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Kent Bertilsson Muhammad Amir Yousaf 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 Sinusoidal Waveform Unit of measurement for horizontal axis can be time , degrees or radians. Kent Bertilsson Muhammad Amir Yousaf Sinusoidal Waveform Unit of measurement for horizontal axis can be time , degrees or radians. 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 Kent Bertilsson Muhammad Amir Yousaf Sinusoidal Waveform Mathematically it is represented as: Kent Bertilsson Muhammad Amir Yousaf 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 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 fs Kent Bertilsson Muhammad Amir Yousaf 2 fs 3 fs 4 fs 5 fs f 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, fs=1/T And multiples of the fundamental f1,f2,f3,…=1xfs ,2xfs ,2xfs V V T=1/fs t fs Kent Bertilsson Muhammad Amir Yousaf 2 fs 3 fs 4 fs 5 fs f 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 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 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 Derivative of sinusoidal Kent Bertilsson Muhammad Amir Yousaf Response of R to Sinusoidal Voltage or Current Resistor at a particular frequency Kent Bertilsson Muhammad Amir Yousaf Response of L to Sinusoidal Voltage or Current Inductor at a particular frequency Kent Bertilsson Muhammad Amir Yousaf Response of C to Sinusoidal Voltage or Current Capacitor at a particular frequency Kent Bertilsson Muhammad Amir Yousaf Frequency Response of R,L,C How varying frequency affects the opposition offered by R,L and C Kent Bertilsson Muhammad Amir Yousaf Complex Numbers Real and Imaginary axis on complex plane Polar Form Rectangular Form Kent Bertilsson Muhammad Amir Yousaf Conversion between Forms Real and Imaginary axis on complex plane Kent Bertilsson Muhammad Amir Yousaf Phasors • The radius vector, having a constant magnitude (length) with one end fixed at the origin, is called a phasor when applied to electric circuits. Kent Bertilsson Muhammad Amir Yousaf R,L,C and Phasors How to determine phase changes in voltage and current in reactive circuits Kent Bertilsson Muhammad Amir Yousaf R,L,C and Phasors How to determine phase changes in voltage and current in reactive circuits Kent Bertilsson Muhammad Amir Yousaf 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. Circuits combining different types of elements will have total impedances that extend from 90° to -90° Kent Bertilsson Muhammad Amir Yousaf R,L,C in series Kent Bertilsson Muhammad Amir Yousaf Voltage Divide Rule Kent Bertilsson Muhammad Amir Yousaf Frequency response of R-C circuit Kent Bertilsson Muhammad Amir Yousaf 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 dB angle H f Kent Bertilsson Muhammad Amir Yousaf Decibel, dB decibel, dB is very useful measure to compare two levels of power. P Out APdB 10 log AP 10 log PIn V2 P V I R 2 VOut VOut POut R APdB 10 log 10 log 10 log 2 PIn VIn VIn 2 VOut 20 log VIn R VOut AVdB 20 log AV 20 log VIn It is used for expressing amplification (and attenuation) Kent Bertilsson Muhammad Amir Yousaf Bode Plot for High-Pass RC Filter Kent Bertilsson Muhammad Amir Yousaf Sketching Bode Plot for High-Pass RC Filter High-Pass R-C Filter Voltage gain of the system is: Av 1 / 1 j ( fc / f ) In magnitude and phase form Av Av 1 / 1( fc / f )^ 2 tan^1( fc / f ) AvdB AvdB 20 log 10 Av f 20 log 10 fc AvdB 0 Kent Bertilsson Muhammad Amir Yousaf For f << fc For fc << f 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 Asymptotic Bode Plot amplitude response Plotting eq below for higher frequencies: AvdB 20 log 10 For f= fc/10 AvdB For f= fc/4 AvdB For f= fc/2 AvdB For f= fc AvdB f fc = -20dB = -12dB = -6dB = 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. Kent Bertilsson Muhammad Amir Yousaf Actual Bode Plot Amplitude Response For actual plot using equation Av 20 log(1 / 1( fc / f )^ 2 ) For f >> fc , fc / f = 0 AvdB = 0dB For f = fc , fc / f = 01AvdB = -3dB Av 20 log(1 / 1( 0)^ 2 ) 20 log(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. Kent Bertilsson Muhammad Amir Yousaf 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 Kent Bertilsson Muhammad Amir Yousaf 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. Kent Bertilsson Muhammad Amir Yousaf Actual Bode Plot Phase Response At f = fc/10 tan^1( fc / f ) tan^1( fc / fc / 10) tan^1(10) 84.29 90 – 84.29 = 5.7 At f = 10fc tan^1( fc / 10 fc) 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 Kent Bertilsson Muhammad Amir Yousaf Bode Plot for RC low pass filter Kent Bertilsson Muhammad Amir Yousaf Draw an asymptotic bode diagram for the RC filter. Bode Plot for RC low pass filter Draw an asymptotic bode diagram for the RC filter. Zc VOut Zc Z R V In 1 / jwC 1 / jwC R 1 1 jwRC 1 1 j 2 fRC 1 1 jf / f c Kent Bertilsson Muhammad Amir Yousaf In terms of poles and Zeros: 1 1 jwRC S jw 1 1 sRC 1 1 s wc Pole at wc Bode diagram for multiple stage filter According to logarithmic laws Atot A1 A2 A3 Atot A1 A2 A3 dB dB dB dB angle Atot angle A1 angle A2 angle A3 Kent Bertilsson Muhammad Amir Yousaf Bode diagram for multiple stage filter Kent Bertilsson Muhammad Amir Yousaf Bode diagram for multiple stage filter Kent Bertilsson Muhammad Amir Yousaf Bode diagram Kent Bertilsson Muhammad Amir Yousaf Bode diagram Kent Bertilsson Muhammad Amir Yousaf Exercise Draw an asymptotic bode diagram for the shown filter. R VIn Kent Bertilsson Muhammad Amir Yousaf R2 C R3 VOut Amplifier • Voltage amplification PIN IIN IOut VIn VOut POut VOut AV VIn • Current amplification I Out AI I In • Power amplification POut AP PIn Kent Bertilsson Muhammad Amir Yousaf Amplifier model • The amplifier model is often sufficient describing how an amplifier interacts with the environment ROut VIn RIn AVVIn VOut • RIn – Input impedance • AV – Voltage gain • ROut – Output impedance Kent Bertilsson Muhammad Amir Yousaf Amplifier model Kent Bertilsson Muhammad Amir Yousaf Bandwidth The bandwidth is the frequency range where the transferred power are more than 50%. AP 0.5 AP max H(f) AVmax 0.707AVmax AV 2 AV max 0.707 AV max B f 2 f1 f1 Kent Bertilsson Muhammad Amir Yousaf f2 f Distortion A nonlinear function between UIn and UOut distorts the signal An amplifier that saturates at high voltages A diode that conducts only in the forward direction Kent Bertilsson Muhammad Amir Yousaf 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 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 References • Introductory Circuit Analysis By Boylestad Kent Bertilsson Muhammad Amir Yousaf