Analog Circuits and Systems
Prof. K Radhakrishna Rao
Lecture 6
Synthesis of Amplifiers using Nullators ans
Norators
Immittance Matrix
pr ⎤
⎡ pi +pS
[p] = ⎢ p p +p ⎥
o
L⎦
⎣ f
Δ= ( pi +pS )( po +p L ) [1-g l ]
⎛ -pf ⎞⎛ -p r ⎞
Loop gain = g l = ⎜
⎟⎜
⎟
⎝ po +p L ⎠⎝ pi +pS ⎠
pin = ( pi +pS ) [1-g l ]
pout = ( p o +p L ) [1-g l ]
Passive Network
Yr =Yf
g r =-g f
Zr =Zf
h r =-h f
Active Network
Yr ≠ Yf
g r ≠ -g f
Z r ≠ Zf
h r ≠ -h f
Linear Signal processing functions of Two-port Networks
Two-port networks can perform
—  attenuation
—  amplification
—  addition
—  subtraction
—  integration
—  differentiation
—  filtering
Attenuation
Can be performed using a wide range of two-port networks
—  Resistive
—  Capacitive
—  Inductive
—  Resistive-Capacitive
—  Resistive-Capacitive-Inductive
Resistive Voltage Attenuator
V2
is the forward transfer function
V1
Voltage Attenuator is represented in g-parameters
i1=g i v1+g ri2
v 2 =g f v1+g oi2
Ra
Ra
R aR b
1
with g i =
, gf =
, g r =and g o =
R a +R b
R a +R b
R a +Rb
R a +R b
Resistive Current Attenuator
I2
is the forward transfer function
I1
Current attenuator is characterized by h-parameters
v1=hii 1+hr v 2
i2 =hf i1+ho v 2
The h-parameters are given by
R aR b
-R a
Ra
1
hi =
;hf =
;hr =
and ho =
R a +R b
R a +R b
R a +R b
R a +R b
Capacitive Voltage Attenuator
Cb
v2 =
v1
Ca +Cb
Macro - model of voltage attenuator
i1 = gi v1 +g r i 2
v 2 = g f v1 +g oi 2
sCa Cb
Cb
Cb
1
gi =
;g r =;g f =
;g o =
Ca +Cb
Ca +Cb
Ca +Cb
s ( Ca +Cb )
Inductive Current Attenuator
Current attenuator is characterized by h - parameters
v1 = h ii1 + h r v 2
i 2 = h f i1 + h o v 2
sLa Lb
-La
La
1
hi =
; hf =
; hr =
and h o =
La + L b
La + L b
La + L b
s ( La + L b )
RC Attenuator
Outputs of the two two - port networks
Cb
Ra
Vo1 =
and Vo2 =
Cb +Ca
R a +R b
If Vo1 = Vo2 at all times the two
output nodes can be shorted
Cb
Ra
As Vo1 = Vo2 ;
=
leads to Ca R a = Cb R b
Cb +Ca R a +R b
RC attenuator is commonly used as an attenuator
(1/10 being the most commonly used attenuation
factor) probe supplied with oscilloscopes.
Input of the oscilloscope is a resistance shunted
by a capacitance (typically 1MΩ with10 pF)
Example
Design a probe suitable for an
oscilloscope with R =1 MΩ
shunted by C =10 pF for an
attenuation by a factor of 10.
Probe network
Ra
1
=
R b = 9 MΩ
R a +R b 10
6
Ca R a = Cb R b =10 x10x10
Cb =1.11 pF
-12
Transformers
—
—
—
An ideal transformer is a
simple passive two-port
network
It attenuates (steps-down)
voltage or current
simultaneously amplifying
(stepping-up) current or
voltage
An ideal transformer is
characterized by NI11 =N2I0 and VI11 =V0I0
Transformers (contd.,)
—
—
—
Power gain in an ideal transformer is unity
Transformers are primarily used for impedance matching
and stepping up/stepping down voltages and currents
Represented by a g- or h-matrix
⎡
⎢ 0
[g ] = ⎢
⎢ N2
⎢N
⎣ 1
N2 ⎤
⎡
0
⎥
⎢
N1
⎥ or [ h ] = ⎢
⎥
⎢ N1
0 ⎥
⎢- N
⎦
⎣ 2
N1 ⎤
⎥
N2
⎥
⎥
0 ⎥
⎦
Transformer with load and source
2
2
⎛ N1 ⎞ 1
⎛ N2 ⎞
g in = ⎜
or h in = ⎜
⎟
⎟ RL
⎝ N2 ⎠ R L
⎝ N1 ⎠
For maximum power transfer from the source to the load
2
⎛ N2 ⎞
R S = h in = ⎜
⎟ RL
⎝ N1 ⎠
Ladder Network as a Voltage Attenuator
—
—
—
—
A two-port resistor (R-2R) or capacitor (2C-C) ladder network as
an attenuator can generate binary weighted values of a reference
voltage
Digital-to-Analog converters use such ladder networks
The impedance seen looking into the ladder at any node is R
The node voltages are VR , VR , VR and VR
2
4
8
Ladder Network as a Current Attenuator
—
—
Voltage attenuating ladder network is used with a current source,
IR, at the output port with the input port shorted
Current divides according to binary weights, as
IR IR
IR
IR , ,
and
2 4
8
Example
Determine R in the ladder network so
that the voltage source delivers 1/5th
of its value at the output port?
What is the current through10 kΩ
resistor due to the current source?
Vo 1
1
= =
Vi 5 1+ 10 + 15 + 10
5 R R
R = 12.5 kΩ
Current through the 10 kΩ resistor
is 1/5th of the current from the current
source 0.4 sin400t
Attenuator
Amplification
Ideal amplifiers
—  have zero input power
—  deliver finite output power
providing infinite power gain
Input power is zero if the
—  input current to the amplifier
is zero or
—  input voltage to the amplifier
is zero
Ideal amplifiers at their input
should be
—  voltage controlled or
—  current controlled
Output of ideal amplifiers could
be
—  ideal voltage sources or
—  current sources
Types of ideal amplifiers as two-port networks
—
VCCS (Voltage Controlled Current Source) (Trans-conductance
amplifier)
—
CCVS (Current Controlled Voltage Source) (Trans-resistance
amplifier)
—
VCVS (Voltage Controlled Voltage Source) (Voltage amplifier)
—
CCCS (Current Controlled Current Source) (Current amplifier)
Parameters associated with ideal amplifier
—
Trans-conductance (Gf) for VCCS
—
Trans-resistance (Rf) for CCVS
—
Voltage gain (gf) for VCVS
—
Current gain (hf) for CCCS
VCCS and CCVS functions
—
—
VCCS and CCVS functions are more fundamental
CCCS and VCVS and functions can be performed by
suitably cascading the VCCS and CCVS functions
VCCS
⎡ 0
[Y ] = ⎢G
⎣ f
0⎤
⎥
0⎦
CCVS
⎡0
[Z ] = ⎢ R
⎣ f
0⎤
⎥
0⎦
VCVS
⎡0
[g] = ⎢g
⎣ f
0⎤
⎥
0⎦
CCCS
⎡0
[ h] = ⎢ h
⎣ f
0⎤
⎥
0⎦
Example
What is the voltage gain of the network
0 ⎤
⎡ 0
whose [ Y ] = ⎢
?
⎥
⎣10mS 0.1 mS⎦
It is VCCS with transconductance of 10 mS feeding
a load of 0.1mS
It produces an output current
Io =10 x Vi mA.
This current flowing through a load of 0.1mS
produces an output voltage Vo = -100Vi
Voltage gain = -100
Example
Design a voltage amplifier with a voltage gain of 1000
using only transconductance and transresistance amplifiers?
Voltage amplifier with a gain of 1000 can be
obtained by cascading a VCCS and a CCVS
VCCS is realized by a two - port network represented by
0⎤
⎡ 0
[Y ] = ⎢10mS 0⎥
⎣
⎦
CCVS is realized by a two - port network represented by
⎡ 0
[ Z] = ⎢100kΩ
⎣
0⎤
⎥
0⎦
Example
1)
2)
3)
4)
⎡ 2 mS -1 mS⎤
Ideal VCCS
[Y ] = ⎢-1 mS 3mS ⎥ a)
⎣
⎦
⎡ 0 -10 ⎤
b)
Passive Network
[g ] = ⎢10 0 ⎥
⎣
⎦
0⎤
⎡ 0
c)
Ideal Transformer
[Y ] = ⎢1 mS 0⎥
⎣
⎦
⎡ 0 0⎤
d) Ideal Current Amplifier
[h ] = ⎢100 0⎥
⎣
⎦
Conclusion
31