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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<1 = =n<1V R+R I R+R
27
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 0
Qω
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