Lowpass RC circuit

Linear Wave Shaping - Lowpass RC Circuit

1. Linear Wave shaping: Low pass RC circuit Electronic Devices and Circuits UNIT – I SVEC19

2. Linear wave shaping M.Balaji , Department of ECE, SVEC 2 A linear network is a circuit made up of linear elements only. The process where a non- sinusoidal signal is altered by transmission through a linear network is called linear wave shaping. There are a number of waveforms, which appear very frequently. The most important of these are sinusoidal, step, pulse, square wave and ramp waveforms. Low-pass RC circuit High-pass RC circuit

3. The low-pass RC Circuit When f= 0; X C = ∞ (Capacitor is open circuited); V o (t) = V i (t) ; then gain A= . When f increases , X C decreases , then V o (t ) = decreases ; then gain . When f= ∞; X C = (Capacitor is short circuited); V o (t ) = ; then gain A .   M.Balaji, Department of ECE, SVEC 3 The circuit which transmits only low- frequency signals and attenuates or stops high frequency signals.

4. Sinusoidal input M.Balaji, Department of ECE, SVEC 4 Laplace transformed low- pass RC circuit and its frequency response

5. Squaring both sides and equating the denominators,   M.Balaji , Department of ECE, SVEC 5

6. :. The upper cut-off frequency, So The angle θ by which the output leads the input is given by   M.Balaji, Department of ECE, SVEC 6

7. Step Voltage input A step signal is one which maintains the value zero for all times t < 0 ; and maintains the value V for all times t > 0 . M.Balaji, Department of ECE, SVEC 7 t = 0 - t = 0 + t < 0 t > 0 t = 0 The transition between the two voltage levels takes place at t = 0. V i = 0 , immediately before t = 0 (to be referred to as time t= 0 - ) and V i = V , immediately after t= 0 (to be referred to as time t= 0 + ).

8. M.Balaji, Department of ECE, SVEC 8

9. Charge Q = C V ------------------------ (1) Where i = instantaneous current through the capacitor C = Capacitance (Farads) dV / dt = instantaneous rate of change of voltage (Volts / Sec) Integrate equation (1) on both sides we get,   M.Balaji, Department of ECE, SVEC 9

10. Let V ' be the initial voltage across the capacitor Writing KVL around the loop, we get Differentiating this equation with respect to t, we get   We know, for a step input   M.Balaji, Department of ECE, SVEC 10

11. Taking the Laplace transform on both sides, ----------- (1) The initial current is given by From equation (1) we get,   M.Balaji, Department of ECE, SVEC 11 Step voltage, V Initial voltage across the capacitor is V'

12. Taking the inverse Laplace transform on both sides , Where V’ is the initial voltage across the capacitor ( V initial ) and V is the final voltage ( V final ) to which the capacitor can charge .   M.Balaji, Department of ECE, SVEC 12

13. The expression for the voltage across the capacitor of an RC circuit excited by a step input is given by If the capacitor is initially uncharged   M.Balaji, Department of ECE, SVEC 13

14. Expression for rise time M.Balaji, Department of ECE, SVEC 14

15. Expression for rise time At   At   M.Balaji, Department of ECE, SVEC 15  

16. Relation between rise time and upper 3-dB frequency M.Balaji, Department of ECE, SVEC 16 The upper 3-dB frequency of a low-pass circuit (same as bandwidth) is Thus the rise time is inversely proportional to the upper 3-dB frequency. The time constant ( τ =RC) of a circuit is defined as the time taken by the output to rise to 63.2% of the amplitude of the input step.  

17. Pulse input The pulse is equivalent to a positive step followed by a delayed negative step. M.Balaji, Department of ECE, SVEC 17 Pulse waveform Pulse w aveform in terms of step

18. M.Balaji, Department of ECE, SVEC 18

19. A pulse shape will be preserved if the 3-dB frequency is approximately equal to the reciprocal of the pulse width To pass a 0.25µs pulse reasonably well requires a circuit with an upper cut-off frequency of the order of 4MHz. M.Balaji, Department of ECE, SVEC 19

20. Square wave input A square wave is a periodic waveform which maintains itself at one constant level V’ with respect to ground for a time T 1 , and then changes abruptly to another level V’’, and remains constant at that level for a time T 2 , and repeats itself at regular intervals of time T 1 +T 2 . A square wave may be treated as a series of positive and negative steps. M.Balaji, Department of ECE, SVEC 20

21. (a) Square wave input waveform (b) output waveform for RC   M.Balaji, Department of ECE, SVEC 21

22. (c) Output waveform for RC ≈ T and (d) RC   M.Balaji, Department of ECE, SVEC 22

23. M.Balaji, Department of ECE, SVEC 23

24. In the above figure when RC ≈ T, the equation for the rising portion is Where V 2 is the voltage across the capacitor at t=0, and V’ is the maximum level to which the capacitor can charge . The equation for the falling portion is Where V 1 is the voltage across the capacitor at t= T 1 and V’’ is the minimum level to which the capacitor can discharge.   M.Balaji, Department of ECE, SVEC 24

25. Setting -------- (1)   M.Balaji, Department of ECE, SVEC 25 Setting ----- (2)  

26. Substituting this value of V 2 ( eq 2) in the expression for V 1 (1), we get Similarly substituting the value of V 1 in the expression for V 2 , we get   M.Balaji, Department of ECE, SVEC 26

27. For a symmetrical square wave with zero average value, So, V 2 will be equal to – V 1   M.Balaji, Department of ECE, SVEC 27

28. Where is the period of the square wave. Now ,   M.Balaji, Department of ECE, SVEC 28

29. Ramp input M.Balaji, Department of ECE, SVEC 29 When a low- pass RC circuit is excited with a ramp input, i.e is the slope of the ramp We have, For the frequency domain circuit of a low-pass RC circuit, the output is given by  

30. = Put s= 0; then B= α Put ; then C= α RC To get the value of A,   M.Balaji, Department of ECE, SVEC 30

31. Equating the coefficients of s 2 on both sides A+C=0 Therefore A= - C As C= α RC then A= - α RC Substituting the values of A, B and C, we get M.Balaji, Department of ECE, SVEC 31  

32. Taking the inverse Laplace transform on both sides, we get If the time constant RC is very small, then   M.Balaji, Department of ECE, SVEC 32

33. When the time constant is very small relative to the total ramp time T, the ramp will be transmitted with minimum distortion Response of a low-pass RC circuit for a ramp circuit for (a) RC/T<< 1 (b) Response of a low-pass RC circuit for a ramp circuit for (a) RC/T>> 1 M.Balaji, Department of ECE, SVEC 33

34. Expanding in to an infinite series in t/RC in the above equation for V (t), The above expression shows a quadratic response obtained for a linear input and hence the circuit acts as an integrator for RC/T >> 1   M.Balaji, Department of ECE, SVEC 34

35. Low pass RC circuit as an integrator For the low-pass RC circuit to behave as an integrator, RC >>T . For T to be small when compared to RC , the frequency has to be high. At high frequencies, XC is very small when compared to R. Therefore , the voltage drop across R is very large when compared to the drop across C. Hence , vi ∼= iR M.Balaji, Department of ECE, SVEC 35 RC>>T When f is high, then T will be small then RC will be much greater than T At high frequencies, X c = Drop across C is very less. Then the entire drop will be at R  

36. The output is proportional to the integral of the input signal   M.Balaji, Department of ECE, SVEC 36

37. If the time constant of an RC low-pass circuit is very large in comparison with the time required for the input signal to make an appreciable change, the circuit acts as an integrator. The low-pass circuit acts as an integrator provided the time constant of the circuit RC > 15T, where T is the period of the input signal. When RC > 15T, the input sinusoid will be shifted at least by 89.4 ⁰ when it is transmitted through the network. M.Balaji, Department of ECE, SVEC 37