Analog Circuits and Systems
Prof. K Radhakrishna Rao
Lecture 5
Analog Signal Processing using One Port
Networks, Passive Two Ports and Ideal Amplifiers
1
One Port Devices
—
—
Passive devices like R, L, C and diodes
Active Device - Negative resistance – Characteristics and
applications
2
Differential Equations: Solutions
First Order Differential Equation
3
Differential Equations: Solutions (contd.,)
Second Order Differential Equation
4
Linear Two-port Networks
Aim
—
Two port network parameters - Y, Z, g and h
—
Immittance Matrix parameters – p
—
Characteristics of two port passive networks
—
Characteristics of two port active networks
—
Controlled sources/amplifiers
5
Nature of Two-port Networks
6
Input port
Inputs can be
— active one-port network like a transducer
— output of another signal processing two-port network
Input source can be represented as
— ideal voltage source in series with impedance ZS
— ideal current source in shunt with admittance YS
7
Output port
Output may be connected to
— a load
— the input port of another signal processing network
8
Two-port Network
Characterized by four variables: two at the input port as input voltage
Vi and input current Ii
— Any two variables can be independent and the other two will
become dependent variables
— Two equations between the two dependent variables and the two
independent variables characterize the two-port network
9
Two-port Network Parameters
—
—
—
—
—
Six combinations of independent/dependent variables are possible.
The parameters associated with these six combinations are
Y, Z, g, h and ABCD and S
If one is restricted to use one independent variable at the input and
one at the output port
then only the parameters Y, Z, g and h are of interest
ABCD and S parameters are primarily used for studying microwave
networks.
10
g, h,Y and Z parameters
—
—
g and h are hybrid parameters involving admittance, impedance,
voltage ratio and current ratio Y has all admittance (short-circuit)
parameters
Z has all impedance (open-circuit) parameters
11
Matrix representation of Two-port networks
⎡ Ii ⎤ ⎡ g i
=
⎢ V ⎥ ⎢g
⎣ o⎦ ⎣ f
g r ⎤ ⎡ Vi ⎤
,
⎥
⎢
⎥
g o ⎦ ⎣ Io ⎦
⎡ Vi ⎤ ⎡ h i
=
⎢ I ⎥ ⎢h
⎣ o⎦ ⎣ f
⎡ Ii ⎤ ⎡ Yi
=
⎢I ⎥ ⎢Y
⎣ o⎦ ⎣ f
Yr ⎤ ⎡ Vi ⎤ ⎡ Vi ⎤ ⎡ Zi
,
=
⎥
⎢
⎥
⎢
⎥
⎢
Yo ⎦ ⎣ Vo ⎦ ⎣ Vo ⎦ ⎣ Zf
h r ⎤ ⎡ Ii ⎤
⎥
⎢
⎥
h o ⎦ ⎣ Vo ⎦
Z r ⎤ ⎡ Ii ⎤
⎥
⎢
⎥
Zo ⎦ ⎣ I o ⎦
12
Immittance Parameters
Y, Z, g and h can be generally represented as p- immittance
parameters
— Input current relating to input voltage is input self-immittance pi
— Output current related to output voltage is known as output selfimmittance po
— Input related to output is called forward transfer parameter pf
— Output related to input is known as reverse transfer parameter pr
13
Parameters of Two-port Passive Networks
—
—
—
Forward parameter is the same as reverse parameter
it is exactly the same in the case of Z and Y parameters
in the case of h and g parameters the sign is reversed
14
Ideal controlled sources
are known as amplifiers
— have zero input power and deliver finite output power
— have infinite power gain
For the input power to be zero it should have
— Zero input current to the amplifier (zero input admittance)
— Zero input voltage to the amplifier (zero input impedance)
—
15
Ideal Amplifiers
—
—
Input is voltage controlled (zero input admittance) or current
controlled (zero input impedance)
Output is a voltage sources (zero output impedance) or current
source (zero output admittance)
16
Types of amplifiers
—
—
—
—
Voltage Controlled Voltage Source (Voltage Amplifiers) (VCVS)
Current Controlled Current Source (Current Amplifier) (CCCS)
Voltage Controlled Current Source (Transconductance Amplifier)
(VCCS)
Current Controlled Voltage Source (Transresistance Amplifier)
(CCVS)
17
Ideal amplifiers
Can be represented only by one of the four parameters
⎡0
VCVS by g-parameter ⎢
⎣g f
⎡0
CCCS by h-parameter ⎢
⎣h f
⎡0
VCCS by Y-parameter ⎢
⎣G f
0⎤
0 ⎥⎦
⎡0
CCVS by Z-parameter ⎢
⎣R f
0⎤
0 ⎥⎦
0⎤
0 ⎥⎦
0⎤
0 ⎥⎦
18
Ideal Amplifiers
⎡0
Immittance matrix = ⎢
p
⎣ f
0⎤
⎥
0⎦
They all have zero reverse transmission
parameters ( pr ) , which means the
device is unilateral.
19
Two-port networks:Y-parameters
Ii =Vi Yi +Vo Yr
Io =Vi Yf +Vo Yo
20
Example 1
⎡ Ii ⎤ ⎡ Yf -Yf ⎤ ⎡ Vi ⎤
⎢ I ⎥ = ⎢-Y Y ⎥ ⎢ V ⎥
f ⎦⎣ o⎦
⎣ o⎦ ⎣ f
Forward transfer admittance
= reverse transfer admittance
21
Example 2
1+sCR 2
⎡
⎢ R +R +sCR R
1
2
1 2
⎢
[Y] =
-1
⎢
⎢ R +R +sCR R
⎣ 1 2
1 2
-1
⎤
⎥
R1 +R 2 +sCR1R 2
⎥
1+sCR1
⎥
⎥
R1 +R 2 +sCR1R 2 ⎦
22
Example 3: Load and source immittances
Ii =Vi Yi +Vo Yr ; Io =Vi Yf +Vo Yo
Io
Vo =; Io =-Vo YL ; -Vo YL =Yf Vi +Yo Vo and
YL
Vo
Yf
=Vi Yo +YL
23
Example 3 (contd.)
Yf
Yf Yr
Ii =Yi Vi Yr Vi ; Ii Vi =Yin =input admittance =Yi Yo +YL
Yo +YL
1
⎡ 1
+
⎢R
RS
f
[Y] = ⎢
⎢ -1
⎢ R
⎣
f
⎤
⎥
⎥
1
1 ⎥
+
⎥
Rf RL ⎦
-1
Rf
Io
=output admittance for a source admittance YS
Vo
Io
IS 1
Yf Yr
1
1
Yout =
=Yo =
; Yin = =
+
Vo
Yi +YS R f +R S
Vi R S R f +R L
24
Example 3 (contd.)
Voltage gain of the two port (for a given YL )
1 Rf )
Vo
(
-Yf
1
Av =
=
=
=
Vi YL +Yo (1 R L ) + (1 R f ) 1+ ( R f R L )
1 RL )
Io -A v YL
(
-1
Current gain = =
=
= −1
Ii
Yin
1+ ( R f R L ) (1 ( R f +R L ) )
25
Two-port network - Z-Parameters
Vi =Zi Ii +Zr Io
Vo =Zf Ii +Zo Io
26
Example 1
⎡ Vi ⎤ ⎡ Z Z⎤ ⎡ Ii ⎤
=
⎢ V ⎥ ⎢ Z Z⎥ ⎢ I ⎥
⎦⎣ o⎦
⎣ o⎦ ⎣
27
Example 2: p network
⎡ R1 ( R 2 +R 3 )
⎢
R
+R
+R
1
2
3
⎢
[ Z] = ⎢
R1R 3
⎢
⎢⎣ R1 +R 2 +R 3
⎤
R1R 3
⎥
R1 +R 2 +R 3 ⎥
R 3 ( R1 +R2 ) ⎥
⎥
R1 +R 2 +R 3 ⎥⎦
28
Source and Load Immittances
Z r Zf
Zin =Zi Zo +ZL
Z r Zf
Zout =Zo Zi +ZS
Io
-Zf
A I = = current gain =
Ii
ZL +Zo
Io ZL
ZL
Av =
=A I
Ii Zin
Zin
29
Example 3
⎡ R
⎡⎣ Z ⎤⎦ = ⎢
⎣ R
2
RR L
R
Zin =R=
R+R L R+R L
Io
R
=I i R+R L
R ⎤
⎥
R ⎦
2
RR S
R
Zout =R=
R+R S R+R S
Vo
=1
Vi
30
Two-port network with g– parameters
Ii =gi Vi +g r Io
Vo =g f Vi +g o Io
31
Example
⎛ R1 ⎞
Voltage attenuator ⎜
⎟ as g f
⎝ R1 +R 2 ⎠
Its two - port g - parameters are
⎡ 1
⎢ R +R
1
2
⎢
[g ] =
⎢ R1
⎢ R +R
⎣ 1 2
-R1 ⎤
R1 +R 2 ⎥
⎥
1
1 ⎥
+
⎥
R1 R 2 ⎦
g f = −g r
An ideal active device which can be represented by
g - parameters is the voltage controlled voltage
source or VCVS.
32
Two-port network - h-Parameters
Vi =h i Ii +h r Vo
Io =h f Ii +h o Vo
The h - parameter characterization is the dual of g - parameter
characterization of a two - port network.
⎡ Ii ⎤ ⎡ g i
⎢ V ⎥ = ⎢g
⎣ o⎦ ⎣ f
g r ⎤ ⎡ Vi ⎤
⎥
⎢
⎥
g o ⎦ ⎣ Io ⎦
33
Two-port network - h-Parameters (contd.,)
When the independent and dependent variables are interchanged
the matrix representation of the two - port network becomes
⎡ g o -g r ⎤
⎡ g r gf ⎤
⎡ Vi ⎤ ⎢ Δg Δg ⎥ ⎡ Ii ⎤
⎢
⎥
=
where Δg = gi g o - g r g f = g i g o ⎢1⎥
⎢ I ⎥ -g
⎢
⎥
⎣ o ⎦ ⎢ f gi ⎥ ⎣ Vo ⎦
⎣ gi g o ⎦
⎢ Δg Δg ⎥
⎣
⎦
gf
where is the forward transfer admittance,
go
gr
and is reverse transfer impedance
gi
34
Example
⎡
R
R
1
2
⎢
[h ] = ⎢
⎢ -R1
⎢ R +R
⎣ 1 2
R1 ⎤
⎥
R1 +R 2
⎥
1 ⎥
⎥
R1 +R 2 ⎦
h f =-h r
35
Example (contd.,)
—
—
—
An ideal active device which can be represented by h-parameters is
the current controlled current source or CCCS.
In an ideal amplifier the input power is zero: Vi=0 and hi=0.
The current at the input Ii appears as current at the output (hf Ii)
where hf is the forward current ratio called current gain
36
Two port characterization using Y-parameters
Ii =Vi Yi +Vo Yr
Io =Vi Yf +Vo Yo
Io = -YL Vo = Vi Yf + Vo Yo
37
Two port characterization using Y-parameters (contd.,)
Input admittance
Ii
Yr Yf
Yin = =Yi Vi
Yo +YL
Io
Yr Yf
Output admittance Yout = =Yo Vo
Yi +Ys
Voltage gain
Vo
Yf
=Vi Yo +YL
Current gain
Io
Yf YL
=
Ii Yin ( Yo +YL )
38
Two port characterization using h-parameters
Vi =h i Ii +h r Vo
Io =h f Ii +h o Vo
Io
As
=-Vo then -Vo h L =h f Ii +h o Vo
hL
39
Two port characterization using h-parameters (contd.,)
Vo
-h f
Forward Trans - impedance
=
Ii h o +h L
Input impedance
Vi
hf hr
h in = =h i Ii
h o +h L
Output admittance
Io
hr hf
h out =
=h o Vo
h o +h s
Io
hf hL
Forward Trans - admittance
=
Vi h in ( h o +h L )
40
Two port characterization using h-parameters (contd.,)
pr pf
Input immittance pin = pi p O +p L
⎛
pr pf ⎞
Total Input immittance = ( pS + pin ) = ⎜ pS +pi ⎟
pO +p L ⎠
⎝
⎛
⎞
pr pf
= ( pS + pi ) ⎜⎜1⎟⎟
⎝ ( p O +p L )( pS + pi ) ⎠
pr pf
Output immittance p out = p O pi +p S
⎛
pr pf ⎞
Total Output immittance = ( p L + p out ) = ⎜ p L +p O ⎟
pi +pS ⎠
⎝
⎛
⎞
pr pf
= ( p L + p O ) ⎜⎜1⎟⎟
⎝ ( p O +p L )( pS + pi ) ⎠
41
Two-port characterization in immittance parameters
pf
Forward Transfer Immittance pf1 = pO +p L
pf p L
Forward Transfer Immittance pf2 =
pin ( pO +p L )
pf
Forward Transfer Immittance = pO +p L
pr
Reverse Transfer Immittance = pi +p S
42
Loop Gain: gl
p r pf
gl =
( pi +pS )( po +pL )
negative for negative feedback
positive for positive feedback
43
The composite immittance matrix
pr ⎤
⎡ pi +pS
[p] = ⎢ p p +p ⎥
O
f⎦
⎣ f
44
Determinant of the Immittance Matrix
( Δ ) = ( pi +pS )( po +pL ) -pr pf
= ( pi +pS )( po +p L ) (1-g l )
(1-gl ) characteristic polynomial
Input immittance = ( pi +pS ) (1-g l )
Output immittance = ( po +p L ) (1-g l )
45
Conclusion
46