Analog Circuits and Systems

Prof. K Radhakrishna Rao

Lecture 4 Analog Signal Processing One-Port Networks

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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

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Revision of Pre-requisite course material

  Networks and Systems   One-port Networks   Two-port Networks   Passive Networks   Active Networks

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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

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Network Elements

  Passive network elements are not capable of power amplification

  Active network elements can provide power amplification

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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

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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)

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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.

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Linear one-port network

  has two terminals   only one independent source

should be connected between the terminals

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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)

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One-port Network Elements

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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

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v-i relationship of Resistor

  If ‘i’ is considered as the independent variable

v=Ri

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v-i relationship of Resistor (contd.,)

  If ‘v’ is considered as the independent variable

i =Gv

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Resistor (conductor)

  Performs the analog signal processing function of multiplying a variable by a constant

  Used extensively in realizing attenuation and data conversion operations

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Capacitor

1v= idtC ∫

dvi =Cdt

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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∫

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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 ∫

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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)

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Negative Resistance

  If ‘i’ is considered as the independent variable

v= -Ri

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Negative Resistance (contd.,)

  If v is considered as the independent variable

i=-Gv

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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

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Signal Processing Functions of

One-port Networks

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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

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Nature of one-port networks

  A voltage source should not be shorted

  A current source should not be opened

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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).

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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

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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

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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

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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.

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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

′

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟′⎝ ⎠ ⎝ ⎠

′ ′

∫

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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

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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

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Example 2

  Design a diode-resistor one-port network with V-I characteristic

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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

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Diode-resistor network

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Voltage across the port

10mS 20mS

(-2/3)

(4/3)

(2/3)

(-1/3)

mA

time

1V

1.5V

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