1. 1 Eeng 224 Chapter 14 Frequency Response Huseyin Bilgekul Eeng 224 Circuit Theory II Department of Electrical and Electronic Engineering Eastern Mediterranean University Chapter Objectives:  Understand the Concept of Transfer Functions.  Be Familiar with the Decibel Scale.  Learn how to make Bode Magnitude and Phase plots.  Learn about series and parallel resonant RLC circuits.  Know Different Types of Passive and Active Filters and their Characteristics.  Understand the use of scaling in circuit analysis.  Be Able to use PSpice to obtain frequency response.  Apply what is learnt to radio receiver and touch-tone telephone.

2. 2 Eeng 224 DECIBEL SCALE  The DECIBEL provides us with a unit of less magnitude.

3. 3 Eeng 224 DECIBEL SCALE Magnitude HDecibel Value H dB 0.001-60 0.01-40 0.1-20 0.5-6 1/√2-3 10 √23 26 1020 26 10040  The DECIBEL value is a logarithmic measurement of the ratio of one variable to another of the same type.  Decibel value has no dimension.  It is used for voltage, current and power gains.

4. 4 Eeng 224 Typical Sound Levels and Their Decibel Levels.

5. 5 Eeng 224 SEMILOG SCALE

6. 6 Eeng 224 BODE PLOTS  Bode plots are APPROXIMATE semilog plots of the magnitude (in Decibels) and phase (in degrees) of a transfer function versus frequency. They are much easier to plot. High pass filter circuit magitude response. Bode plots are APPROXIMATE plots of the magnitude and phase responses.

7. 7 Eeng 224 BODE PLOTS Actual Response BODE PLOT

8. 8 Eeng 224 BODE PLOTS Bode plot and actual magnitude and phase responses of the RC high pass filter circuit.

9. 9 Eeng 224 BODE PLOTS  Bode plots are semilog plots of the magnitude (in Decibels) and phase (in degrees) of a transfer function versus frequency. Bode plots carry same information. They are much easier to plot. Logarithmic axis Decibel values Total Magnitude Response

10. 10 Eeng 224 BODE PLOTS  This representation is called the STANDARD FORM. It has several different factors. We can draw the Bode plots by plotting each of the terms of the transfer function separately and then adding them. The different factors of the transfer function are 1.) Gain term K. 2.) A pole (j  ) -1 or a zero (j  ) at the origin. 3.) A simple pole or zero 4.) A quadratic pole or zero  A transfer function may be written in terms of factors that have real and imaginary parts.

11. 11 Eeng 224 BODE PLOTS, The Decade  A DECADE is an interval between 2 frequencies with a ratio of 10 (between 10 Hz and 100 Hz or between 500 Hz and 5000 Hz). 20 dB/decade means that magnitude changes 20 dB whenever the frequency changes tenfold or one decade. The DC value (ω→ 0) does not appear on Bode plots ( Log0 = - ∞).  Slopes are expressed in dB/decade. One Decade 20 dB

12. 12 Eeng 224 BODE PLOTS  To plot the Bode plots of a given transfer function. 1.) Put the transfer function in STANDARD FORM. 2.) Write the Magnitude and phase equations from the STANDARD FORM. 3.) Plot the magnitude of each term separately. 4.) Add all magnitude terms to obtain the magnitude transfer function. 5.) Repeat 2-4 for the phase response. 6.) The total magnitude response in Decibel units is the summation and subtraction of the responses of different terms. 7.) The total phase response in degrees is the summation and subtraction of the phase responses of different terms.

13. 13 Eeng 224 BODE PLOTS  We examine how to plot different terms that may appear in a transfer function. The total response will be obtained by adding all the responses. CONSTANT TERM

14. 14 Eeng 224 BODE PLOTS ZERO AT THE ORIGIN POLE AT THE ORIGIN -20 -90 

15. 15 Eeng 224 SIMPLE ZERO  Approximate the magnitude response of a simple zero by two linear curves before and after ω=z 1 Approximate the phase response of a simple zero by three linear curves before ω=0.1z 1 after ω=10z 1 and between ω=0.1z 1 and ω=10z 1 CORNER FREQUENCY BREAK FREQUENCY 3 dB FREQUENCY

16. 16 Eeng 224 SIMPLE POLE -20 -90   Approximate the magnitude response of a simple pole by two linear curves before and after ω=p 1 Approximate the phase response of a simple pole by three linear curves before ω=0.1p 1 after ω=10p 1 and between ω=0.1p 1 and ω=10p 1  NOTICE the pole and zero responses are in opposite directions. CORNER FREQUENCY

17. 17 Eeng 224 EXACT RESPONSE OF QUADRATIC POLE BODE PLOTS

18. 18 Eeng 224 BODE PLOT OF QUADRATIC ZERO  The EXACT responses can be approximated by BODE plots in terms of the corner frequency ω n

19. 19 Eeng 224 BODE PLOT OF QUADRATIC POLE

20. 20 Eeng 224 Express transfer function in Standard form. Find the Pole and Zero frequencies. Express the magnitude and phase responses. Sketch each term of the magnitude and phase responses. Add each term of magnitude response to find total magnitude response. Add each term of phase response to find total phase response. Another Procedure Zeros cause an increase and poles cause a decrease in the slope. Start with the lowest frequency of the Bode plot. Move along the frequency axis and increase or decrease the slope at each corner frequency. Repeat the same procedure for both the magnitude and the phase. Procedure for Bode plot Construction

21. 21 Eeng 224 Express transfer function in Standard form. Express the magnitude and phase responses. Two corner frequencies at ω=2, 10 and a zero at the origin ω=0. Sketch each term and add to find the total response. EXAMPLE 14.3 Construct Bode plots for

22. 22 Eeng 224 EXAMPLE 14.3 Construct Bode plots for 26 dB XX XX

23. 23 Eeng 224 EXAMPLE 14.3 Continued: Let us calculate  H  and  at  =50 rad/sec graphically. 26 dB  =50

24. 24 Eeng 224 EXAMPLE 14.4 Construct Bode plots for Express transfer function in Standard form. Express the magnitude and phase responses. Two corner frequencies at ω=5, 10 and a zero at ω=10. The pole at ω=5 is a double pole. The slope of the magnitude is -40 dB/decade and phase has slope -90 degree/decade. Sketch each term and add to find the total response.

25. 25 Eeng 224 EXAMPLE 14.4 Construct Bode plots for XO XO

26. 26 Eeng 224 PRACTICE PROBLEM 14.6 Obtain the transfer function for the Bode plot given.

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28. 28 Eeng 224