Chapter5 CMOS_Distributedamp_v244

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.

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
2M
3M
BR
LRAR
CR
LC
LL
inV
CR
AR
BR
LR
LL
LC
1dsg
1dsg
1 1m gsg V
1 1m gsg V
3 3m gsg V
2gsC
2gdC
1dbC
1gdC
1gsC
3gsC
3gdC
3dsg
3dbC
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 2
1 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
9
9.5 10 oxide thicknessoxT m−
= × =
(1)
( )8 2
460 10 / sec mobility of NMOSUon cm V carrier= × × =
(2)
( )8 2
115 10 / sec mobility of PMOSUop cm V carrier= × × =
(3)
11
3.45 10 /oxide F mε −
= ×
(4)
15
2
Oxide Capacitance =3.63 10ox
F
C
mµ
−
= ×
(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)
1/2
0.5 [V ] body effect parameter of NMOS threshold voltagegamman γ= = =
(9)
1/2
0.45 [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 C
kp Uop C
= ×
= ×
(13)
6
0.08 10 lateral diffusion into the channel from source to drain diffusion regions of NMOSLovn m−
= × =
(14)
6
0.09 10 lateral diffusion into the channel from the source to drain diffusion regions of PMOSLovp m−
= × =
(15)

7
min
min
2
2
effN
effP
L L Lovn
L 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)
1
1 / 1
MJ
db
db a
V
C CJ AD
PB
  
=× +  
  
(2.1)
( ) 1
1 / 1
MJSW
db
db b
V
C CJSW PD
PB
  
= × +  
  
(2.2)
2
3 3gd gda C C=
(2.3)
( ) ( ) ( )2
2 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 2
1
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 C
a
C g g g g g C g
 − − + + +
 =
 + − − − 
(2.5)
( )0 2 2 2 2 3
1
mb m ds m ds
B
a g g g g g
R
 
= − − + 
 
(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 ds
B
L db gs gd gd ds L m gd
L
L C C C g
R
b
L C C C C g L g C
R
  
+ + +  
  =   
 + + + + + + 
   
(2.8)
1 2 3
1 1
L ds ds
L B
b L g g
R R
  
= + +  
  
(2.9)
( )0 3 3 2 3 2
1
ds db gs gd gd
B
b g C C C C
R
 
= + + + + + 
 
(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.
ind
outd
w
s
( )a
( )b
( )c ( )d
ind
outd
s
w
ind
outd
w s
ind
outd
s
w
Fig. 2.2 Silicon Inductor with various shapes
(a) Square (b) octagonal (c) hexagonal (d) circular

9
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 0
21
avg
MW
n d
L K
K
µ
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 s
l
ρ ρ
+ −
< <
2
in out
avg
d d
d
+
=
For square silicon inductor, if someone want to design 1 nanohenry with
modified Wheeler how can he approximate , avgdρ
( )( )
( )
( )
( ) ( )
42 13
9 7
1 0
2
4 4
4
6
300 10 8821.59 10
1 10 2.34 4 10
1 1 2.75 1 2.75
1 2.75 8821.59 10 8821.59 10 1 2.75 0.9 2.475
3.475
3.93
8821.59 10
1 2.75 8821.59 10 8821.59 10
avg
MW
nn d n
L K
K
n n
n
n n
µ π
ρ ρ ρ
ρ
ρ
− −
− −
− −
−
−
 ×  ×  = =× = × =
   + + +   
+= × → × −= =
= =
×
+ = × → ×( ) ( )6
4
1 2.75 0.1 0.275
1.275
1.44
8821.59 10
n
−
−
−= =
= =
×
(2.3.2)

10
( ) ( ) ( )( )
( )
6 6
5 5
5
3.93 14 10 2.93 4 101
0.9=
5.502 10 1.172 10
7.415 10
0.9
nw n s
l 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.
2
1 22
3 4ln
2
avg
GMD
n d c c
L c c
µ
ρ ρ
ρ
   
 = + +       
(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 6
2 9
13 2 9
4
2
4 10 300 10 1.27 2.07
ln 0.18 0.13 1 10
2
if 0.9
2393.89 10 0.8329 0.162 0.1053 10
10
3.7968 1.9485 2
2633.7577
GMD
GMD
n
L
L n
n n
π
ρ 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 4
mono out avgL d w d n sα αα α α
β=
(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

11
( )
( )
( )0
0
0
tanh
tanh
L
in
L
Z Z l
Z Z
Z Z l
γ
γ
+
=
+
( ) ( )
( ) ( )
( ) ( )0 0 0tanh tanh
j l j l
in j l j l
e e
Z Z l Z j l Z
e e
α β α β
α β α β
γ α β
+ − +
+ − +
 −
= = + =    
+  
( )
( ) ( )( ) ( )
( ) ( )( )
( )
( ) ( )( ) ( )
( ) ( )( )
0
cos sin cos sin
cos sin cos sin
l l
in l l
e l j l e l j l
Z Z
e l j l e l j l
α α
α α
β β β β
β β β β
−
−
 + − −
 =
 + + − 
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
2 3 4 5 2 3 4 5
0 2 3 4 5 2 3 4 5
1 1
2 3! 4! 5! 2 3! 4! 5!
1 1
2 3! 4! 5! 2 3! 4! 5!
in
l l l l l l l l
l l
Z Z
l l l l l l l l
l l
γ γ γ γ γ γ γ γ
γ γ
γ γ
    − − − −
    + + + + + − − + + + +
       =  
    − − − −
   + + + + + + − + + + + 
        
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
2 3 4 5 2 3 4 5
0 2 3 4 5 2 3 4 5
1 1
2 3! 4! 5! 2 3! 4! 5!
1 1
2 3! 4! 5! 2 3! 4! 5!
in
l l l l l l l l
l l
Z Z
l l l l l l l l
l l
γ γ
γ γ
    
    + + + + + − − + − + −
       =  
    
   + + + + + + − + − + − 
        
( )3 51 2 4 9 3 1.21 0.147 2.40 1.78 0.030
10 1.62 10mono out avg out avgL d w d n s d w d n sα αα α α
β − − − − −
= = = ×
(2.3.7)
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
9 3 1.21 0.147 2.40 1.78 0.030
log10 log 1.62 10 9
9 2.790 1.21 log 0.147 log 2.40log 1.78log 0.030log
out avg
out avg
d w d n s
d w d n s
− − − − − =× =−
 
− =− − − + + −
(2.3.8)
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 l
l l
n n R j L
Z Z Z
l l l l l l
n n
γ γ γ γ γ γ
γ γ
ω
γ
γγ γ γ γ γ γ
= =
= =
      
      + + + + + + + +
        +   = =     
        
   + + + + + + + +   
            
( )
( )
( ) ( ) ( )
( ) ( ) ( )
2 4
2 4
1 ...
3! 5! !
1 ...
2! 4! !
n even
n even
l l l
n
l
l 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 l
n
Z Rl j Ll
l l l
n
γ γ γ
ω
γ 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.
1L 2L
1C
2C
1R 2R
DDV
1M
2M
1L 2L
1C
2C
1R 2R
DDV
1 2mg V
1dsg
2 1mg V
1gsC
2gsC
1gdC 2gdC
2dsg
1V
2V
1V
2V
inV
inI
( )a ( )b
Figure 3.1 (a) Simple Cross Couple Oscillator
(b) Input Impedance Analysis of figure 3.1 (a)
( )2
1 2 1 2 2
2
4 3 2
4 3 2 1
1
1
1
x ds
in
in
in
sL s L C sL g
RV
Z
I s a s a s a sa
  
+ + +   
  = =
+ + + +
(3.2.1)
1 2 1 1 2 2
2 2 2 1 1 1
x db gs gd gd
x gs gd db gd
C C C C C C
C C C C C C
= + + + +
= + + + +
(3.2.2)

14
( )
( )
2
4 1 2 2 1 1 2 1 2
3 1 2 2 2 1 2 1 1 1 2 2 1 2
2 1
2 2
2 1 2 2 1 1 2 1 2 2 2
1 2
1 1 1 2 2
1 2
0
1 1
2
1 1
1 1
1
x x gd gd
x ds x ds m gd gd
x x ds ds m
ds ds
a L C L C L C C C
a L C L g L L C g L L g C C
R R
a L C L C L L g g L g
R R
a L g L g
R R
a
= − +
   
= + + + + +   
   
  
= + + + + −  
  
   
= + + +   
   
=
(3.2.3)
( )
( )
( ) ( )
3 2
1 2 1 1 2 2
2
4 2 3
4 2 1 3
1
1
1
x ds
in
in
in
j L L C L L g
RV
Z s j
I a a j a a
ω ω
ω
ω ω ω ω
  
− − + +   
  = = =
− + + −
(3.2.4)
Multiply both numerator and denominator with ( ) ( )4 2 3
4 2 1 31a a j a aω ω ω ω− + − − which is
complex conjugate of denominator so that we can separate symbolic real part and
symbolic imaginary part of the input impedance
( )
( )
( ) ( )
( ) ( )
( ) ( )
3 2
4 2 31 2 1 1 2 2
4 2 1 32
4 2 3 4 2 3
4 2 1 3 4 2 1 3
1
1
1
1 1
x ds
in
j L L C L L g
a a j a aR
Z j
a a j a a a a j a a
ω ω
ω ω ω ω
ω
ω ω ω ω ω ω ω ω
   
− + − +        − + − −     ×
 − + + − − + − −
 
(3.2.5)
( )
( ) ( ) ( )( )
( ) ( )
3 2 4 2 3
1 2 1 1 2 2 4 2 1 3
2
2 24 2 3
4 2 1 3
1
1 1
1
x ds
in
j L L C L L g a a j a a
R
Z j
a a a a
ω ω ω ω ω ω
ω
ω ω ω ω
   
− + − + − + − −        =
− + + −
(3.2.6)
( )
( )( ) ( )
( )( ) ( )
( ) ( )
3 3 2 4 2
1 2 1 1 3 1 2 2 4 2
2
3 4 2 2 3
1 2 1 4 2 1 2 2 1 3
2
2 24 2 3
4 2 1 3
1
1 1
1
1 1
1
x ds
x ds
in
L L C a a L L g a a
R
j L L C a a L L g a a
R
Z j
a a a a
ω ω ω ω ω ω
ω ω ω ω ω ω
ω
ω ω ω ω
   
− − + − + − +    
   
 
    − − + + − + −     
     =
− + + −
(3.2.7)
From equation (3.2.7) we can separate symbolic resistance and symbolic reactance
which are a function of frequency as following

15
( )
( )( ) ( )
( ) ( )
3 3 2 4 2
1 2 1 1 3 1 2 2 4 2
2
2 24 2 3
4 2 1 3
1
1 1
1
x ds
in
L L C a a L L g a a
R
R
a a a a
ω ω ω ω ω ω
ω
ω ω ω ω
   
− − + − + − +    
    =
− + + −
(3.2.8)
( )
( )( ) ( )
( ) ( )
3 4 2 2 3
1 2 1 4 2 1 2 2 1 3
2
2 24 2 3
4 2 1 3
1
1 1
1
x ds
in
j L L C a a L L g a a
R
X
a a a a
ω ω ω ω ω ω
ω
ω ω ω ω
    
− − + + − + −     
      =
− + + −
(3.2.9)
The second methodology is based on feedback model concept which can be
drawn as following figure
1L 2L
1C
2C
1R 2R
DDV
1M
2M
2L
2C
2R
DDV
2 1mg V
2gsC
2gdC
2dsg
1V
2V
2V
inV
inI
( )a
1L
1C
1R
1 2mg V
1dsg 1gsC
1gdC1V
1V
( )b
Figure 3.2 (a) Simple Cross Coupled Oscillator
(b)Transfer function of simple cross coupled Oscillator
Gain stage transfer function can be derived as following
( )
( )
gd m
gd ds
sC g sLV
A
V L
s C C L s g L
R
−
= =
 
+ + + + 
 
2 2 22
1 2 2
2 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 L
s C C L s g L
R
β
−
= =
 
+ + + + 
 
1 1 11
2 2 1
1 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
L
s C C L s g L
RV A
V A
sC g sL sC g sL
L L
s C C L s g L s C C L s g L
R R
β
−
 
+ + + + 
 = =
+    
   
− −   +
      
   + + + + + + + +   
         
2 2 2
2 2
2 2 2 2 2
22
1 1 1 2 2 2
2 21 2
1 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
1C
2C
1M
3M2M
DDV
3R 4R
2L
1R
1L
1C
2R
2C
3R
4R
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
inV
1m ing V
1outR
2outR
2 1m outg V
1outC
2outC
1outV
probeZ
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 −
CL
outV
DDV
SSV
inV
1m ing V
1outR
2outR
CL
2 1m outg V
1outC
2outC
1outV
probeZ
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 2
1 1 2 1 1
4 3 1 1 1
1 1 1 2 1 1 1 2
2 1 1
2 1 1 1
1 1 1 2 1 2
1 2
1 1
m C m m C
probe probeout
in C C C
C C C C
out in out
C C C
C C C m
probe out out
s g L g s g L
Z ZV
V L L L
s L C L C s L C L C
r Z r
L L L
s L C L C L g
Z r r
      
   − − + −         
      = −
  
+ + +  
   
 
+ + + + −  
 
1 1 1
1 2
2C C C
probe out out
L L L
s
Z 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
-30
-25
-20
-15
-10
-5
0
5
System: sys
Frequency (Hz): 3.05e+09
Magnitude (dB): 0.382
Magnitude(dB)
10
9
10
10
45
90
135
180
225
270
Phase(deg)
Bode Diagram
Frequency (Hz)

22
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
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
System: sys
Magnitude(dB)
10
9
10
10
45
90
135
180
225
270
Phase(deg)
Bode Diagram
Frequency (Hz)

23
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.
inV
gC gC gC gC gC gC
gL gL gL gL
pLpLpL pL
pC pC pC pC pC
B + B +4
4
output
3
3
1 2
21
Fig 5.1 Basic distributed amplifier based on TWT

24
5.2 Complementary Input Regulated Cascode amplifier
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.
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
7M
8M
1BR
2BR
3, 2, 7D G D
1V
3V
2V2V
2V
4V
4V
4V
8 4mg V
4V4V
2V
7 2mg V
inI
2dsg
( )2 10mbg V−
DDV
1BR
2BR
8dsg
5 3mg V
5dsg
3 1mg V3dsg
7dsg
2V
( )a
( )b
Fig 5.2 (a) Complementary Input Regulated Cascode Amplifier
with current mirror bias
(b) Small signal Low Frequency Equivalent circuit of (a)
5.2.1 Small signal DC gain derivation
Small signal dc gain is derived as following
6 9
1
11
10 9
2
11
m x
m
xout
in x x
ds
x
g g
g
gV
V g g
g
g
 
− 
 =
 
− 
 
(5.2.1)

25
7 8
11
6
2 8
10
6
2 3
9
2
x x
x
x
ds x
x
x
m m
x
x
g g
g
g
g g
g
g
g g
g
g
=
=
=
(5.2.2)
4 5
8 4
1
2 3
7 5
2
4 5
6 4 4
1
m m
x x
x
m m
x x
x
m m
x m mb
x
g g
g g
g
g g
g g
g
g g
g g g
g
= +
= +
 
= − − 
 
(5.2.3)
1 8 5 8
2
2 7 3 7
1
3 1 2 2 2
4 6 4 4 4
5 2 2 2
1
1
x ds ds m
B
x ds ds m
B
x ds ds m mb
x ds ds m mb
x m mb ds
g g g g
R
g g g g
R
g g g g g
g g g g g
g g g g
= + + −
= + + +
= + + +
= + − −
= + +
(5.2.4)
From computer simulation with MATLAB, its maximum dc gain is approximately 100
times of the input at 0.5 micron process.

26
5.2.2 Derivation of Input Impedance of the MRGC 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
2V2V
2V
4V
4V
4V
8 4mg V
4V4V
2V
8 8 5gs db dbC C C+ +
7 2mg V
inI
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 V
5dsg
3 1mg V
3dsg
6gsC 6gdC
Fig 5.3 (a) Complementary Input Regulated Cascode Amplifier
with current mirror bias
(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)
(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 g
C 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 4
2
0
1
ds m m gs gd
ds gs db db out gd
B
V g g V g V V V s C C
V g V s C C C V V sC
R
− + + + − +
 
= + + + + + − 
 
KCL at node Vout
(5.2.9)
( ) ( )
( ) ( ) ( )( )
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 g
g g g g
g 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)
(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
0
out 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 3
1
2 2 2 1 2 3
1
0
m gs db db ds m ds
B
out gd gs gd
g V V s C C C V g g V V g
R
V V sC V V s C C
 
+ + + + + + + 
 
+ − + − + =
( )
( )( ) ( )
2 7 7 3 7 7 3 2 2 3
1
1 3 2 3 2
1
m ds ds gs db db gd gs gd
B
m gs gd out gd
V g g g s C C C C C C
R
V g s C C V sC
 
+ + + + + + + + + 
 
= − + + +
( )( ) ( )1 3 2 3 2
2
7 6
6 7 7 3 2 2 3
7 7 7 3
1
1
m gs gd out gd
x x
x gs db db gd gs gd
x m ds ds
B
V g s C C V sC
V
g sC
C C C C C C C
g g g g
R
− + + +
=
+
= + + + + +
= + + +
( )( ) ( )
[ ]
3 8 4 5 4
4
6 5
5 8 8 5 4 5 4
6 8 5 8
1
m gs gd out gd
x x
x gs db db gs gd gd
x ds ds m
B
V g s C C V sC
V
g sC
C C C C C C C
g g g g
R
+ + +
=
+
= + + + + +
= + + −
( )
( )8 4 5
1
6 5
m gs gd
x x
g s C C
H s
g sC
+ +
=
+
( )( )
( )
( )
8 5 8
23 8 4 5 4 4
8 8 5 4 5 4
1
ds ds m
Bm gs gd out gd
gs db db gs gd gd
g g g
RV g s C C V V sC
s 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)
(5.2.20)
(5.2.21)
Intermediate transfer function can be define to make the path to finish derivation
shorter.
(5.2.22)

29
( ) 4
2
6 5
gd
x x
sC
H s
g sC
=
+
(5.2.23)
( )
( )3 2 3
3
7 6
m gs gd
x x
g s C C
H s
g sC
− + +
=
+
(5.2.24)
( ) 2
4
7 6
gd
x x
sC
H 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 2
7 6
m gs gd
x x m gs
x x
g s C C
H s g sC g sC
g sC
 − + +
 = + − +
 +
 
(5.2.26b)
( )
( )( ) ( )( )( )
( )
1 2 7 6 3 2 3 2 2
5
7 6
x x x x m gs gd m gs
x x
g sC g sC g s C C g sC
H s
g sC
+ + − − + + +
=
+
(5.2.26c)
( )
( ) ( )
( )( ) ( )( )
( )
2
1 7 2 7 6 1 2 6
2
3 2 2 3 2 2 3 2 3 2
5
7 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 C
H s
g sC
+ + +
− − + + − + +
=
+
(5.2.26d)
( )
( )
( ) ( )
( ) ( )( )
2
11 11 11
5
7 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 c
H s
g 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)
( ) ( )2
6 2 2 2
7 6
gd
ds m gs
x x
sC
H s g g sC
g sC
 
=− + 
+ 
(5.2.27b)
( )
2
2 2 2 2
6 2
7 6
gd gs gd m
ds
x x
s C C sC g
H s g
g sC
 +
 = −
 + 
(5.2.27c)
( )
( )2 2
2 2 6 2 2 2 2 7 21 11 01
6
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 g
H s
g sC g sC
C 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
10
5
gd m m gd xsC g H s g sC g
H s
H s
− − −
=
(5.2.31)
( ) ( ) ( )( )
( ) ( )( )( )
( )
6 3 2 2 3
11 8 4 2 2
5
m gd x
m gd
H s H s g sC g
H s H s H s g sC
H s
− −
= − − −
(5.2.32)
( ) ( ) ( ) ( )( )( ) ( )
( )
7
12 11 2 4 4 5 4
9
m gs gd ds
H s
H s H s H s g s C C g
H s
 
= + − + −   
 
(5.2.33)

31
( )
( ) ( )
( )
( )
6 6 7
13 10
9
gd msC g H s
H s H s
H s
+
= −
(5.2.34)
( )
( ) ( )
2 2 2 2
6 6 6 1 1 1
14 1
9 5
gd gd m gd gd m
x
s C sC g s C sC g
H s sC
H s H s
   + −
   =− −
   
   
(5.2.35)
( )
( )
( )
( )
( )
( ) ( )( )
( )
( )
2 4 4 5 41 613
15 6
12 5 9
m gs gd dsgd
gd
H s g s C C gsC H sH s
H s sC
H s H s H s
  − + −   = −         
(5.2.36)
( ) ( )14 15
1in
in
in
V
Z
I 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 2
6 5
m gs gd
gd m x
x x
g s C C
H s sC g g
g sC
 + +
 = + −
+  
(5.2.38)
( )
2
22 12 02
7
6 5
y y y
x x
s C sC g
H 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 03
8
5 6
y y y
x x
s C sC g
H 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 04
9
6 5
y y y
x x
s C sC g
H s
g sC
+ +
=
+
(5.2.46)
( )
( )( )
2
24 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
( )
( )( )
( )
2
1 1 25 15 057 6
10 2
6 711 11 11
gd m y y yx x
x x
sC g s C sC gg sC
H s
sC 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 06
10 2
11 11 11
y y y ys C s C sC g
H 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 2
5
m gd x
m gd
H s H s g sC g
H s H s H s g sC
H s
− −
= − − −
(5.2.32)
( ) ( ) ( )( ) ( )
( )
( )6
11 8 4 2 2 16
5
m gd
H s
H s H s H s g sC H s
H s
 
= − − −  
 
(5.2.54)
( ) ( )( )( )
2
23 13 03
16 3 2 2 3
7 6
m gd x
x x
s d sd d
H s H s g sC g
g 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 s
H s
can be defined as following
( )
( )
( )
2 2
21 11 01 21 11 016 7 6
17 2 2
5 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 sC
H s
H 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 03
18 17 16 2
7 611 11 11
y y y
x x
s C sC g s d sd d
H s H s H s
g sCs a sb c
 − + +  + +
 = =     ++ +   
(5.2.58)
( )
4 3 2
44 34 24 14 04
18 3 2
35 25 15 05
s d s d s d sd d
H s
s 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 C
d b C a g
d b g c C
d 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 2
19 4 2 2
6 7
gd gd m
m gd
x x
s C sC g
H 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 2
61 51 41 31 21 11 01
6 5 4 3 2
62 52 42 32 22 12
6 5 4 3 2
63 53 43 33 23 13 03
11 3 2
5 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
H 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)
( )
( )
( )
( )
2
62 5 35
2 2
52 35 2 2 5 6 25 5
2 2
42 35 2 2 6 25 2 2 5 6 15 5
2 2
32 25 2 2 6 15 2 2 5 6 05 5
2
22 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 d
f 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
f 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 4
61 62 63 51 52 53 41 42 43
3 2
31 32 33 21 22 23 11 12 13 01 03
11 3 2
5 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 f
H s
 + − + − − + − −
 
 + − − + − − + − − + − =
+ + + + +
(5.2.67)

36
( )
( )
( )( )( )
6 5 4 3 2
64 54 44 34 24 14 04
11 3 2
5 6 6 7 35 25 15 05x x x x
H s
+ + + + + +
=
+ + + + +
(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 f
f f f f
f f f f
f f f f
f f f f
f f f f
f 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 04
11 5 4 3 2
55 45 35 25 15 05
H 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 d
f 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 d
f d g g
=
= + +
= + + +
= + + +
= + +
=
(5.2.71)

37
( ) ( ) ( ) ( )( )( ) ( )
( )
7
12 11 2 4 4 5 4
9
m gs gd ds
H s
H 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
( )
( )
( )
2
22 12 027
19 2
9 24 14 04
y y y
y y y
s C sC gH s
H s
H s s C sC g
+ +
= =
+ +
(5.2.72)
( )
( ) ( ) ( ) ( )( )
2
4 4 5 4 4
20 2 4 4 5
5 6
gd gs gd gd m
m gs gd
x x
s C C C s C g
H s H s g s C C
sC g
− + +
= = − +
+
(5.2.73)
( )
( ) ( )
( ) ( )( )( )
4 42
4 4 5 4 6
4 5
21 2 4 4 5 4
5 6
gd m
gd gs gd ds x
ds x
m gs gd ds
x x
C g
s C C C s g g
g C
H s H s g s C C g
sC g
 
− + + − 
 − = = − + −
+
(5.2.74)
( ) ( ) ( )
4 3 2
41 31 21 11 01
22 21 19 3 2
32 22 12 02
s g s g s g sg g
H s H s H s
s 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)
( ) ( ) ( )
( )
6 5 4 3 4 3 2
64 54 44 34 41 31 21
2
24 14 04 11 01
12 11 22 5 4 3 3 2
55 45 35 32 22 12 02
2
25 15 05
s f s f s f s f s g s g s g
s f sf f sg g
H s H s H s
s 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 2
32 22 12 022
24 14 04
5 4 34 3 2
55 45 3541 31 21
2
11 01 25 15 05
12 5 4 3
55 45 35 3 2
32 222
25 15 05
s f s f s f s f
s g s g sg g
s f sf f
s f s f s fs g s g s g
sg g s f sf f
H s
s f s f s f
s g s g sg
s 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 03
12 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 g
H s
s f s f s f s f sf f s g s g sg g
+ + + + + + + + +
=
+ + + + + + + +
(5.2.80)
93 64 32 41 55
83 64 22 54 32 41 45 31 55
73 64 12 54 22 44 32 41 35 31 45 21 55
63 64 02 54 12 44 22 34 32 41 25 31 35 21 45 11 55
53 54 02 44 12 34 22 24 32 41 15 31 25 2
g f g g f
g f g f g g f g f
g f g f g f g g f g f g f
g f g f g f g f g g f g f g f g f
g f g f g f g f g g f g f g
= +
= + + +
= + + + + +
= + + + + + + +
= + + + + + + 1 35 11 45 01 55
43 44 02 34 12 24 22 14 32 41 05 31 15 21 25 11 35 01 45
33 34 02 24 12 14 22 04 32 31 05 21 15 11 25 01 35
23 24 02 14 12 04 22 21 05 11 15 01 25
13 14 02
f g f g f
g f g f g f g f g g f g f g f g f g f
g f g f g f g f g g f g f g f g f
g f g f g f g g f g f g f
g f g f
+ +
= + + + + + + + +
= + + + + + + +
= + + + + +
= + 04 12 11 05 01 15
03 04 02 01 05
g g f g f
g f g g f
+ +
= +
(5.2.81)

39
Multiply two brackets of denominator polynomial in (5.2.80), we will get
( )
( )
( )
9 8 7 6 5 4 3 2
93 83 73 63 53 43 33 23 13 03
12 8 7 6 5 4 3 2
84 74 64 54 44 34 24 14 04
s g s g s g s g s g s g s g s g sg g
H s
s g s g s g s g s g s g s g sg g
+ + + + + + + + +
=
+ + + + + + + +
(5.2.82)
84 55 32
74 55 22 45 32
64 55 12 45 22 35 32
54 55 02 45 12 35 22 25 32
44 45 02 35 12 25 22 15 32
34 35 02 25 12 15 22 05 32
24 25 02 15 12 05 22
14 15 02 05 12
04 05 02
g f g
g f g f g
g f g f g f g
g f g f g f g f g
g f g f g f g f g
g f g f g f g f g
g f g f g f g
g f g f g
g f g
=
= +
= + +
= + + +
= + + +
= + + +
= + +
= +
=
(5.2.83)
( )
( ) ( )
( )
( ) ( ) ( )
6 6 7
13 10 23 10
9
gd msC g H s
H s H s H s H s
H s
+
= − = −
(5.2.84)
( )
( ) 2 3 2
6 6 22 12 02 35 25 15 05
23 2 2
24 14 04 24 14 04
gd m y y y
y y y y y y
sC g s C sC g s g s g sg g
H s
s C sC g s C sC g
 + + + + + + =
   + + + +   
(5.2.85)
Coefficients of numerator of equation (5.2.85) can be defined as following
35 6 22
25 6 12 6 22
15 6 02 6 12
05 6 02
gd y
gd y m y
gd y m y
m y
g C C
g C C g C
g C g g C
g g g
=
= +
= +
=
(5.2.86)
Substitute equation (5.2.85) and (5.2.52) into equation (5.2.84) as following

40
( ) ( ) ( )
3 23 2
36 26 16 0635 25 15 05
13 23 10 2 2
24 14 04 11 11 11
y y y y
y y y
s C s C sC gs g s g sg g
H s H s H s
s C sC g s a sb c
   + + ++ + +
   = − = −
   + + + +   
(5.2.87)
Multiply both numerator and denominator with ( )( )2 2
24 14 04 11 11 11y y ys C sC g s a sb c+ + + +
( )
( )( )
( )( )
( )( )
3 2 2
35 25 15 05 11 11 11
3 2 2
36 26 16 06 24 14 04
13 2 2
24 14 04 11 11 11
y y y y y y y
y y y
s g s g sg g s a sb c
s C s C sC g s C sC g
H s
s C sC g s a sb c
+ + + + +
− + + + + +
=
+ + + +
(5.2.88)
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
5 4 3
35 11 35 11 25 11 35 11 25 11 15 11
2
25 11 15 11 05 11 15 11 05 11 05 11
5 4 3
36 24 36 14 26 24 36 04 26 14 16 24
2
26 04 16 14 06 04 16
13
y y y y y y y y y y y y
y y y y y y y
s g a s g b g a s g c g b g a
s g c g b g a s g c g b g c
s C C s C C C C s C g C C C C
s C g C C g g s C
H s
 + + + + +
 
 + + + + + + 
+ + + + +
−
+ + + +
=
( ) ( )
( ) ( ) ( )
( ) ( )
04 06 14 06 04
4 3 2
14 11 24 11 14 11 24 11 14 11 04 11
14 11 04 11 04 11
y y y y y
y y y y y y
y y y
g g C g g
s C a s C b C a s C c C b g a
s C c g b g c
 
 
 + +
 
+ + + + +
+ + +
(5.2.89)
Coefficient of numerator in the first bracket of equation (5.2.89) can be defined as
following
56 35 11
46 35 11 25 11
36 35 11 25 11 15 11
26 25 11 15 11 05 11
16 15 11 05 11
06 05 11
g g a
g g b g a
g g c g b g a
g g c g b g a
g g c g b
g g c
=
= +
= + +
= + +
= +
=
(5.2.90)
Coefficient of numerator in the second bracket of equation (5.2.89) can be defined as
following

41
57 36 24
47 36 14 26 24
37 36 04 26 14 16 24
27 26 04 16 14 06 24
17 16 04 06 14
07 06 04
y y
y y y y
y y y y y y
y y y y y y
y y y y
y y
g C C
g C C C C
g C g C C C C
g C g C C g C
g C g g C
g g g
=
= +
= + +
= + +
= +
=
(5.2.91)
Coefficient of denominator in the bracket of equation (5.2.89) can be defined as
following
48 14 11
38 24 11 14 11
28 24 11 14 11 04 11
18 14 11 04 11
08 04 11
y
y y
y y y
y y
y
g C a
g C b C a
g C c C b C a
g C c g b
g g c
=
= +
= + +
= +
=
(5.2.92)
( )
( ) ( )
2 2 2 2
6 6 6 1 1 1
14 1
9 5
gd gd m gd gd m
x
s C sC g s C sC g
H s sC
H s H s
   + −
   =− −
   
   
(5.2.93)
Substitute (5.2.26e) and (5.2.46) into (5.2.93), we will get
( ) ( ) ( )
2 2 2 2
6 6 6 1 1 1
14 1 6 5 7 62 2
24 14 04 11 11 11
gd gd m gd gd m
x x x x x
y y y
s C sC g s C sC g
H s sC g sC g sC
s C sC g s a sb c
   + −
   = − + − +
   + + + +   
(5.2.94)
Multiply both numerator and denominator of equation (5.2.94)
With ( )( )2 2
24 14 04 11 11 11y y ys C sC g s a sb c+ + + +

42
( ) ( )( )
( )( )( )
( )( )( )
2 2
14 1 24 14 04 11 11 11
2 2
6 6 6 2 2
6 5 24 14 04 11 11 112
24 14 04
2 2
1 1 1 2 2
7 6 24 14 04 11 11 112
11 11 11
x y y y
gd gd m
x x y y y
y y y
gd gd m
x x y y y
H s sC s C sC g s a sb c
s C sC g
g sC s C sC g s a sb c
s C sC g
s C sC g
g sC s C sC g s a sb c
s a sb c
= + + + +
 +
 − + + + + +
 + + 
 −
 − + + + + +
 + + 
(5.2.95)
( )
( )( )
( )( )( )
( )( )( )
( )( )
2 2
1 24 14 04 11 11 11
2 2 2
6 6 6 6 5 11 11 11
2 2 2
1 1 1 7 6 24 14 04
14 2 2
24 14 04 11 11 11
x y y y
gd gd m x x
gd gd m x x y y y
y y y
sC s C sC g s a sb c
s C sC g g sC s a sb c
s C sC g g sC s C sC g
H s
s C sC g s a sb c
 + + + +
 
 − + + + +
 
 
− − + + +  =
+ + + +
(5.2.96)
( )
( ) ( )
( )
( ) ( )
( ) ( )( )
5 4
1 24 11 1 24 11 1 14 11
3
1 24 11 1 14 11 1 04 11
2
1 14 11 1 04 11 1 04 11
5 2 4 2 2
6 6 11 6 6 11 11 6 6 6 6 5
3 2 2
6 6 11 6 6
14
x y x y x y
x y x y x y
x y x y x y
gd x gd x gd x gd m x
gd x gd x
s C C a s C C b C C a
s C C c C C b C g a
s C C c C g b s C g c
s C g a s C g b a C g C g C
s C g c C g C
H s
 + +
 
 + + +
 
 + + +
 
+ + +
+ + +
−
=
( ) ( )( )
( )
( )
( ) ( )( )
( )
6 6 6 11 6 6 6 11
2 2
6 6 6 6 5 11 6 6 6 11
6 6 6 11
5 2 4 2 2
1 6 24 1 6 14 1 7 1 1 6 24
3 2 2
1 6 04 1 7 1 1 6 14 1
gd m x gd m x
gd x gd m x gd m x
gd m x
gd x y gd x y gd x gd m x y
gd x gd x gd m x y gd
g C b C g g a
s C g C g C c C g g b
s C g g c
s C C C s C C C C g C g C C
s C C g C g C g C C C g
 
 
 
+ 
 
  + + +
  
 + 
+ + −
+ + − −
−
( )( )
( ) ( )( )
( )
( )( )
1 7 24
2 2
1 7 1 1 6 04 1 1 7 14
1 1 7 04
2 2
24 14 04 11 11 11
m x y
gd x gd m x y gd m x y
gd m x y
y y y
g C
s C g C g C g C g g C
s C g g g
s C sC g s a sb c
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
  
  
  
  
  + − −
  
  + −   
+ + + +
(5.2.97)
After this step, you can group and define new coefficients as a group of small signal
parameters as following

43
( )
( )
( )
( )
5 2 2
1 24 11 6 6 11 1 6 24
2 2
1 24 11 1 14 11 6 6 11 11 6 6 6 6 5
4
2 2
1 6 14 1 7 1 1 6 24
2
1 24 11 1 14 11 1 04 11 6 6 11
3
6
14
x y gd x gd x y
x y x y gd x gd x gd m x
gd x y gd x gd m x y
x y x y x y gd x
gd
s C C a C g a C C C
C C b C C a C g b a C g C g C
s
C C C C g C g C C
C C c C C b C g a C g c
s C
H s
− −
 + − − +
 +
  − − −
 
+ + −
+ −
=
( ) ( )
( ) ( )( )
( )
( ) ( )( )
2
6 6 6 6 11 6 6 6 11
2 2
1 6 04 1 7 1 1 6 14 1 1 7 24
2
1 14 11 1 04 11 6 6 6 6 5 11 6 6 6 11
2
2
1 7 1 1 6 04 1 1 7 14
x gd m x gd m x
gd x gd x gd m x y gd m x y
x y x y gd x gd m x gd m x
g C g C b C g g a
C C g C g C g C C C g g C
C C c C g b C g C g C c C g g b
s
C g C g C g C g g C
 
 
 
+ − 
 
− + − − 
 
 + − + −
+
− − −

( )
( )( )
1 04 11 6 6 6 11 1 1 7 04
2 2
24 14 04 11 11 11
x y gd m x gd m x y
y y y
s C g c C g g c C g g g
s C sC g s a sb c
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
  
 
 + − + 
+ + + +
(5.2.98)
Let us define new coefficients of the numerator polynomial as following
( )
( )
( )
2 2
59 1 24 11 6 6 11 1 6 24
2 2
1 24 11 1 14 11 6 6 11 11 6 6 6 6 5
49
2 2
1 6 14 1 7 1 1 6 24
2
1 24 11 1 14 11 1 04 11 6 6 11
2
39 6
x y gd x gd x y
x y x y gd x gd x gd m x
gd x y gd x gd m x y
x y x y x y gd x
gd
g C C a C g a C C C
C C b C C a C g b a C g C g C
g
C C C C g C g C C
C C c C C b C g a C g c
g C
= − −
 + − − +
 =
  − − −
 
+ + −
= −( ) ( )
( ) ( )( )
( )
( ) ( )( )
6 6 6 6 11 6 6 6 11
2 2
1 6 04 1 7 1 1 6 14 1 1 7 24
2
1 14 11 1 04 11 6 6 6 6 5 11 6 6 6 11
29
2
1 7 1 1 6 04 1 1 7 14
x gd m x gd m x
gd x gd x gd m x y gd m x y
x y x y gd x gd m x gd m x
g C g C b C g g a
C C g C g C g C C C g g C
C C c C g b C g C g C c C g g b
g
C g C g C g C g g C
 
 
 
+ − 
 
 − + − −
 
 + − + −
=
− − −

( )19 1 04 11 6 6 6 11 1 1 7 04x y gd m x gd m x yg C g c C g g c C g g g


 
 

= − +
(5.2.99)

44
From equation (5.2.36) , it can be seen that there are additional two new variables
( )
( )
( )
( )
( )
( ) ( )( )
( )
( )
( )
( )
( )
( ) ( ) ( )
( ) ( ) ( )
2 4 4 5 41 613
15 6
12 5 9
5 4 3 8
56 57 46 47 36 37 8
2
26 27 16 17 06 0713
24 4 3 2
12 48 38 28 18 08
m gs gd dsgd
gd
H s g s C C gsC H sH s
H s sC
H s H s H s
s g g s g g s g g s g
s g g s g g g gH s
H s
H s s g s g s g sg g
  − + −   = −          
 − + − + −
 
 + − + − + −
= = × 
+ + + + 
  
 
( )
( )
( )
( )
7 6 5 4
4 74 64 54 44
3 2
34 24 14 04
9 8 7 6 5
93 83 73 63 53
4 3 2
43 33 23 13 03
2
21 11 01
1
7 61 6
25 2
5 11 11 11
7 6
y y y
gd
x xgd
x x
s g s g s g s g
s g s g sg g
s g s g s g s g s g
s g s g s g sg g
s C sC g
sC
g sCsC H s
H s
H s s a sb c
g sC
 + + + +
 
 + + + +
 
+ + + + 
 
+ + + + + 
 − + +
 
 +
 = =
 + +
 + 
3 2
1 21 1 11 1 01
2
11 11 11
gd y gd y gd ys C C s C C sC g
s a sb c
− + +
=
+ +

(5.2.100)
The results of multiplication of numerator of ( )24H s can be seen as following
( )
13 12 11 10 9 8 7
131 121 111 101 91 81 71
6 5 4 3 2
61 51 41 31 21 11 01
24 4 3 2 9 8 7 6 5
48 38 28 93 83 73 63 53
4 3 2
18 08 43 33 23 13 03
1
s h s h s h s h s h s h s h
s h s h s h s h s h sh h
H s
s g s g s g s g s g s g s g s g
sg g s g s g s g sg g
 + + + + + + 
 
 + + + + + + + = × + + + + + + 
  + + + + + + +   





(5.2.101)
The coefficients in numerator polynomial of equation (5.2.101) can be defined as
following
( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( )
131 56 57 84
121 56 57 74 46 47 84
111 56 57 64 46 47 74 36 37 84
101 56 57 54 46 47 64 36 37 74 26 27 84
91 56 57 44 46 47 54 36 37 64 26 27 74 16 17 84
81 56 57 34
h g g g
h g g g g g g
h g g g g g g g g g
h g g g g g g g g g g g g
h g g g g g g g g g g g g g g g
h g g g g
= −
= − + −
= − + − + −
= − + − + − + −
= − + − + − + − + −
= − + ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )
( )
46 47 44 36 37 54 26 27 64 16 17 74
06 07 84
71 56 57 24 46 47 34 36 37 44 26 27 54 16 17 64
06 07 74
61 56 57 14 46 47 24 36 37 34 26 27 44 16 17 54
06 07 64
g g g g g g g g g g g
g g g
g g g
g g g
− + − + − + −
+ −
= − + − + − + − + −
+ −
= − + − + − + − + −
+ −
(5.2.102)

45
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
51 56 57 04 46 47 14 36 37 24 26 27 34 16 17 44
06 07 54
41 46 47 04 36 37 14 26 27 24 16 17 34 06 07 44
31 36 37 04 26 27 14 16 17 24 06 07 34
21 26 27 04 16 17 14
g g g
h g g g g g g g g g g g g
h g g g g g g
= − + − + − + − + −
+ −
= − + − + − + − + −
= − + − + − + −
= − + − + ( )
( ) ( )
( )
06 07 24
11 16 17 04 06 07 14
01 06 07 04
g g g
h g g g g g g
h g g g
−
= − + −
= −
(5.2.103)
The results of multiplication of denominator of ( )24H s can be seen as following
( )
13 12 11 10 9 8 7
131 121 111 101 91 81 71
6 5 4 3 2
61 51 41 31 21 11 01
24 13 12 11 10 9 8 7
132 122 112 102 92 82 72
6 5 4 3 2
62 52 42 32 22 12 02
H s
 + + + + + +
 
 + + + + + + +
=  
+ + + + + + 
 
+ + + + + + + 
(5.2.104)
The coefficients in denominator polynomial of equation (5.2.104) can be defined as
following
132 48 93
122 48 83 38 93
112 48 73 38 83 28 93
102 48 63 38 73 28 83 18 93
92 48 53 38 63 28 73 18 83 08 93
82 48 43 38 53 28 63 18 73 08 83
72 48 33 38 43 28 53 18 63 08 73
h g g
h g g g g
h g g g g g g
h g g g g g g g g
h g g g g g g g g g g
=
= +
= + +
= + + +
= + + + +
= + + + +
= + + + +
62 48 23 38 33 28 43 18 53 08 63
52 48 13 38 23 28 33 18 43 08 53
42 48 03 38 13 28 23 18 33 08 43
32 38 03 28 13 18 23 08 33
22 28 03 18 13 08 23
12 18 03 08 13
02 08 03
h g g g g g g g g
h g g g g g g
h g g g g
h g g
= + + + +
= + + + +
= + + + +
= + + +
= + +
= +
=
(5.2.105)
It can be seen that the numerator polynomial from the right hand side of equation
(5.2.100) can be define as new variable as following
( ) ( )( )
( )2
4 4 4 4 5 6 4 4 54
26 4 4 5 4
6 5 6 5
gd m gd gs gd x ds ds xgd
m gs gd ds
x x x x
sC g s C C C g g sg CsC
H s g s C C g
g sC g sC
− + − − 
= − + −= 
+ + 
(5.2.106)

46
( ) ( ) ( )
( )
( )
( ) ( ) ( ) ( )( )
( )
( )
( )
( )
( )
26
15 24 25 6 24 25 27
9
3 2
33 23 13
27 2
24 14 04
33 4 6 4 5
23 6 4 4 4 5
13 6 6 4
3 2
1 21 1 11 1 01
25 2
11
gd
y y y
gd gd gs gd
gd gd m ds x
gd x ds
gd y gd y gd y
H s
H s H s H s sC H s H s H s
H s
s h s h sh
H s
s C sC g
h C C C C
h C C g g C
h C g g
s C C s C C sC g
H s
s a sb
  
= − = −    
  
+ +
=
+ +
=− +
= −
= −
− + +
=
+ 11 11c+
(5.2.107)
Equation (5.2.107) can be rewritten again as following
( ) ( )
3 2 3 2
1 21 1 11 1 01 33 23 13
15 24 2 2
11 11 11 24 14 04
gd y gd y gd y
y y y
s C C s C C sC g s h s h sh
H s H s
s a sb c s C sC g
 − + + + +
 −
 + + + + 
(5.2.108)
Multiply both numerator and denominator of polynomial with
( )( )2 2
11 11 11 24 14 04y y ys a sb c s C sC g+ + + +
( ) ( )
( )( )
( )( )
( )( )
3 2 2
1 21 1 11 1 01 24 14 04
3 2 2
33 23 13 11 11 11
15 24 2 2
11 11 11 24 14 04
gd y gd y gd y y y y
y y y
s C C s C C sC g s C sC g
s h s h sh s a sb c
H s H s
s a sb c s C sC g
 − + + + +
 
 − + + + +
 =
 + + + +
 
 
  
(5.2.109)

47
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
5 4
1 21 24 1 11 24 1 21 14
3
1 11 14 1 01 24 1 21 04
2
1 11 04 1 01 14 1 01 04
5 4 3
33 11 33 11 23 11 33 11 23 11 13 11
2
23
15 24
gd y y gd y y gd y
gd y y gd y y gd y y
s C C C s C C C C C C
s C C C C g C C C g
s C C g C g C s C g g
s h a s h b h a s h c h b h a
s h
H s H s
 − + −
 
 + + −
 
 + + +
 
+ + + + +
−
+
=
( ) ( )
( ) ( )
( )
( )
11 11 11 13 11
4 3
11 24 11 14 11 24
2
11 04 11 14 11 24
11 04 11 14 11 04
y y y
y y
y y y
c h b s h c
s a C s a C b C
s a g b C c C
s b g c C c g
 
 
 
 
 
 
      + +  
 + +
 
 + + +
 
 + + + 
 
 
 
  
(5.2.110)
The new coefficients of equation (5.2.110) can be defined as following
( ) ( )
( ) ( )
( ) ( )
( ) ( )
5 4 3 2
55 45 35 25 15
15 24 4 3 2
46 36 26 16 06
55 1 21 24 33 11
45 1 11 24 1 21 14 33 11 23 11
35 1 11 14 1 01 24 1 21 04 33 11 23 11 13 11
2
gd y y
gd y y gd y
s h s h s h s h sh
H s H s
s h s h s h sh h
h C C C h a
h C C C C C C h b h a
h C C C C g C C C g h c h b h a
h
 + + + +
=  
+ + + + 
=− −
= − − +
= + − − + +
( ) ( )
( ) ( )
( )
( )
( )
( )
5 1 11 04 1 01 14 23 11 11 11
15 1 01 04 13 11
46 11 24
36 11 14 11 24
26 11 04 11 14 11 24
16 11 04 11 14
06 11 04
gd y y gd y y
gd y y
y
y y
y y
y y
y
C C g C g C h c h b
h C g g h c
h a C
h a C b C
h a g b C c C
h b g c C
h c g
= + − +
= −
=
= +
= + +
= +
=
(5.2.111)

48
Substitute ( )24H s from equation (5.2.104) into equation (5.2.111), we get
( )
13 12 11 10
131 121 111 101
9 8 7 6 5
91 81 71 61 51
4 3 2
41 31 21 11 01
15 13 12 11 10
132 122 112 102
9 8 7 6 5
92 82 72 62 52
4 3 2
42 32 22 12 02
s h s h s h s h
s h s h s h s h s h
s h s h s h sh h
H s
s h s h s h s h
s h s h s h s h s h
s h s h s h sh h
 + + +
 
 + + + + +
 
+ + + + + =  + + +
 
 + + + + +

 + + + + + 
5 4 3 2
55 45 35 25 15
4 3 2
46 36 26 16 06
s h s h s h s h sh
s h s h s h sh h
 + + + +
 
+ + + + 



(5.2.112)
The results of these numerator and denominator polynomial multiplication or
convolution can be written as following
( )
18 17 16 15 14 13 12 11 10 9
187 177 167 157 147 137 127 117 107 97
8 7 6 5 4 3 2
87 77 67 57 47 37 27 17
15 17 16 15 14 13 12 11 10 9 8
178 168 158 148 138 128 118 108 98
s h s h s h s h s h s h s h s h s h s h
s h s h s h s h s h s h s h sh
H s
s h s h s h s h s h s h s h s h s h s
+ + + + + + + + +
+ + + + + + + +
=
+ + + + + + + + 88
7 6 5 4 3 2
78 68 58 48 38 28 18 08
h
s h s h s h s h s h s h sh h
 
 
 
 
 
 + + + + + + + + 
(5.2.113)
The coefficients of numerator polynomial of equation (5.2.113) can be defined as
following
187 131 55
177 131 45 121 55
167 131 35 121 45 111 55
157 131 25 121 35 111 45 101 55
147 131 15 121 25 111 35 101 45 91 55
137 121 15 111 25 101 35 91 45 81 55
127 111 15 101 2
h h h
h h h h h
h h h h h h h
h h h h h h h h h
h h h h h h h h h h h
h h h h h
=
= +
= + +
= + + +
= + + + +
= + + + +
= + 5 91 35 81 45 71 55
117 101 15 91 25 81 35 71 45 61 55
107 91 15 81 25 71 35 61 45 51 55
97 81 15 71 25 61 35 51 45 41 55
87 71 15 61 25 51 35 41 45 31 55
77 61 15 51 25 41 3
h h h h h h
h h h h h h h
+ + +
= + + + +
= + + + +
= + + + +
= + + + +
= + + 5 31 45 21 55
67 51 15 41 25 31 35 21 45 11 55
57 41 15 31 25 21 35 11 45 01 55
47 31 15 21 25 11 35 01 45
37 21 15 11 25 01 35
27 11 15 01 25
17 01 15
h h h h
h h h h h h h h h
h h h h h h h
h h h h h
h h h
+ +
= + + + +
= + + + +
= + + +
= + +
= +
=
(5.2.114)

49
The coefficients of denominator polynomial of equation (5.2.113) can be defined as
following
178 132 46
168 132 36 122 46
158 132 26 122 36 112 46
148 132 16 122 26 112 36 102 46
138 132 06 122 16 112 26 102 36 92 46
128 122 06 112 16 102 26 92 36 82 46
118 112 06 102 1
h h h
h h h h h
h h h h h h h
h h h h h h h h h
h h h h h
=
= +
= + +
= + + +
= + + + +
= + + + +
= + 6 92 26 82 36 72 46
108 102 06 92 16 82 26 72 36 62 46
98 92 06 82 16 72 26 62 36 52 46
88 82 06 72 16 62 26 52 36 42 46
78 72 06 62 16 52 26 42 36 32 46
68 62 06 52 16 42 26
h h h h h h
h h h h h h h
+ + +
= + + + +
= + + + +
= + + + +
= + + + +
= + + 32 36 22 46
58 52 06 42 16 32 26 22 36 12 46
48 42 06 32 16 22 26 12 36 02 46
38 32 06 22 16 12 26 02 36
28 22 06 12 16 02 26
18 12 06 02 16
08 02 06
h h h h
h h h h h h h h h
h h h h h h h
h h h h h
h h h
+ +
= + + + +
= + + + +
= + + +
= + +
= +
=
(5.2.115)
49 24 11
39 24 11 14 11
29 24 11 14 11 04 11
19 24 11 14 11
09 04 11
y
y y
y y y
y y
y
h C a
h C b C a
h C c C b g a
h C c g b
h g c
=
= +
= + +
= +
=
(5.2.115b)
substitute equation (5.2.98) and (5.2.113) into equation (5.2.37)
( )( )
18 17 16 15
187 177 167 157
14 13 12 11
147 137 127 117
10 9 8 7
107 97 87 77
5 4 3 2 6 5 4 3 2
59 49 39 29 19 67 57 47 37 27 17
172 2
17824 14 04 11 11 11
1
in
y y y
Z
s h s h s h s h
s h s h s h s h
s h s h s h s h
s g s g s g s g sg s h s h s h s h s h sh
s hs C sC g s a sb c
=
+ + +
+ + + +
+ + + +
 + + + + + + + + + +  +
+ + + +
16 15 14
168 158 148
13 12 11 10
138 128 118 108
9 8 7 6 5
98 88 78 68 58
4 3 2
48 38 28 18 08
s h s h s h
s h s h s h s h
s h s h s h s h s h
s h s h s h sh h
 
 
 
 
 
 
 
 + + +
 
 + + + +
 
 + + + + +
 
 + + + + + 
(5.2.116)

50
( )
17 16 15 14
178 168 158 148
13 12 11 10
138 128 118 108
4 3 2 9 8 7 6
49 39 29 19 09 98 88 78 68
5 4 3 2
58 48 38 28
18 08
5 4 3 2
59 49 39 29 19
in
s h s h s h s h
s h s h s h s h
s h s h s h sh h s h s h s h s h
s h s h s h s h
sh h
Z
s g s g s g s g sg
 + + +
 
 + + + +
 
+ + + + + + + + 
 
+ + + + 
 + + 
 =
 + + + +
17 16 15 14
178 168 158 148
13 12 11 10
138 128 118 108
9 8 7 6
98 88 78 68
5 4 3 2
58 48 38 28
18 08
18 17 16 15
187 177 167 157
14 13 12
147 137 127
s h s h s h s h
s h s h s h s h
s h s h s h s h
s h s h s h s h
sh h
s h s h s h s h
s h s h s h s
 + + +
 
+ + + + 
 
 + + + + 
 
+ + + + 
 + +
  
+ + +
+ + + +
+ ( )
11
117
10 9 8 7 4 3 2
107 97 87 77 49 39 29 19 09
6 5 4
67 57 47
3 2
37 27 17
h
s h s h s h s h s h s h s h sh h
s h s h s h
s h s h sh
 
 
 
 
+ + + + + + + + 
 
+ + + 
 
+ + +  
(5.2.117)
After numerator polynomial multiplication in equation (5.2.117), we got the following
21 20 19 18 17 16 15 14
211 201 191 181 171 161 151 141
13 12 11 10 9 8 7 6
131 121 111 101 91 81 71 61
5 4 3 2
51 41 31 21 11 01
5 4 3 2
59 49 39 29 19
in
s k s k s k s k s k s k s k s k
s k s k s k s k sk k
Z
s g s g s g s g sg
 + + + + + + +
 
 + + + + + + + +
 
 + + + + + + =
 + + + + 
17 16 15 14
178 168 158 148
13 12 11 10
138 128 118 108
9 8 7 6
98 88 78 68
5 4 3 2
58 48 38 28
18 08
18 17 16 15
187 177 167 157
14 13 12 11
147 137 127
s h s h s h s h
s h s h s h s h
s h s h s h s h
s h s h s h s h
sh h
s h s h s h s h
s h s h s h s
 + + +
 
+ + + + 
 
+ + + + 
 
+ + + + 
 + +
  
+ + +
+ + + +
+ ( )
117
10 9 8 7 4 3 2
107 97 87 77 49 39 29 19 09
6 5 4
67 57 47
3 2
37 27 17
h
s h s h s h s h s h s h s h sh h
s h s h s h
s h s h sh
 
 
 
 
+ + + + + + + + 
 
+ + + 
 
+ + +  
(5.2.118)

51
The coefficients of numerator polynomial of equation (5.2.118) can be defined as
following
211 49 178
201 49 168 39 178
191 49 158 39 168 29 178
181 49 148 39 158 29 168 19 178
171 49 138 39 148 29 158 19 168 09 178
161 49 128 39 138 29 148 19 158 09 168
151 49 118 39
k h h
k h h h h
k h h h h h h
k h h h h h h h h
k h h h h h h h h h h
k h h h
=
= +
= + +
= + + +
= + + + +
= + + + +
= + 128 29 138 19 148 09 158
141 49 108 39 118 29 128 19 138 09 148
131 49 98 39 108 29 118 19 128 09 138
121 49 88 39 98 29 108 19 118 09 128
111 49 78 39 88 29 98 19 108 09 118
1
h h h h h h h
k
+ + +
= + + + +
= + + + +
= + + + +
= + + + +
01 49 68 39 78 29 88 19 98 09 108
91 49 58 39 68 29 78 19 88 09 98
81 49 48 39 58 29 68 19 78 09 88
71 49 38 39 48 29 58 19 68 09 78
61 49 28 39 38 29 48 19 58 09 68
51 49 18
h h h h h h h h h h
k h h
= + + + +
= + + + +
= + + + +
= + + + +
= + + + +
= 39 28 29 38 19 48 09 58
41 49 08 39 18 29 28 19 38 09 48
31 39 08 29 18 19 28 09 38
21 29 08 19 18 09 28
11 19 08 09 18
01 09 08
h h h h h h h h
k h h h h h h h h
k h h h h h h
k h h h h
k h h
+ + + +
= + + + +
= + + +
= + +
= +
=
(5.2.119)
After denominator polynomial multiplication in equation (5.2.118), we got the following

52
21 20 19 18 17 16 15 14
211 201 191 181 171 161 151 141
13 12 11 10 9 8 7 6
131 121 111 101 91 81 71 61
5 4 3 2
51 41 31 21 11 01
22 21 20 19
222 212 202 192
in
Z
s k s k s k s k
 + + + + + + +
 
 + + + + + + + +
 
 + + + + + + =
+ + + + 18 17 16 15
182 172 162 152
14 13 12 11 10 9 8 7
142 132 122 112 102 92 82 72
6 5 4 3 2
62 52 42 32 22 12
18 17 16 15
187 177 167 157
14 13 12 11
147 137 127 117
s k s k s k s k
s k s k s k s k s k sk
s h s h s h s h
s h s h s h s h
 + + +
 
 + + + + + + + +
 
 + + + + + + 
+ + +
+ + + +
+ ( )4 3 2
49 39 29 19 0910 9 8 7
107 97 87 77
6 5 4 3 2
67 57 47 37 27 17
s h s h s h sh h
s h s h s h s h
s h s h s h s h s h sh
 
 
 
+ + + + 
+ + + + 
 
+ + + + + + 
(5.2.120)
The coefficients of first brackets of denominator polynomial of equation (5.2.120) can
be defined as following
222 59 178
212 59 168 49 178
202 59 158 49 168 39 178
192 59 148 49 158 39 168 29 178
182 59 138 49 148 39 158 29 168 19 178
172 59 128 49 138 39 148 29 158 19 168
162 59 118 49
k g h
k g h g h
k g h g h g h
k g h g h g h g h
k g h g h g h g h g h
k g h g
=
= +
= + +
= + + +
= + + + +
= + + + +
= + 128 39 138 29 148 19 158
152 59 108 49 118 39 128 29 138 19 148
142 59 98 49 108 39 118 29 128 19 138
132 59 88 49 98 39 108 29 118 19 128
122 59 78 49 88 39 98 29 108 19 118
1
h g h g h g h
k
+ + +
= + + + +
= + + + +
= + + + +
= + + + +
12 59 68 49 78 39 88 29 98 19 108
102 59 58 49 68 39 78 29 88 19 98
92 59 48 49 58 39 68 29 78 19 88
82 59 38 49 48 39 58 29 68 19 78
72 59 28 49 38 39 48 29 58 19 68
62 59 1
g h g h g h g h g h
k g h
= + + + +
= + + + +
= + + + +
= + + + +
= + + + +
= 8 49 28 39 38 29 48 19 58
52 59 08 49 18 39 28 29 38 19 48
42 49 08 39 18 29 28 19 38
32 39 08 29 18 19 28
22 29 08 19 18
12 19 08
g h g h g h g h
k g h g h g h g h
k g h g h g h
k g h g h
k g h
+ + + +
= + + + +
= + + +
= + +
= +
=
(5.2.121)
After denominator polynomial multiplication in the right hand side of equation
(5.2.120), we got the following

53
21 20 19 18 17 16 15 14
211 201 191 181 171 161 151 141
13 12 11 10 9 8 7 6
131 121 111 101 91 81 71 61
5 4 3 2
51 41 31 21 11 01
22 21 20 19
222 212 202 192
in
Z
s k s k s k s k
 + + + + + + +
 
 + + + + + + + +
 
 + + + + + + =
+ + + + 18 17 16 15
182 172 162 152
14 13 12 11 10 9 8 7
142 132 122 112 102 92 82 72
6 5 4 3 2
62 52 42 32 22 12
22 21 20 19 18 17 16 15
223 213 203 193 183 173 163 153
s k s k s k s k
 + + +
 
 + + + + + + + +
 
 + + + + + + 
+ + + + + + +
+ 14 13 12 11 10 9 8 7
143 133 123 113 103 93 83 73
6 5 4 3 2
63 53 43 33 23 13
 
 
 + + + + + + + +
 
 + + + + + + 
(5.2.122)
The coefficients of first brackets of denominator polynomial of equation (5.2.122) can
be defined as following
223 187 49
213 187 39 177 49
203 187 29 177 39 167 49
193 187 19 177 29 167 39 157 49
183 187 09 177 19 167 29 157 39 147 49
173 177 09 167 19 157 29 147 39 137 49
163 167 09 15
k h h
k h h h h
k h h h h h h
k h h h h h h h h
k h h h
=
= +
= + +
= + + +
= + + + +
= + + + +
= + 7 19 147 29 137 39 127 49
153 157 09 147 19 137 29 127 39 117 49
143 147 09 137 19 127 29 117 39 107 49
133 137 09 127 19 117 29 107 39 97 49
123 127 09 117 19 107 29 97 39 87 4
h h h h h h h
+ + +
= + + + +
= + + + +
= + + + +
= + + + + 9
113 117 09 107 19 97 29 87 39 77 49
103 107 09 97 19 87 29 77 39 67 49
93 97 09 87 19 77 29 67 39 57 49
83 87 09 77 19 67 29 57 39 47 49
73 77 09 67 19 57 29 47 39 37 49
63
k
= + + + +
= + + + +
= + + + +
= + + + +
= + + + +
= 67 09 57 19 47 29 37 39 27 49
53 57 09 47 19 37 29 27 39 17 49
43 47 09 37 19 27 29 17 39
33 37 09 27 19 19 29
23 27 09 17 19
13 17 09
h h h h h h h h h h
k h h h h h h h h
k h h h h h h
k h h h h
k h h
+ + + +
= + + + +
= + + +
= + +
= +
=
(5.2.123)

54
Fig. 5.4 Magnitude and Phase response of modified CRGC
amplifier
Fig. 5.5 Magnitude and Phase response of modified CRGC amplifier
-200
-150
-100
-50
0
50
100
System: Zin
Magnitude(dB)
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
11
10
12
180
270
360
450
540
630
720
Phase(deg)
Bode Diagram
Frequency (Hz)
-350
-300
-250
-200
-150
-100
-50
0
50
100
150
System: Zin3 = 1500uA
Magnitude (dB): -13.1
Magnitude(dB)
10
4
10
6
10
8
10
10
10
12
-180
-90
0
90
180
270
360
450
Phase(deg)
Bode Diagram
Frequency (Hz)
Zin = 400uA
Zin2 = 600uA
Zin3 = 1500uA

55
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
2V2V
2V
4V
4V
4V
8 4mg V
4V4V
2V
7 2mg V
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 V
5dsg
3 1mg V
3dsg
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
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
− + − + −
= − + + − + − + − + +
(5.2.127)
( )
( )
( ) ( )
4 5 5 6 6
6 6 3 4 4 4 5 4
6 4 4 4
gs gd gs db gd
in gd m m gs gd out ds
ds ds m mb
s C C C C C
V 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 1
2 2 3 2 1
1 2 2 2
2 1 1
gs gd gd gs db
gs gd m
ds m mb ds
out ds in m gd
s C C C C C
V 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 7
1
2 1 2 3 2 13 2 2
1
0
m gs db db m ds
B
gs gd ds out gd
g V V s C C C g V V g
R
V V s C C V g V V sC
 
 + + + + + +  
 
 + − + + + − =
 
(5.2.136)
( ) ( )
7 7 3
1
2 1 3 2 3 2
7 3 7
2 2 3
1
0
m ds ds
B
m gs gd out gd
gs db db
gd gs gd
g g g
R
V V g s C C V sC
C C C
s
C C C
 
+ + + 
   + − + − =   + + 
 +   + + +   
(5.2.137)
( ) ( ) ( )2 5 5 1 3 2 3 2
5 7 3 7 2 2 3
5 7 7 3
1
0
1
x x m gs gd out gd
x gs db db gd gs gd
x m ds ds
B
V g s C V g s C C V sC
C C C C C C C
g g g g
R
  + + − + − =   
= + + + + +
= + + +
(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 4
2
0
1
m m ds gs gd
ds gs db db out gd
B
g V g V V g V V s C C
V g V s C C C V V sC
R
 + + − + − +
 
 
 = + + + + + −   
 
(5.2.139)
( ) ( ) ( )3 5 4 5 4
8 5 8
4 8 5 8
4 5 42
1
m gs gd out gd
gs db db
ds ds m
gs gd gdB
V g s C C V sC
C C C
V g g g s
C C CR
 + + +
 
 + +  
= + + − +   + + +   
(5.2.140)
( ) ( ) ( ) [ ]3 5 4 5 4 4 6 6
6 8 5 8
2
6 8 5 8 4 5 4
1
m gs gd out gd x x
x ds ds m
B
x gs db db gs gd gd
V g s C C V sC V g sC
g g g g
R
C C C C C C C
 + + + = +
 
= + + −
= + + + + +
(5.2.141)

58
From equation (5.2.126)
1 6
1 3
1 1
gd gd
in
x x
sC sC
V V V
sC sC
   
= −   
   
(5.2.126b)
[ ] ( ) ( )4 4 4 53 2 2 4
6 6 6 6 6 6
m gs gdx x out ds
in
gd m gd m gd m
V g s C CV sC g V g
V
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 2
4
6 6
4 4 5
3
6 6
4
5
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 g
H s
sC g
g s C C
H s
sC g
g
H s
sC g
= + −
+
=
+
 − +
 =
+
=
+
(5.2.129c)
( ) ( ) [ ]2 2 3 2 1 4 4 2
1 1 1 1 1 1
gs gd m x x out ds
in
m gd m gd m gd
V s C C g V s C g V g
V
g sC g sC g sC
 + +  +   = − +
− − −
(5.2.135b)
( ) ( )3 5 4 5 4
4
6 6 6 6
m gs gd out gd
x x x x
V g s C C V sC
V
sC g sC g
 + +
 = +
+ +
(5.2.141b)

59
( ) ( )
( )
( )
( )
( )
4 3 1 2
5 4 5
1
6 6
4
2
6 6
out
m gs gd
x x
gd
x x
V V H s V H s
g s C C
H s
sC g
sC
H 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)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )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 6
3 4 1 3 2 3 5 1 3
1 1
gd gd
out
x x
sC sC
V H s H s H s V H s H s H s V V
sC sC
   
+ + − = −          
   
(5.2.126c)
( ) ( ) ( )
( ) ( ) ( )
6
4 1 3
2 3 51
1 3
1 1
1 1
gd
x
out
gd gd
x x
C
H s H s H s
H s H s H sC
V V V
C C
C C
 
+ +  −    +
   
   
   
(5.2.126d)
( ) ( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
1 3 11 12
6
4 1 3
1
11
1
1
2 3 5
12
1
1
out
gd
x
gd
x
gd
x
V V H s V H s
C
H s H s H s
C
H s
C
C
H s H s H s
H s
C
C
= +
 
+ + 
 =
 
 
 
−  =
 
 
 
(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)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )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)

61
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
3 13 2 8 14
13 7 1 6 11 9
14 10 2 6 12 9
out outV H s i V H s V H s
H s H s H s H s H s H s
+= +      
= + +
= − −
(5.2.132g)
From equation (5.2.135b),
( ) ( ) ( )
( )
( )
( )
( )
( )
[ ]
2 15 1 16 17
2 3 2
15
1 1
4 4
16
1 1
2
17
1 1
in out
gs gd m
m gd
x x
m gd
ds
m gd
V V H s V H s V H s
s C C g
H s
g sC
s C g
H s
g sC
g
H s
g sC
= − +
 + +
 =
−
 + =
−
=
−
(5.2.135c)
Substitute equation (5.2.126e), into equation (5.2.135c)
( ) ( ) ( ) ( ) ( )2 15 3 11 12 16 17in out outV V H s V H s V H s H s V H s= − + +  
(5.2.135d)
( ) ( ) ( ) ( ) ( ) ( )2 15 3 11 16 17 12 16in outV V H s V H s H s V H s H s H s= − + −      
(5.2.135e)
From equation (5.2.129d), substitute it into (5.2.135e)
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
3 4 1 3 2 3 5
2 15 3 11 16 17 12 16
out
out
V H s H s H s V H s H s H s
V H s V H s H s V H s H s H s
+ + −      
= − + −      
(5.2.135f)
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
17 12 16
3 4 1 3 11 16 2 15
2 3 5
out
H s H s H s
V H s H s H s H s H s V H s V
H s H s H s
− 
+ + = +    
− +  
(5.2.135g)

62
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
3 18 2 15 19
18 4 1 3 11 16
19 17 12 16 2 3 5
outV H s V H s V H s
H s H s H s H s H s H s H s
= +      
= + +
= − − +
(5.2.135h)
From equation (5.2.135h), Let us write
( )
( )
( )
( )
15 19
3 2
18 18
out
H s H s
V V V
H s H s
   
= +      
   
(5.2.135i)
Substitute equation (5.2.135i) into equation (5.2.132g)
( )
( )
( )
( )
( ) ( ) ( )15 19
2 13 2 8 14
18 18
out out out
H s H s
V V H s i V H s V H s
H s H s
    
+ += +              
     
(5.2.132h)
( ) ( )
( )
( )
( ) ( )
( )
( )15 13 19 13
2 8 14
18 18
0out out
H s H s H s H s
V 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)
( ) ( ) ( )( )
( ) ( )( )
( )
12 3 2 3
2 5 5 3 11 3 2 3
2
0
m gs gd
x x m gs gd out
gd
H s g s C C
V g s C V H s g s C C V
sC
 − +
   + + − + + =     − 
(5.2.138c)
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )
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 20
18 18
0x x out m gs gd out
H s H s
V g s C V V H s g s C C V H s
H s H s
    
  + + + − + + =                   
(5.2.138e)
( )
( )
( )
( ) ( )( )
( )
( )
( )
( ) ( )( )
5 5
19
2 20 11 3 2 315
1811 3 2 3
18
0
x x
out m gs gd
m gs gd
g s C
H s
V V H s H s g s C CH s
H sH s g s C C
H s
 +
   
+ + − + =       + − +        
  
(5.2.138f)
( ) ( )
( ) ( )
( )
( )
( ) ( )( )
( ) ( )
( )
( )
( ) ( )( )
2 21 22
15
21 5 5 11 3 2 3
18
19
22 20 11 3 2 3
18
0out
x x m gs gd
m gs gd
V H s V H s
H s
H s g s C H s g s C C
H s
H s
H s H s H s g s C C
H s
+ =      
 
= + + − +  
 
 
= + − +  
 
(5.2.138g)
From equation (5.2.138g), we can write
( )
( )
22
2
21
out
H s
V V
H s
 
= −  
  
(5.2.138h)
Substitute equation (5.2.138h) into equation (5.2.132i)
( )
( )
( ) ( )
( )
( )
( ) ( )
( )
( )22 15 13 19 13
8 14
21 18 18
0out out out
H s H s H s H s H s
V H s V H s i
H s H s H s
    
− − + − + =     
    
(5.2.138i)

64
( )
( )
( ) ( )
( )
( )
( ) ( )
( )
( )
22 15 13
8
21 18
19 13
14
18
out out
H s H s H s
H s
H s H s
V i
H s H s
H s
H s
   
−    
   
= 
  − −  
  
(5.2.138j)
( )
( )
( ) ( )
( )
( )
( ) ( )
( )
( )22 15 13 19 13
8 14
21 18 18
1out
out
out
V
Z
i H s H s H s H s H s
H s H s
H 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
11
1
1
5 4 5 4
1 2
6 6 6 6
4 4 52 2
4 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 g
sC g sC g sC g C
H s
C
C
g s C C sC
H s H s
sC g sC g
g s C CsC g
H s H s
sC g sC
    + + − ++     + +
 + + +
 =
 
 
 
 + +
 = =
+ +
 − ++  = =
+
( )
( )
( )
( ) ( ) ( )
4
5
6 6 6 6
4 4 54 4
6 6 6 6 6 6
12
1
1
, ds
d m gd m
m gs gdgd ds
x x gd m gd m
gd
x
g
H s
g sC g
g s C CsC g
sC g sC g sC g
H s
C
C
=
+ +
  − +
  −
 + + +
 =
 
 
 
(5.2.126f)

65
Multiply both numerator and denominator polynomial with ( )( )6 6 6 6gd m x xsC g sC g+ +
( )
( ) ( )
( )
( ) ( )
2
22 12 02
11 2
21 11 01
6
22 2 6 4 5 6 4 5 6 6 6
1
12 2 6 6 2 4 5 6 6 5 6 4
6
6 4 5 6 6 6 6
1
02 2 6 5 6 4
gd
x x gs gd gd gs gd x x gd
x
x x x x gs gd m gd m x m
gd
x gs gd x m x gd
x
x x m m m x
s a sa a
H s
s a sa a
C
a C C C C C C C C C C
C
a C g C g C C g C g C g
C
g C C C g g C
C
a g g g g g g
+ +
=
+ +
 
= + + − + +  
 
= + + + + +
 
− + + +  
 
= + +
( )
6
6 6 6
1
1
21 6 6
1
1
11 6 6 6 6
1
1
01 6 6
1
gd
x m
x
gd
gd x
x
gd
gd x x m
x
gd
m x
x
C
g g
C
C
a C C
C
C
a C g C g
C
C
a g g
C
 
+  
 
 
=  
 
 
= +  
 
 
=  
 
(5.2.142)
( )
( ) ( ) ( )
( ) ( )
( )
( )
( )
2
4 5 4 4 4 4 6 4 6
12
1 1 12
6 6 6 6 6 6 6 6
1 1 1
2
25 15 05
12 2
26 16 06
25 4 5 4
15 4 4 4 6
05 4
gs gd gd gd m ds x ds x
gd gd gd
x 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 g
H s
C C C
s C C s C g C g g g
C C C
s a sa a
H s
s a sa a
a C C C
a C g g C
a g g
 − + + − −
 =
     
+ + +     
     
− + −
=
+ +
= +
= −
= ( )
( )
( )
6
1
26 6 6
1
1
16 6 6 6 6
1
1
06 6 6
1
x
gd
x gd
x
gd
x m gd x
x
gd
x m
x
C
a C C
C
C
a C g C g
C
C
a g g
C
 
=  
 
 
= +  
 
 
=  
 
(5.2.143)

Chapter5 CMOS_Distributedamp_v244

Chapter5 CMOS_Distributedamp_v244

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