Analog Circuits and Systems
Prof. K Radhakrishna Rao
Lecture 4 Analog Signal Processing One-Port Networks
1
Analog Signal Processing Functions
ASP Mathematical Functions Amplification Multiplication by a constant Filtering Solution of Differential Equation Oscillation Solution of 2nd Order Differential
Equation Mixing, Modulation, Demodulation, Phase Detection, Frequency Multiplication
Multiplication
D-A Conversions Multiplication Pulse Width Modulation Multiplication A-D Conversions Comparison
2
Revision of Pre-requisite course material
Networks and Systems One-port Networks Two-port Networks Passive Networks Active Networks
3
One-port networks for analog signal processing
Aim
§ Review properties and the signal processing functions of linear
passive and active one-port and two-port networks
4
Network Elements
Passive network elements are not capable of power amplification
Active network elements can provide power amplification
5
One-port Network Elements
One-port passive network elements • resistors • capacitors • inductors • diodes (nonlinear)
One-port active network elements • Negative resistance • Independent current and voltage sources
6
Two-port Network Elements
Two-port passive network elements ◦ Transformers ◦ Gyrators
Two-port active network elements ◦ Controlled voltage sources ◦ Controlled current sources ◦ Comparators (nonlinear) ◦ Controlled switches (nonlinear) ◦ Multipliers (nonlinear)
7
Networks
one-port passive networks are interconnections of R, L, C and diodes
one-port active networks are interconnections of –R and R, L, C or diodes
two-port passive networks are interconnections of R, L, C, transformers and diodes
two-port active networks are interconnections of R, L, C, transformers, gyrators, diodes, independent voltage and current sources, controlled voltage and current sources and multipliers.
8
Linear one-port network
has two terminals only one independent source
should be connected between the terminals
9
Linear One-port Network Characteristics
Immittance (admittance/ impedance) between its two terminals
Admittance between the two terminals
G(ω) > 0 ω Y(jω)G(ω) 0 ω, Y(jω) ≤
If for all then represents a stable networkIf for any then represents an unstable network
Y(jω)=G(ω)+ jB(ω)
10
One-port Network Elements
11
Resistor
v is voltage across the resistor in volts i is the current through the resistor in amps R the resistance in Ohms (W) of the resistor G is the conductance of the element in Siemens (S) One of the variables (voltage and current) can be considered as
independent variable, while the other one becomes dependent variable.
v=Ri i =Gvand
12
v-i relationship of Resistor
If ‘i’ is considered as the independent variable
v=Ri
13
v-i relationship of Resistor (contd.,)
If ‘v’ is considered as the independent variable
i =Gv
14
Resistor (conductor)
Performs the analog signal processing function of multiplying a variable by a constant
Used extensively in realizing attenuation and data conversion operations
15
Capacitor
1v= idtC ∫
dvi =Cdt
16
Capacitor (contd.)
is the charge Q in Coulombs stored in the capacitor
A capacitor can perform integration of a variable and its inverse
function of differentiating a variable.
Energy is stored in a capacitor as charge in electrostatic form and is
given by 0.5CV2.
idt∫
17
Inductors
Li is the flux linkages associated with the inductor Inductors store energy in electromagnetic form - 0.5Li2
Inductor performs integration of a variable and its inverse function of differentiating a variable
div=Ldt
1i= vdtL ∫
18
Diode (Controlled Switch)
Current i is the independent variable in the forward direction (i > 0; v=0)
Voltage is the independent variable when the diode is reverse biased (v < 0; i=0)
19
Negative Resistance
If ‘i’ is considered as the independent variable
v= -Ri
20
Negative Resistance (contd.,)
If v is considered as the independent variable
i=-Gv
21
Negative Resistance (contd.,)
v is voltage across the resistor in volts i is the current in amps through the resistor R the resistance in Ohms (W) G is the conductance in Siemens (S) A negative resistor (conductor) can multiply a variable by a negative
constant, and is used for loss compensation, amplification and oscillation
22
Signal Processing Functions of
One-port Networks
23
Signal processing
If voltage is the dependent variable current becomes independent
variable and vice-versa in one-port networks
Different relationships between independent variable and
dependent variable can be created using different combinations of
network elements
24
Nature of one-port networks
A voltage source should not be shorted
A current source should not be opened
25
Conversion of variable (v to i and i to v)
A resistor (R) converts a current into a voltage as long as its value does not go to infinity (open circuit).
A conductor (G) converts a voltage into current as long as its value is not infinity (short circuit).
26
Attenuation
If the voltage and current sources have finite source resistances
This is equivalent to multiplying the independent variable by a constant less than one
o o s
s s s s
V I RR= =n
Integration and Differentiation
28
Filtering
( )
i
o
O OS
O O S
O
S
iv v dv+C =iR dtdv v i+ =dt RC C
V R=I 1+sCR
is the independent variable and
is the dependent variable.
The driving point impedance function of the RC network is given as
The RC network acts as a low -pa .ss filter
29
Parallel RC network with negative resistance –R1
o o oS
1
o o o S
1
v v dv- +C =iR R dtdv v v i+ - =dt RC R C C
30
Parallel RC network with negative resistance –R1 (contd.,)
The driving point impedance function If R < R1 it becomes a low-pass
filter If R > R1 the transfer function has
negative real part, and the impulse response of the dependent variable grows unbounded with time making the circuit unstable.
( )/
o/
S
/ 1
1
V R=I 1+sCR
RRR =R -R
where
1
o
S
R = R V 1=I sC
If
and the circuit becomes an ideal integrator.
31
Parallel RLC one-port network with negative resistance –R1
o o oo S
1
1
o2
2S20 0
00
v v dv 1- +C + v dt=iR R dt L1 1 1If = -R R R
V sL sL= =sLI s ss LC+ +1 + +1R ω ω Q
1 R Cω = andQ= =Rω L LLC
where
′
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟′⎝ ⎠ ⎝ ⎠
′ ′
∫
32
Parallel RLC one-port network with negative resistance –R1 (contd.,)
This driving point impedance function represents a band pass
filter with centre frequency of and a band width of
If R1 = R it is sine wave oscillator of frequency If R > R1 the
circuit becomes unstable (oscillations grow without bound in
amplitude)
0ω 0Qω
33
Example 1
Design an amplifier using negative resistance for a voltage gain of 10. The voltage source has a source resistance of 1 k ohms and the load resistance is 2 k ohms.
The circuit may be simplified as
R 2= =102 3R-3
5R=7
Voltage gain
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
Ωk
34
Example 2
Design a diode-resistor one-port network with V-I characteristic
35
Example 2 (contd.,)
Plot the voltage across the port when the current is of triangular waveform
10mS 20mS
(-2/3)
(2/3)
(1/3)
(-1/3)
mA
time
36
Diode-resistor network
37
Voltage across the port
10mS 20mS
(-2/3)
(4/3)
(2/3)
(-1/3)
mA
time
1V
1.5V
38