Analog Circuits and Systems

Transcription

1 Analog Circuits and Systems Prof. K Radhakrishna Rao Lecture 4 Analog Signal Processing One-Port Networks 1

2 Analog Signal Processing Functions ASP Amplification Filtering Oscillation Mixing, Modulation, Demodulation, Phase Detection, Frequency Multiplication D-A Conversions Pulse Width Modulation A-D Conversions Mathematical Functions Multiplication by a constant Solution of Differential Equation Solution of 2 nd Order Differential Equation Multiplication Multiplication Multiplication Comparison 2

3 Revision of Pre-requisite course material Networks and Systems One-port Networks Two-port Networks Passive Networks Active Networks 3

4 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

5 Network Elements Passive network elements are not capable of power amplification Active network elements can provide power amplification 5

6 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

7 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

8 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

9 Linear one-port network has two terminals only one independent source should be connected between the terminals 9

10 Linear One-port Network Characteristics Immittance (admittance/ impedance) between its two terminals Admittance between the two terminals Y(jω) = G(ω) + jb(ω) If G(ω) > 0for all ω then Y(jω) represents a stable network If G(ω) 0 for any ω, then Y(jω) represents an unstable network 10

11 One-port Network Elements 11

12 Resistor v = Ri andi = Gv 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. 12

13 v-i relationship of Resistor If i is considered as the independent variable v=ri 13

14 v-i relationship of Resistor (contd.,) If v is considered as the independent variable i=gv 14

15 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

16 i=c dt Capacitor v = 1 idt C dv 16

17 Capacitor (contd.) idt 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 0.5CV 2. 17

18 Inductors di v=l dt 1 i= vdt L Li is the flux linkages associated with the inductor Inductors store energy in electromagnetic form - 0.5Li 2 Inductor performs integration of a variable and its inverse function of differentiating a variable 18

19 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

20 Negative Resistance If i is considered as the independent variable v=-ri 20

21 Negative Resistance (contd.,) If v is considered as the independent variable i=-gv 21

22 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

23 Signal Processing Functions of One-port Networks 23

24 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

25 Nature of one-port networks A voltage source should not be shorted A current source should not be opened 25

26 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

27 Attenuation If the voltage and current sources have finite source resistances This is equivalent to multiplying the independent variable by a constant less Vo R Io R = =n<1 = s =n<1 V R+R I R+R s s s s than one 27

28 Integration and Differentiation 28

29 Filtering i i v v o O O S dv O +C =is R dt dvo vo is + = dt RC C The driving point impedance function V I is the independent variable and is the dependent variable. of the RC network is given as = R 1+sCR ( ) The RC network acts as a low -pa ss filter. 29

30 Parallel RC network with negative resistance R1 v v dv o - o +C o =i R R dt 1 dv v v i o + o - o = S dt RC R C C 1 S 30

31 Parallel RC network with negative resistance R1 (contd.,) The driving point impedance function If R < R 1 it becomes a low-pass filter If R > R 1 the transfer function has negative real part, and the impulse response of the dependent variable grows unbounded with time making the circuit unstable. V I o S = R ( / 1+ scr ) / 1 where R = R-R 1 If S R = R Vo 1 = I sc 1 / RR and the circuit becomes an ideal integrator. 31

32 Parallel RLC one-port network with negative resistance R1 vo vo dvo 1 - +C + vodt=i R R dt L If = - R R R 1 Vo sl sl = = 2 IS 2 sl s LC+ +1 s s R ω0 ω0q where 1 R C ω 0 = and Q= =R LC ωl L S 0 32

33 Parallel RLC one-port network with negative resistance R1 (contd.,) This driving point impedance function represents a band pass filter with centre frequency of ω ω 0 and a band width of 0 Q If R 1 = R it is sine wave oscillator of frequency If R > R 1 the circuit becomes unstable (oscillations grow without bound in amplitude) 33

34 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 Voltage gain 5 R= 7 kω R 2 = =10 2 R

35 Example 2 Design a diode-resistor one-port network with V-I characteristic 35

36 Example 2 (contd.,) Plot the voltage across the port when the current is of triangular waveform ma (2/3) (1/3) time 10mS 20mS (-1/3) (-2/3) 36

37 Diode-resistor network 37

38 Voltage across the port ma (4/3) 1.5V (2/3) 1V time 10mS 20mS (-1/3) (-2/3) 38