1. 1 Electrical Engineering BA (B), Analog Electronics, Lecture 2 ET065G 6 Credits ET064G 7.5 Credits Muhammad Amir Yousaf

2. Frequency Response of R,L,C  How varying frequency affects the opposition offered by R,L and C Muhammad Amir Yousaf 2

3. Impedance Diagram Muhammad Amir Yousaf 3

4. Impedance Diagram  The resistance appears on the positive real axis, the inductive reactance on the positive imaginary axis, and the capacitive reactance on the negative imaginary axis. Muhammad Amir Yousaf  Circuits combining different types of elements will have total impedances that extend from 90° to -90° 4

5. AC Circuit Analysis Muhammad Amir Yousaf 5

6. Complex Numbers A complex number represents a point in a two-dimensional plane located with reference to two distinct axes. This point can also determine a radius vector drawn from the origin to the point. Muhammad Amir Yousaf 6

7. Complex Numbers  Rectangular and Polar forms Muhammad Amir Yousaf  Rectangular Form  Polar Form 7

8. Conversion between Forms Muhammad Amir Yousaf 8

9. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS Muhammad Amir Yousaf 9

10. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS Muhammad Amir Yousaf Complex Conjugate simply changing the sign of the imaginary part 10

11. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS Muhammad Amir Yousaf Reciprocal 11

12. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS Muhammad Amir Yousaf AdditionSubtraction Addition or subtraction cannot be performed in polar form unless the complex numbers have the same angle u or unless they differ only by multiples of 180°. 12

13. MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS Muhammad Amir Yousaf MultiplicationDivision 13

14. Phasors 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. It should be pointed out that in phasor notation, the sine wave is always the reference, and the frequency is not represented. 14

15. Phasors Muhammad Amir Yousaf Phasor algebra for sinusoidal quantities is applicable only for waveforms having the same frequency. 15

16. R,L,C in series Muhammad Amir Yousaf 16

17. Voltage Divide Rule Muhammad Amir Yousaf 17

18. Frequency response of series R-C circuit Muhammad Amir Yousaf 18

19. 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. Muhammad Amir Yousaf 19

20. Decibel, dB  decibel, dB is very useful measure to compare two levels of power.  It is used for expressing amplification (and attenuation) Muhammad Amir Yousaf 20

21. Bode Plot for a RC Circuit Muhammad Amir Yousaf 21

22. Bode Plot for a RC Circuit Muhammad Amir Yousaf This gives an idealized bode plot. 22

23. Bode Plot for a RC Circuit Muhammad Amir Yousaf Note that as the frequency of interest approaches f c, the dB gain becomes less negative and approaches the final normalized value of 0 dB. The resulting plot is a straight line intersecting the 0 dB line at fc. It increases to the right at a rate of 6 dB per octave or 20 dB per decade. At higher frequencies: 23

24. Bode Plot for a RC Circuit Muhammad Amir Yousaf The phase response can also be sketched using straight-line asymptotes by considering a few critical points in the frequency spectrum. An asymptote at theta = 90 for f > 10fc and an asymptote from fc/10 to 10fc that passes through theta = 45 at f= fc. 24

25. Bode diagram for multiple stage filter  According to logarithmic laws Muhammad Amir Yousaf 25

26. Bode diagram for multiple stage filter Muhammad Amir Yousaf 26

27. Bode diagram for multiple stage circuit Muhammad Amir Yousaf 27

28. Bode diagram Muhammad Amir Yousaf 28

29. Bode diagram Muhammad Amir Yousaf 29

30. Exercise Muhammad Amir Yousaf 30

31. Exercise Muhammad Amir Yousaf Draw a detailed asymptotic bode-diagram for a system’s gain. Both the amplitude and phase should be clearly visualized. 31

32. Exercise  Derive to get Bode plot format equation for the system shown in the figure Muhammad Amir Yousaf Z1Z1 Z2Z2 Gain = -Z 2 /Z 1 32

33. Thank You Muhammad Amir Yousaf 33

34. References Introductory Circuit Analysis By Boylestad Muhammad Amir Yousaf 34