C a g aa g
− − + + + + +
= − + + + +
= + − + +
=
= +
= +
= 6 06a
(5.2.151)
70
( ) ( ) ( ) ( ) ( ) ( )
( )
( )( )
( )( )
( )
( )
18 4 1 3 11 16
2 24
6 6
5 4 51
6 6
4 4 53
6 6
222 12 02
11 221 11 01
4 416
1 1
x x
gd m
m gs gd
x x
m gs gd
gd m
x x
m gd
H s H s H s H s H s H ssC g
H ssC g
g s C CH s
sC g
g s C CH s
sC g
s a sa aH s
s a sa a
sC gH s
g sC
= + +
+=
+
+ + = + − + = + + +
= + + +
= −
(5.2.152)
( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( )
18 4 1 3 11 16
25 4 5 4 4 52 2 22 12 02 4 4
18 26 6 6 6 6 6 1 121 11 01
m gs gd m gs gdx x x x
gd m x x gd m m gd
H s H s H s H s H s H s
g s C C g s C CsC g s a sa a sC gH s
sC g sC g sC g g sCs a sa a
= + +
+ + − + + + + + = + + + + + −+ +
(5.2.153)
( )
( )( )( )( )( )( ) ( )( )( )( )
( )( )( )( )
( )( )( )( )
22 2 6 6 21 11 01 1 1
25 4 5 4 4 5 21 11 01 1 1
222 12 02 4 4 6 6 6 6
18 26 6 6 6 21 11 01 1 1
x x x x m gd
m gs gd m gs gd m gd
x x gd m x x
gd m x x m gd
sC g sC g s a sa a g sC
g s C C g s C C s a sa a g sC
s a sa a sC g sC g sC gH s
sC g sC g s a sa a g sC
+ + + + −
+ + + − + + + −
+ + + + + +=
+ + + + −
(5.2.154)
71
( )( )( )( )( )
( )( ) ( )
( )
5 4 3 251 41 31 21 11 01
18 26 6 6 6 21 11 01 1 1
251 22 4 6 6 2 6 21 1 4 5 21 1
41 22 4 6 6 6 6 22 4 12 4 6 6
2 6 21 1 11 1 2 6
gd m x x m gd
x gd x x x gd gs gd gd
x gd x m x x x gd x
x x m gd x x
s b s b s b s b sb bH s
sC g sC g s a sa a g sC
b a C C C C C a C C C a C
b a C C g g C a g a C C C
C C a g a C C g
+ + + + +=
+ + + + −
= − + +
= + + +
+ − − +( )
( )( ) ( ) ( )( ) ( )( )
( )( ) ( ) ( )
2 6 21 1
24 5 4 5 21 1 4 5 21 1 11 1
31 2 6 11 1 01 1 2 6 2 6 21 1 11 1 2 6 21 1
221 1 4 5 4 5 4 5 21 1 11 1 4 5 11 1
x x gd
m m gs gd gd gs gd m gd
x x m gd x x x x m gd x x gd
gd m m m m gs gd m gd gs gd m
g C a C
g g C C a C C C a g a C
b C C a g a C C g g C a g a C g g a C
a C g g g g C C a g a C C C a g a
− − + − + −
= − + + − −
− + − + − − + − ( )( )( ) ( )
( )( ) ( )( ) ( )( ) ( )
01 1
22 4 6 6 22 4 12 4 6 6 6 6 12 4 02 4 6 6
21 2 6 01 1 2 6 2 6 11 1 01 1 2 6 21 1 11 1
24 5 01 1 4 5 4 5 11 1 01 1 4 5 21 1
gd
x m x x x gd x m x x x gd x
x x m x x x x m gd x x m gd
gs gd m m m gs gd m gd m m m
C
a C g g a g a C C g g C a g a C C C
b C C a g C g g C a g a C g g a g a C
C C a g g g C C a g a C g g a g a
+ + + + + +
= + + − + −
− + + − + − + − ( )( ) ( )( )
( ) ( )( )( ) ( )( ) ( )
11 1
22 4 12 4 6 6 12 4 02 4 6 6 6 6 02 4 6 6
11 2 6 2 6 01 1 2 6 11 1 01 1
4 5 4 5 01 1 11 1 01 1 4 5
12 4 02 4 6 6 02 4 6 6 6 6
01
gd
x x m x x x gd x m x x gd x
x x x x m x x m gd
m m gs gd m m gd m m
x x m x x gd x m x
C
a g a C g g a g a C C g g C a g C C
b C g g C a g g g a g a C
g g C C a g a g a C g g
a g a C g g a g C g g C
b
+ + + + + +
= + + −
+ − + + −
+ + + +
2 6 01 1 4 5 01 1 02 4 6 6x x m m m m x m xg g a g g g a g a g g g= + +
(5.2.155)
Recall equation (5.2.135h)
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( )
( ) [ ]
( )( )
( )( )
( ) ( )
19 17 12 16 2 3 5
24 4 225 15 05
12 16 1721 1 1 126 16 06
4 4 54 42 3 5
6 6 6 6 6 6
, ,
, ,
x x ds
m gd m gd
m gs gdgd ds
x x gd m gd m
H s H s H s H s H s H s H s
s C g gs a sa aH s H s H s
g sC g sCs a sa a
g s C CsC gH s H s H s
sC g sC g sC g
= − − +
+− + − = = =− −+ +
− + = = =+ + +
(5.2.156)
( )( ) ( ) ( ) ( )
( )
2 4 4 544 4 42 25 15 0519 2
1 1 1 1 6 6 6 6 6 626 16 06
225 15 05
12 226
− + + − + − = − − + − − + + ++ +
− + −=
+
m gs gdgdx x dsds
m gd m gd x x gd m gd m
g s C CsCs C g gg s a sa aH s
g sC g sC sC g sC g sC gs a sa a
s a sa aH s
s a sa( )
( )( ) [ ]
( )( )
( )( )
( ) ( )
4 4 216 17
1 1 1 116 06
4 4 54 42 3 5
6 6 6 6 6 6
, ,
, ,
+ = =− −+
− + = = =+ + +
x x ds
m gd m gd
m gs gdgd ds
x x gd m gd m
s C g gH s H s
g sC g sCa
g s C CsC gH s H s H s
sC g sC g sC g
(5.2.157)
72
( ) ( )( ) ( )
( )
( )( ) ( )( ) ( )( )
225 15 05
20 3 2 3 2226 16 06
225 15 05
12 226 16 06
2 226 16 06 3 2 3 2 26 16 06
20 226 16 06
− + −= − + −
+ +
− + −=
+ +
+ + − + − + +=
+ +
m gs gd gd
m gs gd gd
s a sa aH s g s C C sC
s a sa a
s a sa aH s
s a sa a
s a sa a g s C C sC s a sa aH s
s a sa a
( )( ) ( ) ( ) ( )
( )
( )
2 4 4 544 4 42 25 15 0519 2
1 1 1 1 6 6 6 6 6 626 16 06
22 26 16 06 6
19
− + + − + − = − − + − − + + ++ +
+ + +
=
m gs gdgdx x dsds
m gd m gd x x gd m gd m
ds x
g s C CsCs C g gg s a sa aH s
g sC g sC sC g sC g sC gs a sa a
g s a sa a sC
H s
( )( )( ) ( ) ( )( )( ) ( ) ( )( )( )( )( )( )
6 6 6
225 15 05 4 4 6 6 6 6
24 4 4 5 1 1 26 16 06
21 1 26 16 06 6 6 6 6
+
− − + − + + +
− − + − + + − + + + +
x gd m
x x x x gd m
gd m gs gd m gd
m gd x x gd m
g sC g
s a sa a s C g sC g sC g
sC g s C C g sC s a sa a
g sC s a sa a sC g sC g
(5.2.158)
( )( )( )( )( )( )
( ) ( )
5 4 3 252 42 32 22 12 02
19 21 1 26 16 06 6 6 6 6
52 4 4 5 1 26 25 4 6 6
42 26 6 6 2 15 4 25 4 6 6 25 4 6 6 6 6
4 4 1 26 4 4
+ + + + +=
− + + + +
= + −
= + − − +
− − +
m gd x x gd m
gd gs gd gd x x gd
x gd ds x m x gd x x m x gd
gd m gd gd gs
s b s b s b s b sb bH s
g sC s a sa a sC g sC g
b C C C C a a C C C
b a C C g a C a g C C a C C g g C
C g C a C C( )( )( )
( )( ) ( )
( ) ( )( )( )
5 26 1 1 16
32 26 6 6 2 26 6 16 6 6 2
15 4 25 4 6 6 6 6 25 4 6 6 15 4 05 4 6 6
4 4 26 1 1 16 4 4 5 16 1 1 06
22 26 6 16 6 6 2 16 6
−
= + +
+ − + − + −
+ − − + −
= + +
gd m gd
x m ds x x gd ds
x x x m x gd x x m x x x ds
gd m m gd gd gs gd m gd
x x m ds x
C a g C a
b a C g g a g a C C g
a C a g C g g C a C g g a g a C C g
C g a g C a C C C a g C a
b a g a C g g a g( )( ) ( )( )
( ) ( )( )
( ) ( )
06 6 6 2
15 4 25 4 6 6 15 4 05 4 6 6 6 6 05 4 6 6
4 4 16 1 1 06 4 4 5 06 1
12 16 6 06 6 6 2 06 6 6 2
15 4 05 4 6 6 05 4 6 6 6 6
+
+ − + − + −
+ − − +
= + +
+ − − + +
x gd ds
x x x m x x x m x gd x x gd
gd m m gd gd gs gd m
x x m ds x gd ds
x x x m x x m x gd
a C C g
a C a g g g a g a C C g g C a g C C
C g a g C a C C C a g
b a g a C g g a g C g
a g a C g g a g C g g C
( )4 4 06 1
02 06 6 6 2 05 4 6 6= −
gd m m
x m ds x x m
C g a g
b a g g g a g g g
(5.2.159)
Recall equation (5.2.138d) for convenience
(5.2.160)
73
( )
( )( )( )
3 233 23 13 03
20 226 16 06
33 26 2 3 2 26
23 26 3 16 2 3 2 16
13 16 3 06 2 3 2 06
03 06 3
+ + +=
+ +
= − + −
= − + −
= − + −
=
gs gd gd
m gs gd gd
m gs gd gd
m
s b s b sb bH s
s a sa a
b a C C C a
b a g a C C C a
b a g a C C C a
b a g
(5.2.161)
( ) ( )
( ) ( ) ( )( ) ( ) ( )( )
( ) ( ) ( )( ) ( ) ( )( )
( )
( )( )
( )
2 21 22
1521 5 5 11 3 2 3
18
1922 20 11 3 2 3
18
222 12 02
11 221 11 01
2 3 215
1 1
5 451 41
18
0+ =
= + + − +
= + − +
+ +=
+ +
+ + =−
+ +=
out
x x m gs gd
m gs gd
gs gd m
m gd
V H s V H s
H sH s g s C H s g s C C
H s
H sH s H s H s g s C C
H s
s a sa aH s
s a sa a
s C C gH s
g sC
s b s b sH s
( )( )( )( )
( )( )( )( )( )
( )
3 231 21 11 01
26 6 6 6 21 11 01 1 1
5 4 3 252 42 32 22 12 02
19 21 1 26 16 06 6 6 6 6
3 233 23 13 03
20 226 16 06
+ + +
+ + + + −
+ + + + +=
− + + + +
+ + +=
+ +
gd m x x m gd
m gd x x gd m
b s b sb bsC g sC g s a sa a g sC
s b s b s b s b sb bH s
g sC s a sa a sC g sC g
s b s b sb bH s
s a sa a
(5.2.162)
From equation (5.2.162), we saw that, it has many function inside this function so, you should separate group of function to perform polynomial multiplication as a smaller group
c c c c c c c c c c cd c c c c c c c c c c c c c c c c c c c cd c c c c c c c c c c c c c c c c c c c cd c c c79 14 69 24 59 34 49 44 39 54 29 64 19 74 09 84
145 143 03 133 13 123 23 113 33 103 43 93 53 83 63 73 73 63 83 53 93 43 103 d f d f d ff d f d f d f d f d f d f d f d f d f d f d f d f d f d ff d f d f d f d f d f d f d f d f d
125 123 03 113 13 103 23 93 33 83 43 73 53 63 63 53 73 43 f d f d f d f d ff d f d f d f d f d f d f d f d f d f d f d f d ff d f d f d f d f d f d f d f d f d f d f
105 103 03 93 13 83 23 73 33 63 43 53 53 43 63 33 73 23 83 13 9 d ff d f d f d f d f d f d f d f d f d f d ff d f d f d f d f d f d f d f d f d ff d f d f d f d f d f d f d f
236 232 03 222 13 212 23 202 33 192 43 182 53 172 63 162 73 152 83 142 93 13 f d f d f d ff d f d f d f d f d f d f d f d f d f d f d f d f d f d ff d f d f d f d f d f d
156 152 03 142 13 132 23 122 33 112 43 102 53 92 63 82 73 72 83 62 93 52 10 d f d f d ff d f d f d f d f d f d f d f d f d f d f d f d f d f d ff d f d f d f d f d f d f d f d f
136 132 03 122 13 112 23 102 33 92 43 82 53 72 63 62 73 d f d f d f d f d f d ff d f d f d f d f d f d f d f d f d f d f d f d f d ff d f d f d f d f d f d f d f
116 112 03 102 13 92 23 82 33 72 43 62 53 52 d f d f d f d f d ff d f d f d f d f d f d f d f d f d f d f d ff d f d f d f d f d f d f d f d f d f d ff d
97 94 03 84 13 74 23 64 33 54 43 44 53 34 63 d f d f df f d f d f d f d f d f d f d f d f df f d f d f d f d f d f d f d f df f d f d f d f d f d f d f d
There are at least 10 circuit techniques in Distributed Amplifier. This section discuss about circuit techniques which should be useful to extend gain per stage and bandwidth of the CMOS distributed amplifier. The first paper to be review is published by Ghadiri [8] since November 2010. The authors of this paper add additional circuit called negative capacitance cell (NCC) to conventional distributed amplifier with artificial transmission line which is believed to be the best technique for highest gain per stage with the same current consumption.
inRF 2gL
0Z2dL
2dL
dL dL
gL gL
2gL
oZ1M 2M 3M
1C−2C−
3C− oZ
DDV
outV
1LR 2LR
inV
inIinZ
1L1CM
2CM
1L
1LR
2LR
inV
inIinZ
, 1gs McC
( ), 1 , 1m Mc gs Mcg V
1
2
2
, 2gd McC, 2ds Mcg
, 1 , 1m Mc gs Mcg V
1
, 1ds Mcg
, 2gs McC
, 1gd McC
( ) Conventional Distributed Amplifiera ( ) NCCb( ) Equivalent Circuit of the proposed NCCc
Fig 5.6 Conventional CMOS Distributed Amplifier with additional NCC [8]
Because distributed amplifier concept used two types of coupling between gate terminal and gate terminal and drain terminal and drain terminal. It is called inductive coupling and transmission line coupling. It is good to review classic transmission line theory which appears in many textbook related with microwave engineering. Wave propagation in transmission line can be modeled as second order differential equation as following
( ) ( )2
22 0
d V zV z
dzγ− =
(5.3.1.1)
( ) ( )2
22 0
d I zI z
dzγ− =
(5.3.1.2)
The solution of these two equation can be written as following
( ) z zo oV z V e V eγ γ+ − −= +
(5.3.1.3)
( ) z zo oI z I e I eγ γ+ − −= +
(5.3.1.4)
( )( )j R j L G j Cγ α β ω ω= + = + +
(5.3.1.5)
For lossless transmission line, the attenuation factor can be approximated as zero
104
Thus, equation 5.3.1.5 can be simplified to
j j LCγ α β ω= + =
(5.3.1.6)
Compare imaginary part with imaginary part in equation (5.3.1.6), phase constant can be written as following
LCβ ω=
(5.3.1.7)
For sinusoidal steady state condition, the differential equation of lumped element or telegrapher equation can be written in phase form as
( ) ( ) ( )dV zR j L I z
dzω= − +
(5.3.1.8)
( ) ( ) ( )dI zG j C V z
dzω= − +
(5.3.1.9)
Differentiate equation (5.3.1.4), it can be written as following
( ) ( ) ( ) ( ) ( ) ( )z z z zo o o o
dI zI e I e I e I e G j C V z
dzγ γ γ γγ γ γ ω+ − − − + − = − + = − = − +
(5.3.1.10)
Differentiate equation (5.3.1.3), it can be written as following
( ) ( ) ( ) ( )( ) ( ) ( )z z z zo o o o
dV zV e V e V e V e R j L I z
dzγ γ γ γγ γ γ ω+ − − − + −= − + = − = − +
(5.3.1.11)
( ) ( ) ( )1 z zo oV e V e I z
R j Lγ γγ
ω+ − −
− = +
(5.3.1.12)
Characteristic impedance can be defined as following
( )( )oR j L R j L R j LZ
G j CR j L G j Cω ω ω
γ ωω ω+ + +
= = =++ +
105
(5.3.1.13)
Table5.2.1 Comparison of Transmission Line waves to uniform plane waves [12]
Transmission Line Uniform Plane Waves 2
22 0d V V
dzγ− =
22
2 0xx
d Ek E
dz+ =
22
2 0d I Idz
γ− = 2
22 0y
yd H
k Hdz
+ =
ZYγ = ˆˆjk zy= z z
o oV V e V eγ γ+ − −= + jkz jkzx o oE E e E e+ − −= +
z zo oI I e I eγ γ+ − −= + jkz jkz
y o oH H e H e+ − −= +
o oo
o o
V V ZZYI I
+ −
+ −= = − = ˆ
ˆo o
o o
E E zyH H
η+ −
+ −= = − =
P VI ∗= z x yS E H ∗=
What is uniform plane waves? Uniform plane waves may travel only in one direction without rotation like circular wave or rectangular waves. Such as electric field propagate into the x direction only and magnetic field propagate into the y direction only. Another meaning of uniform plane waves may have constant amplitude.
106
5.3.2 Analysis of Conventional CMOS Distributed Amplifier with Lossless and Lossy Transmission Line Theory [11]
2gL
0Z2dL
2dL
dL dL
gL gL
2gL
oZ1M 2M 3M
1C− 2C−3C− oZ
DDVoutV
( ) Conventional Distributed Amplifiera
1
2gL
0Z
oZ
1C− 2C−3C− oZ
DDV
, 1gs MC
, 1gd MC
1 1m gsg V, 2gs MC
2 2m gsg V
, 2gd MC
, 1ds Mg , 2ds Mg, 2db MC
, 1db MC , 3gs MC , 3gd MC
3 3m gsg V
, 3ds Mg, 3db MC
1V2V 3V 4V
5V 6V 7V 8V
( ) Equivalent Circuit of Conventional CMOS Distributed Amplifierb
inV
inV
1
2dL
2dL 3dL 4
2dL
2gL 3gL 4
2gL
outV
Fig. 5.9 (a) Conventional Distribute Amplifier with NCC [8]
(e) Equivalent Circuit of Conventional Distributed Amplifier with NCC [8]
( ) ( ) ( ) ( )1 1 211 1 1 6 1
1 1 2
2
ings gd
Lg C Lg
V V V VV V sC V V sCZ Z Z− −
= + + + −
(5.3.2.1)
( ) ( ) ( )( )1 1 11 1 1 1 1 2 6 1 1
1 2 21Lg Lg Lg
in Lg gs Lg gd Lg gdC Lg Lg
Z Z ZV V Z sC Z sC V V Z sC
Z Z Z
= + + + + − −
(5.3.2.2)
107
( ) ( ) ( ) ( ) ( )5 6 6 71 6 1 1 1 6 1 1
1 22
gd m ds dbLd Ld
V V V VV V sC g V V g sC
Z Z− −
− + = + + +
(5.3.2.3)
( ) ( ) 1 11 1 1 1 1 5 6 1 1 1 1 1 7
2 22 2 Ld Ld
gd Ld m Ld ds db Ld gd LdLd Ld
Z ZV sC Z g Z V V g sC Z sC Z VZ Z
− + = + + + + −
(5.3.2.4)
5.4 The proposed architecture of CMOS 3 section distributed amplifier
By combine the concept of architecture of distributed amplifier with modified
complementary regulated cascode amplifier. The new architecture of CMOS 3 sections distributed amplifier based on modified complementary regulated cascode amplifie can be drawn in figure 5.6 and figure 5.7
inRF 2gL
0Z2dL
2dL
dL dL
gL gL
2gL
oZ
1C−2C−
3C− oZ
DDV
outV
1LR 2LR
inV
inIinZ
1L1CM
2CM
1L
1LR
2LR
inV
inIinZ
, 1gs McC
( ), 1 , 1m Mc gs Mcg V
1
2
2
, 2gd McC, 2ds Mcg
, 1 , 1m Mc gs Mcg V
1
, 1ds Mcg
, 2gs McC
, 1gd McC
( ) Conventional Distributed Amplifiera ( ) NCCb( ) Equivalent Circuit of the proposed NCCc
CRGCA CRGCA CRGCA
Figure 5.10 The proposed architecture of CMOS 3 section distributed amplifier
108
2gL
0Z2dL
2dL
dL dL
gL gL
2gL
oZ
1C− 2C−3C− oZ
DDVoutV
( ) Conventional Distributed Amplifiera
0Z
oZ
1C− 2C−3C− oZ
DDV
1V2V 3V 4V
5V 6V 7V8V
( ) Equivalent Circuit of Conventional CMOS Distributed Amplifierb
inV
inV
1
2dL
2dL 3dL 4
2dL
2gL 3gL 4
2gL
outV
1
2gL
CRGCA CRGCA CRGCA
1m ing V
( )2 2 1mg V V−
1dsg
( )4 4 3mg V V−
( )4 30mbg V−
4dsg
( )6 0m ing V −
6dsg
outV
1V
3V 5 6gs dbC C+
3 1gs dbC C+
2 4db dbC C+
3, 2, 7D G D
8 4mg V
4V4V
2V
8 8 5gs db dbC C C+ +
7 2mg V
7 7 3gs db dbC C C+ +
2dsg
( )2 10mbg V−
1 7/ /B dsR g
2 8/ /B dsR g
4 5gs gdC C+
2 3gs gdC C+
2gdC
4gdC
1gdC
1gsC
5 3mg V5dsg
3 1mg V3dsg
6gsC 6gdC
1m ing V
( )2 2 1mg V V−
1dsg
( )4 4 3mg V V−
( )4 30mbg V−
4dsg
( )6 0m ing V −
6dsg
outV
1V
3V
5 6gs dbC C+
3 1gs dbC C+
2 4db dbC C+
3, 2, 7D G D
8 4mg V
4V4V
2V
8 8 5gs db dbC C C+ +
7 2mg V
7 7 3gs db dbC C C+ +
2dsg
( )2 10mbg V−
1 7/ /B dsR g
2 8/ /B dsR g
4 5gs gdC C+
2 3gs gdC C+
2gdC
4gdC
1gdC
1gsC
5 3mg V5dsg
3 1mg V3dsg
6gsC 6gdC
1m ing V
( )2 2 1mg V V−
1dsg
( )4 4 3mg V V−
( )4 30mbg V−
4dsg
( )6 0m ing V −
6dsg
outV
1V
3V
5 6gs dbC C+
3 1gs dbC C+
2 4db dbC C+
3, 2, 7D G D
8 4mg V
4V4V
2V
8 8 5gs db dbC C C+ +
7 2mg V
7 7 3gs db dbC C C+ +
2dsg
( )2 10mbg V−
1 7/ /B dsR g
2 8/ /B dsR g
4 5gs gdC C+
2 3gs gdC C+
2gdC
4gdC
1gdC
1gsC
5 3mg V5dsg
3 1mg V3dsg
6gsC 6gdC
Fig. 5.11 The proposed architecture of CMOS 3 section distributed amplifier
(a) Architecture of CMOS 3 section distributed amplifier (b) small signal high frequency equivalent circuit of (a)
5.5 Reference
[1] E. L. Ginzton, W. R. Hewlett, J. H. Jasberg, J. D. Noe, “ Distribute Amplification”, Proceeedings of the I.R.E, August 1948, pp. 956-969
[2] B. J. Hosticka, “ Improvement of the Gain of MOS Amplifiers”, IEEE Journal of Solid-State Circuits, Vol. SC-14, No.6, December 1979, pp. pp. 1111-1114
[3] S. Kimura, Y. Imai, “ 0-40 GHz GaAs MESFET Distributed Basedband Amplifier IC’s for High-Speed Optical Transmission”, IEEE Transactions on Microwave Theory and Techniques, Vol.44, No.11, November 1996, pp. 2076-2082
[4] B. Y. Banyamin, M. Berwick, “ Analysis of the Performance of Four-Cascaded Single-Stage Distributed Amplifiers”, IEEE Transactions on Microwave Theory and Techniques, Vol.48, No.12, December 2000, pp. 2657-2663
109
[5] R. C. Liu, C. S. Lin, K. L. Deng, H. Wang, “Design and Analysis of DC to 14 GHz and 22 GHz CMOS Cascode Distributed Amplifiers”, IEEE Journal Solid State Circuit, Vol.39, No.8, August 2004, pp. 1370-1374
[6] J. C. Chien, L. H. Lu, “ 40 Gb/s High-Gain Distributed Amplifiers with Cascaded Gain stages in 0.18 um CMOS”, IEEE Journal of Solid-State Circuits, Vol.42, No.12, December 2007, pp. 2715-2725
[7] A. Arbabian, A. M. Niknejad, “ Design of a CMOS Tapered Cascaded Multistage Distributed Amplifier”, IEEE Transactions on Microwave Theory and Techniques, Vol. 57, No.4, April 2009, pp. 938-947
[8] A. Ghadiri, K. Moez, “Gain-Enhanced Distributed Amplifier Using Negative Capacitance”, IEEE Transactions on Circuits and Systems I, Regular Papers, Vol.57, No.11, November 2010, pp. 2834-2843
[9] Y. S. Lin, J. F. Chang, S. S. Lu, “ Analysis and Design of CMOS Distributed Amplifier using Inductively Peaking Cascaded Gain Cell for UWB Systems”, IEEE Transactions on Microwave and Techniquesk, Vol.59, No.10, October 2011, pp. 2513-2524
[10] A. Jahanian, P. Heydari, “ A CMOS Distributed Amplifier with Distributed Active Input Balun Using GBW and Linearity Enhancing Techniques”, IEEE Transactions on Microwave Theory and Techniques, Vol. 60, No.5, May 2012, pp. 1331-1341
[11] D. M. Pozar, “ Microwave Engineering”, 2nd edition, copyright 1998, John Wiley &Sons
[12] R. F. Harrington, “ Time-Harmonic Electromagnetic Fields”, copyright 1961, Mcgraw-Hill, pp.61-63
110
Chapter6 Transimpedance amplifier design based on T network
6.1 Literature Review
6.1.1 Introduction
Transimpedance amplifier is the special circuit which converts input current from photodiode to output voltage. There are many topologies which have been proposed in the literature. But there are many basic topologies of transimpedance amplifier, the first topology which should be discussed here is common source based transimpedance amplifier and common source based transimpedance amplifier with resistive feedback. The figure of these circuit can be shown in figure 6.1
PDV DDV
1DRoutV
1MinI 1gsC
1 1m gsg V 1dsg1DR
outV
PDV DDV
2DRoutV
2MinI 2gsC
2 2m gsg V 2dsg2DR
outV1gdCFR
FR
( )a ( )b
( )c ( )d
1dbC
1gdC
2dbC
Figure 6.1 (a) Transimpedance amplifier based on common source
(b) small signal high frequency equivalent circuit of (a)
(c) Transimpedance amplifier amplifier based on common source with resistive feedback
6.1.2 Frequency Response of Transimpedance amplifier based on common souce with and without resistive feedback
It should be interesting to study what are the difference in some of the circuit properties of these two circuit frequency response which is called transimpedance gain and -3dB bandwidth of the circuits. The transimpedace gain of figure 6.1(a) can be derived as following formula
111
( ) ( ) ( )( )2 21 1 1 1 1 1 1 1 1 1 1 1
1in gd m out ds gs gd gd m db gd gs gd gd
DI sC g V s g C C C g s C C C C C
R − = + + + + + + −
(6.1)
( ) ( ) ( )( )2 21 1 1 1 1 1 1 1 1 1 1 1
1in gd m out ds gs gd gd m db gd gs gd gd
DI sC g V s g C C C g s C C C C C
R − = + + + + + + −
(6.2)
( )
( ) ( )( )
( )( )
( )
1 11
2 21 1 1 1 1 1 1 1 1 1
21 1 1 1 1
1 1 1 1 1
1
1
0
gd moutTIA
inds gs gd gd m db gd gs gd gd
D
db gd gs gd gd
ds gs gd gd mD
sC gV ZI
s g C C C g s C C C C CR
a C C C C C
b g C C C gR
c
−= =
+ + + + + + −
= + + −
= + + +
=
(6.3)
This transfer function has denominator which is 2nd order polynomial which can be factored as a two pole frequencies system as following
( ) ( )
( )( )1 1 1 1 1 1 1 1 1 12
1, 2 21 1 1 1 1
1 14
2 2
ds gs gd gd m ds gs gd gd mD D
p pdb gd gs gd gd
g C C C g g C C C gR Rb b acf
a C C C C C
− + + + ± + + +
− ± − = = + + −
(6.4)
It can be seen that one pole is cancelled by itself to zero, as a result, this circuit is single pole system.
( )( )( )
1 1 1 1 12
1, 2 21 1 1 1 1
14 1
2 2
ds gs gd gd mD
p pdb gd gs gd gd
g C C C gRb b acf
a C C C C Cπ
+ + +
− ± − = = − + + −
(6.5)
The transimpedance gain of figure 6.1(c) can be derived as following formula
112
( )( ) ( ) ( )1 121 1 1 1 1 1 1
12 21 11 1 1
11
1 1
1 1 1
gs gddb gd gs gd gs gd ds
D F F
gdin gd m outgd gd m
Fds
D F F m
F
C Cs C C C C s C C g
R R R
CI sC g V s C s C gR
gR R R g
R
+ + + + + + + + − = + −
+ + + − −
(6.6)
( )
( ) ( )
1 12 2 2 2
1 1 1 1 1 1 1 1
1 11 1 1 1 1 1
1 1 1 1 1 1
1
1 1 1 1 1
gd moutTIA
in db gs db gd gd gs gd gd
gs mgs gd ds gd m ds
D F F D F F F
db gs db gd gd gs
gs
sC gVZ
I s C C C C C C C C
C gs C C g C g gR R R R R R R
a C C C C C C
b C
− = = + + + − + + + + + + + + + +
= + +
= +( ) ( )11 1 1 1
11
1 1
1 1 1
gsgd ds gd m
D F F
mds
D F F F
CC g C g
R R R
gc g
R R R R
+ + + +
= + + +
(6.7)
It can be seen that this transimpedance gain has denominator which is 2nd order polynomial which can be factored as a two pole frequencies system as following
( ) ( )
( ) ( )
( )
11 1 1 1 1
21
1 1 1 1 1
11 1 1 1 1 1 12
1, 2
1 1
1 1
1 1 144 1
2 2
gsgs gd ds gd m
D F F
gsgs gd ds gd m
D F F
mdb gs db gd gd gs ds
D F F Fp p
CC C g C g
R R R
CC C g C g
R R R
gC C C C C C gR R R Rb b acf
a π
− + + + + + ±
+ + + + +
− + + + + + − ± − = = ( )1 1 1 1 1 12 db gs db gd gd gsC C C C C C+ +
(6.8)
113
Figure 6.2 Magnitude and Phase Response of Transimpedance amplifier
In figure 6.1(a), 6.1(c) @ 10 microampere
Table 6.1 Circuit parameters from Simulation Results from figure 6.1 (a), 6.1 (c)
1 2 2.387m mg g µ= = 1 2 160D DR R k= = Ω 111 1.4167 10zbω = ×
, 1 , 210D M D MI A Iµ= = 5FR k= Ω
6.1.3 Frequency response of Transimpedance amplifier with and without resistive feedback with parasitic of photo diode and resistive bias circuit
It is well known that photodiode has parasitic capacitance in the range of several hundred femtofarad to several picofarad which depend on the speed of the photodiode. This section will discuss what is the effect of parasitic capacitance of photo diode and resistive bias circuit.
Figure 6.3 (a) Transimpedance amplifier based on common source with bias
circuit and parasitic of photo diode
(b) small signal high frequency equivalent circuit of (a)
(c) Transimpedance amplifier amplifier based on common source with resistive feedback, bias circuit and parasitic of photo diode
(d) small signal high frequency equivalent circuit of (c)
The transimpedace gain of figure 6.3(a) can be derived as following formula
( )
( )( )
( ) ( )
2 21 1 1 1 1
1 1 1 1 1 1 11 2
1 1 11 2
1 1 1
1 1 1
gd db PD gs gd gd
in gd m gd db PD gs gd ds outG G D
ds m gdD G G
s C C C C C C
I sC g s C C C C C g VR R R
g g CR R R
+ + + − − = + + + + + + + + + + +
(6.9)
115
( )
( )
( ) ( )
21 1 1 1 1 1 1 1
1 1 1 1 1 1 11 2
1 1 11 2
1 1 1
1 1 1
gd PD gd gs db PD db gs db gd
in gd m gd db PD gs gd ds outG G D
ds m gdD G G
s C C C C C C C C C C
I sC g s C C C C C g VR R R
g g CR R R
+ + + + − = + + + + + + + + + + +
(6.10)
( )
( )
( ) ( )
1 13
21 1 1 1 1 1 1 1
1 1 1 1 11 2
1 1 11 2
1 1 1 1
1 1 1
1 1 1
gd mTIA
gd PD gd gs db PD db gs db gd
gd db PD gs gd dsG G D
ds m gdD G G
gd PD gd gs db
sC gZ
s C C C C C C C C C C
s C C C C C gR R R
g g CR R R
a C C C C C
−= + + + + + + + + + + + + + + +
= + +( )
( ) ( )
1 1 1 1
1 1 1 1 11 2
1 1 11 2
1 1 1
1 1 1
PD db gs db gd
gd db PD gs gd dsG G D
ds m gdD G G
C C C C C
b C C C C C gR R R
c g g CR R R
+ +
= + + + + + +
= + + +
(6.11)
It can be seen that this transimpedance gain has denominator which is 2nd order polynomial which can be factored as a two pole frequencies system as following
( ) ( )
( ) ( )
( )
1 1 1 1 11 2
2
1 1 1 1 11 2
1 1 1 1 1 1 1 1 11
1, 2
1 1 1
1 1 1
1 1 141
2
gd db PD gs gd dsG G D
gd db PD gs gd dsG G D
gd PD gd gs db PD db gs db gd dsD G
p p
C C C C C gR R R
C C C C C gR R R
C C C C C C C C C C gR R R
fπ
− + + + + + +
+ + + + + +
±
− + + + + + + =
( )1 1
2
1 1 1 1 1 1 1 12
m gdG
gd PD gd gs db PD db gs db gd
g C
C C C C C C C C C C
+ + + + +
(6.12)
116
Figure 6.4 Magnitude and phase response of
Transimpedance amplifier of figure 6.3(a) @ 10 microamperes
It can be seen from the graph that the photodiode parasitic capacitance can make the transimpedance gain more constant but it can be seen that the transimpedance gain at 100MHz reduced from 118dB to 82.4 dB
Table 6.2 Circuit parameters from Simulation Results from figure 6.1 (a), 6.1 (c)
6.1.4 Frequency response of Transimpedance amplifier with and without resistive feedback with parasitic of photo diode and resistive bias circuit and
π type inductor peaking (PIP)
The circuit called π type inductor peaking (PIP) is the circuit technique to extend bandwidth at the input of the transimpedance amplifier which is published by J. J. Jin [5]. Denominator of the transimpedance gain of this circuit can be derived to have third order polynomial. The circuit is redrawn in figure 6.6
inI1L
2L
3L
1R 2R
outV
Figure 6.6 π type inductor peaking (PIP)
( ) ( ) ( )
( )( )
23 23 2 1 2 32 2 2 1 2 2
1 2 3 2 1 2 2 3 2 3 11 1 1 1 1
1 2 2 2 2 1 2
1TIAZ
L L L L LL L R L L Rs L L s L L L L L L L L LR R R R R
Figure 6.8 (a) Transimpedance amplifier based on common source with bias
circuit and PIP
(b) small signal high frequency equivalent circuit of (a)
(c) Transimpedance amplifier amplifier based on common source with resistive feedback, bias circuit and PIP
(d) small signal high frequency equivalent circuit of (c)
Figure 6.9 Magnitude and Phase response of figure 6.8 (a) , 6.8 (c)
102
104
106
108
1010
-225
-180
-135
-90
-45
0
45
90
135
180
Phas
e (d
eg)
Bode Diagram
Frequency (Hz)
80
100
120
140
160
180
200
System: f ig6_8cFrequency (Hz): 9.78Magnitude (dB): 195
System: f ig6_8aFrequency (Hz): 996Magnitude (dB): 173
System: f ig6_8cFrequency (Hz): 3.04e+07Magnitude (dB): 192
Mag
nitu
de (d
B)
f ig68c
fig68a
120
Usually, it is based on cascade common source amplifier. Kim [8] proposed series silicon inductor between input terminal and gate terminal of the transistor.
Input1 485L pH=
1 1R k= Ω2 65R = Ω
195FR = ΩBIASI
DDV
1M 2M
2 165L pH=
3 210L pH=3M
3 65R = Ω4M
4 365L pH=
5 565L pH=BIASV
Fig 6.10 Common source transimpedance amplifier with resistive feedback and inductive degeneration at gate terminal [8]
121
6.1.5 Equivalent input noise voltage response of Transimpedance amplier
There are two types of noise in CMOS technology. The first type is flicker noise which is dominant at low frequency. The second type is thermal noise which is constant as a function of frequency. The flicker noise voltage mean square equation can be rewritten here [2]
2, ker
1n flic
ox
KVC WL f
=
(6.1.4.1)
K is a process dependent constant, f is input frequency, Cox is oxide capacitance of the CMOS process. The flicker noise current mean square can be rewritten here
2 2, ker
1n fli m
ox
KI gC WL f
=
(6.1.4.2)
The thermal noise voltage mean square can be rewritten here as following
2 4n mV kT gγ=
(6.1.4.3)
The thermal noise current mean square can be rewritten here as following
2,
843n thermal m mI kT g kTgγ= =
(6.1.4.4)
122
PDV DDV
outV
1M
PDV DDV
outVFR
( )a ( )b
( )c ( )d
2DR
2M
1DR
DDV
1M
1DR
ac
1
2, Dn RI
1
2,n MI
DDV
FR2DR
2Mac
2
2, Dn RI
2
2,n MI
2, Fn RI
Figure 6.11 (a) Transimpedance amplifier based on common source
(b) Mean squared noise current source of (a)
(c) Transimpedance amplifier amplifier based on common source with resistive feedback
(d) Mean squared noise current source of (c)
123
Table 6.5 Performance Comparison of TIA with different technology Ref Process BW
( )GHz ( )TZ dBΩ GD
(psec) Noise
/pA Hz Supply
(V) Power (mW)
Area
( 2mm )
Figure of Merit
[4] 0.18mµ
9.2 54 2.5 137.5 0.64
[5] 0.18mµ
30.5 51 55.7 1.8 60.1 1.17 0.46×
[6] 0.18mµ
4.3 54.5 1.5 11.5 0.0077
[7] 0.18mµ
6.2-10.5
47.8 1.8 33.3 0.9 0.6×
[8] 0.13mµ
29
50 16 51.8 1.5 45.7 0.4
[10] 65 nm 46.7 30 39.9 [11] 0.18
mµ 8
@ 0.25pF 53 20± 18 1.8 13.5 0.45 0.25×
[12] 0.18mµ
7 @ 0.2 pF
55 65 10± 17.5 1.8 18.6 0.45 0.25×
6.5 References
[1] E. Sackinger, “ Broadband Circuits for Optical Fiber Communication”, John Wiley & Sons, copyright 2005
[2] B. Razavi, “ RF Microelectronics”, Second edition, Prentic-Hall, copyright 2012
[3] A. A. Abidi, “Gigahertz Transresistance Amplifiers in Fine Line NMOS”, IEEE Journal of Solid-State Circuits, Vol. SC-19, No.6, December 1984, pp. 986-994
[4] B. Analui, A. Hajimari, “ Bandwidth Enhancement for Transimpedance Amplifier”, IEEE Journal of Solid-State Circuits, Vol.39, No.8, August 2004, pp. 1263-1270
[5] J. D. Jin, S. S. H. Hsu, “ A 40 Gb/s Transimpedance Amplifier in 0.18 um CMOS Technology”, IEEE Journal of Solid State Cricut, Vol.43, No.6, June 2008, pp. 1449-1457
[6] S. S. H. Hsu, W. H. Cho, S. W. Chen, J. D. Jin, “ CMOS Broadband amplifiers for Optical Communicatinos and Optical Interconnects”, RFIT2011, pp. 105-108
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