Kent Bertilsson Muhammad Amir Yousaf

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2 Today s topics Analog System (Rev) Frequency Domain Signals in Frequency domain Frequency analysis of signals and systems Transfer Function Basic elements: R, C, L Filters RC Filters jw method (Complex impedance method for AC circuits) Bode plot Amplifier

3 Block Diagram Electronic systems is often described by block diagram Antenna Amplifier Filter Analog to digital conversion

4 Signals 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

5 Signals 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 V T=1/f s t f s 2 fs 3 fs 4 fs 5 fs f

6 Time domain vs 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

7 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.

8 Frequency domain Frequency domain is a domain for analysis of mathematical functions and signals with respect to frequency. A time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.

9 Transfer function The transfer function is the relation between the amplitude for the output and input in the frequency domain. H f U U Out H(20kHz)=10 means that for a 20kHz signal the output is ten times larger than the input. H(f) is of course continuous function. in f f H f

10 Basic Elements and jω-method Any combination of passive (R, L, and C) and/or active (transistors or operational amplifiers) elements designed to select or reject a band of frequencies is called a filter. The jω-method is a very powerful tool making it possible performing advanced frequency dependent (alternating current, AC) functions using the same rules that applies for direct current (DC) How these elements behave with alternating voltage and current. Symbol Resistor Capacitor Inductor Reactance X R X 1 jc X jl

11 jω-method Impedance calculations can be performed in the same way as for normal resistances. L j R L Z R Z Z RC j R C j R C j R C Z R Z C Z R Z C Z R Z Z // R L R L

12 Filter Any combination of passive (R, L, and C) and/or active (transistors or operational amplifiers) elements designed to select or reject a band of frequencies is called a filter. In communication system, filters are used to pass frequencies containing the desired information and to reject the remaining frequencies. In stereo systems, filters can isolate particular bands of frequencies for increased or decreased emphasis by the output acoustical system. Filters are used to eliminate any unwanted frequencies, called noise, due to non linear characteristics of electronic deices or signal picked up from surrounding medium.

13 Filters Two classifications of the filters: Passive Filters (R,L,C) Active Filters (R,L,C, transistor, op-amps) Four categories of filters Low Pass Filters High Pass Filters Band Pass Filters Band Stop Filters

14 Filters Cut-off Frequency Pass Band Stop Band Any Frequency (in pass band) with at least 70.7% of max. output voltage will pass through the filter For stop band, the output voltage is 1/1000 of Vmax or -60dB

15 RC - filter Calculate the transfer function H(ω) using jw-method What is the output voltage level and power level at the cut-off frequency?

16 RC - filter Analysis: At low frequencies how capacitor behaves? Xc = 1/ jwc = 1/ 2 pi f C How capacitor behaves at high frequencies?

17 Normalized plot for filter Normalization is a process whereby a quantity such as voltage is divided by a quantity of the same unit of measure to establish a dimensionless level of specific value or range. A normalized plot in filter domain can be obtained by dividing the plotted quantity Vo with applied voltage Vi for the frequency of interest

18 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. H f angle db H f

19 Decibel, db decibel, db is very useful measure to compare two levels of power. A PdB P 10log A PdB Out P In A VdB V 2 PV I R V 2 Out 10log R V 2 In R 10log 20log V Out V In It is used for expressing amplification (and attenuation) A P A V 10log 10log 20log V Out V In P Out P In 2 V 20log Out V In

20 db A V A P

21 Bode Plot for High-Pass RC Filter

22 Sketching Bode Plot for High-Pass RC Filter High-Pass R-C Filter Voltage gain of the system is: A v 1 / 1 j( fc / f ) In magnitude and phase form A v Av 1 / 1( fc/ f )^ 2 tan^1( fc / f ) AvdB 20log10Av f For f << fc AvdB 20log 10 fc A change in frequency by a 2:1 ratio results in a 6dB change in gain. A change in frequency by a 10:1 ratio results in a 20dB change in gain.

23 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

24 Asymptotic Bode Plot amplitude response Plotting eq below for higher frequencies: AvdB 20log 10 f f c For f= fc/10 AvdB = -20dB For f= fc/4 AvdB = -12dB For f= fc/2 AvdB = -6dB For f= fc AvdB = 0dB 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.

25 Actual Bode Plot Amplitude Response For actual plot using equation Av 20log( 1 / 1( fc/ f )^ 2) For f >> fc, fc / f = 0 AvdB = -3dB Av 20log( 1 / 1(0)^ 2) 20log( 1 / 2) 3dB For f = 2fc AvdB = -1dB For f = 1/2fc AvdB = -7dB At f = fc the actual response curve is 3dB down from the idealized Bode plot, whereas at f=2fc and f = fc/2 the acutual response is 1dB down from the asymptotic response.

26 Asymptotic Bode Plot Phase Response Phase response can also be sketched using straight line asymptote by considering few critical points in frequency spectrum. tan^1( fc / f ) Plotting above equation For f << fc, phase aproaches 90 For f >> fc, phase aproches 0 At f = fc tan^-1 (1) = 45

27 Asymptotic Bode Plot Phase Response Must remember rules for sketching Bode Plots: An asymptote at theta = 90 for f << fc/10, an asymptote at theta = 0 for f >> 10fc and an asymptote from fc/10 to 10fc that passes through theta = 45 at f= fc.

28 Actual Bode Plot Phase Response At f = fc/10 tan^1( fc / f ) tan^1( fc / fc / 10) tan^1(10) = 5.7 At f = 10fc tan^1( fc / 10fc) tan^1( fc / fc * 10) tan^1(1 / 10) 5.7 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

29 Bode Plot for RC low pass filter Draw an asymptotic bode diagram for the RC filter.

30 Bode diagram for multiple stage filter According to logarithmic laws A tot A 1 A 2 A 3 A totdb A 1dB A 2dB A 3dB angle A angle A angle A angle A tot 1 2 3

31 Bode diagram for multiple stage filter

32 Bode diagram for multiple stage filter

33 Bode diagram Complicated expressions can be factorized into sub-expressions as Const Differentiator Zero 1 C j j j 1 j 0 Integrator Pole

34 Exercise Draw an asymptotic bode diagram for the shown filter. R R 2 C V In R 3 V Out

35 Amplifier I IN I Out Voltage amplification V AV V In P IN V In V Out P Out Current amplification A I I I Out Out In Power amplification A P P P Out In

36 Amplifier model The amplifier model is often sufficient describing how an amplifier interacts with the environment R Out V In R In A V V In V Out R In Input impedance A V Voltage gain R Out Output impedance

37 Bandwidth The bandwidth is the frequency range where the transferred power are more than 50%. H(f) A Vmax 0.707A Vmax A V A P 2A 0.5A V max B f 2 P max 0.707A f 1 V max f 1 f 2 f

38 Distortion 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

39 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

40 RC Filters in Mindi Design a RC filter in Mindi. Simulate output for diffrent frequencies Analyse the results. db Bode Plots

41 Changes in Timetable

42 References Introductory Circuit Analysis By Boylestad

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