1. Kent Bertilsson Muhammad Amir Yousaf

2. DC and AC Circuit analysis  Circuit analysis is the process of finding the voltages across, and the currents through, every component in the circuit.  For dc circuits the components are resistive as the capacitor and inductor show their total characteristics only with varying voltage or current.  Sinusoidal waveform is one form of alternating waveform where the amplitude alternates periodically between two peaks. Kent Bertilsson Muhammad Amir Yousaf

3. Sinusoidal Waveform  Unit of measurement for horizontal axis can be time, degrees or radians. Kent Bertilsson Muhammad Amir Yousaf

4. Sinusoidal Waveform  Unit of measurement for horizontal axis can be time, degrees or radians. Kent Bertilsson Muhammad Amir Yousaf Vertical projection of radius vector rotating in a uniform circular motion about a fixed point.  Angular Velocity  Time required to complete one revolution is T

5. Sinusoidal Waveform  Mathematically it is represented as: Kent Bertilsson Muhammad Amir Yousaf

6. Frequency of Sinusoidal  Every signal can be described both in the time domain and the frequency domain.  Frequency representation of sinusoidal signal is: Muhammad Amir Yousaf

7. A periodic signal in frequency domain  Every signal can be described both in the time domain and the frequency domain.  A periodic signal is always a sine or cosine or the sum of sines and cosines.  Frequency representation of periodic signal is: V f s 2 fs 3 fs 4 fs 5 fs f Kent Bertilsson Muhammad Amir Yousaf

8. A periodic signal in frequency domain  A periodic signal (in the time domain) can in the frequency domain be represented by:  A peak at the fundamental frequency for the signal, f s =1/T  And multiples of the fundamental f 1,f 2,f 3,…=1 x f s,2 x f s,2 x f s V T=1/f s t V f s 2 fs 3 fs 4 fs 5 fs f Kent Bertilsson Muhammad Amir Yousaf

9. Non periodic signal in frequency domain  A non periodic (varying) signal time domain is spread in the frequency domain.  A completely random signal (white noise) have a uniform frequency spectra V Noise f Kent Bertilsson Muhammad Amir Yousaf

10. Why Frequency Representation?  All frequencies are not treated in same way by nature and man-made systems.  In a rainbow, different parts of light spectrum are bent differently as they pass through a drop of water or a prism.  An electronic component or system also gives frequency dependent response. Kent Bertilsson Muhammad Amir Yousaf

11. Phase Relation  The maxima and the minima at pi/2,3pi/2 and 0,2pi can be shifted to some other angle. The expression in this case would be: Kent Bertilsson Muhammad Amir Yousaf

12. Derivative of sinusoidal Kent Bertilsson Muhammad Amir Yousaf

13. Response of R to Sinusoidal Voltage or Current  Resistor at a particular frequency Kent Bertilsson Muhammad Amir Yousaf

14. Response of L to Sinusoidal Voltage or Current  Inductor at a particular frequency Kent Bertilsson Muhammad Amir Yousaf

15. Response of C to Sinusoidal Voltage or Current  Capacitor at a particular frequency Kent Bertilsson Muhammad Amir Yousaf

16. Frequency Response of R,L,C  How varying frequency affects the opposition offered by R,L and C Kent Bertilsson Muhammad Amir Yousaf

17. Complex Numbers  Real and Imaginary axis on complex plane Kent Bertilsson Muhammad Amir Yousaf  Rectangular Form  Polar Form

18. Conversion between Forms  Real and Imaginary axis on complex plane Kent Bertilsson Muhammad Amir Yousaf

19. Phasors Kent Bertilsson Muhammad Amir Yousaf The radius vector, having a constant magnitude (length) with one end fixed at the origin, is called a phasor when applied to electric circuits.

20. R,L,C and Phasors  How to determine phase changes in voltage and current in reactive circuits Kent Bertilsson Muhammad Amir Yousaf

21. R,L,C and Phasors  How to determine phase changes in voltage and current in reactive circuits Kent Bertilsson Muhammad Amir Yousaf

22. Impedance Diagram  The resistance will always appear on the positive real axis, the inductive reactance on the positive imaginary axis, and the capacitive reactance on the negative imaginary axis. Kent Bertilsson Muhammad Amir Yousaf  Circuits combining different types of elements will have total impedances that extend from 90° to -90°

23. R,L,C in series Kent Bertilsson Muhammad Amir Yousaf

24. Voltage Divide Rule Kent Bertilsson Muhammad Amir Yousaf

25. Frequency response of R-C circuit Kent Bertilsson Muhammad Amir Yousaf

26. Bode Diagram It is a technique for sketching the frequency response of systems (i.e. filter, amplifiers etc) on dB scale. It provides an excellent way to compare decibel levels at different frequencies. Absolute decibel value and phase of the transfer function is plotted against a logarithmic frequency axis. Kent Bertilsson Muhammad Amir Yousaf

27. Decibel, dB  decibel, dB is very useful measure to compare two levels of power.  It is used for expressing amplification (and attenuation) Kent Bertilsson Muhammad Amir Yousaf

28. Bode Plot for High-Pass RC Filter Kent Bertilsson Muhammad Amir Yousaf

29. Sketching Bode Plot for High-Pass RC Filter  High-Pass R-C Filter Voltage gain of the system is: In magnitude and phase form For f << fc Kent Bertilsson Muhammad Amir Yousaf For fc << f

30. Bode Plot Amplitude Response  Must remember rules for sketching Bode Plots: Two frequencies separated by a 2:1 ratio are said to be an octave apart. For Bode plots, a change in frequency by one octave will result in a 6dB change in gain. Two frequencies separated by a 10:1 ratio are said to be a decade apart. For Bode plots, a change in frequency by one decade will result in a 20dB change in gain. True only for f << fc Kent Bertilsson Muhammad Amir Yousaf

31. Asymptotic Bode Plot amplitude response  Plotting eq below for higher frequencies:  For f= f c /10 A vdB = -20 dB  For f= f c /4 A vdB = -12 dB  For f= f c /2 A vdB = -6 dB  For f= f c A vdB = 0 dB  This gives an idealized bode plot.  Through the use of straight-line segments called idealized Bode plots, the frequency response of a system can be found efficiently and accurately. Kent Bertilsson Muhammad Amir Yousaf

32. Actual Bode Plot Amplitude Response  For actual plot using equation  For f >> f c, f c / f = 0 A vdB = 0 dB  For f = fc, fc / f = 01AvdB = -3dB  For f = 2f c A vdB = -1 dB  For f = 1/2f c A vdB = -7dB  At f = f c the actual response curve is 3 dB down from the idealized Bode plot, whereas at f=2f c and f = f c /2 the acutual response is 1 dB down from the asymptotic response. Kent Bertilsson Muhammad Amir Yousaf

33. Asymptotic Bode Plot Phase Response  Phase response can also be sketched using straight line asymptote by considering few critical points in frequency spectrum.  Plotting above equation  For f << f c, phase aproaches 90  For f >> f c, phase aproches 0  At f = f c tan^-1 (1) = 45 Kent Bertilsson Muhammad Amir Yousaf

34. Asymptotic Bode Plot Phase Response  Must remember rules for sketching Bode Plots: An asymptote at theta = 90 for f > 10fc and an asymptote from fc/10 to 10fc that passes through theta = 45 at f= fc. Kent Bertilsson Muhammad Amir Yousaf

35. Actual Bode Plot Phase Response  At f = f c /10  90 – 84.29 = 5.7  At f = 10f c  At f= fc theta = 45 whereas at f=fc/10 and f=10fc, the difference the actual and asymptotic phase response is 5.7 degrees Kent Bertilsson Muhammad Amir Yousaf

36. Bode Plot for RC low pass filter  Draw an asymptotic bode diagram for the RC filter. Kent Bertilsson Muhammad Amir Yousaf

37. Bode Plot for RC low pass filter  Draw an asymptotic bode diagram for the RC filter. Kent Bertilsson Muhammad Amir Yousaf In terms of poles and Zeros: Pole at w c

38. Bode diagram for multiple stage filter  According to logarithmic laws Kent Bertilsson Muhammad Amir Yousaf

39. Bode diagram for multiple stage filter Kent Bertilsson Muhammad Amir Yousaf

40. Bode diagram for multiple stage filter Kent Bertilsson Muhammad Amir Yousaf

41. Bode diagram Kent Bertilsson Muhammad Amir Yousaf

42. Bode diagram Kent Bertilsson Muhammad Amir Yousaf

43. Exercise R R 2 C V In R 3 V Out  Draw an asymptotic bode diagram for the shown filter. Kent Bertilsson Muhammad Amir Yousaf

44. Amplifier Voltage amplification Current amplification Power amplification I IN I Out P IN V In V Out P Out Kent Bertilsson Muhammad Amir Yousaf

45. Amplifier model R In – Input impedance A V – Voltage gain R Out – Output impedance R Out V In R In A V V In V Out The amplifier model is often sufficient describing how an amplifier interacts with the environment Kent Bertilsson Muhammad Amir Yousaf

46. Amplifier model Kent Bertilsson Muhammad Amir Yousaf

47. H(f) A Vmax 0.707A Vmax f 1 f 2 f Bandwidth  The bandwidth is the frequency range where the transferred power are more than 50%. Kent Bertilsson Muhammad Amir Yousaf

48.  A nonlinear function between U In and U Out distorts the signal  An amplifier that saturates at high voltages  A diode that conducts only in the forward direction Distortion Kent Bertilsson Muhammad Amir Yousaf

49. Noise Random fluctuation in the signal Theoretically random noise contains all possible frequencies from DC to infinity Practical noise is often frequency limited to an upper bandwidth by some filter A limited bandwidth from the noisy reduce the noise power Kent Bertilsson Muhammad Amir Yousaf

50. RC Filters in Mindi  Design a RC filter in Mindi.  Simulate output for diffrent frequencies  Analyse the results.  dB  Bode Plots Kent Bertilsson Muhammad Amir Yousaf

51. References Introductory Circuit Analysis By Boylestad Kent Bertilsson Muhammad Amir Yousaf