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## Chapter 6 Frequency Response & Systems Concepts

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**Chapter 6**Frequency Response & Systems Concepts • AC circuit analysis methods to study the frequency response of electrical circuits • Understanding of frequency response aided by the concepts of phasors and impedance. • Filtering – a new concept will be explored Këpuska 2005**Objectives**• Understand significance of frequency domain analysis • Introduction of Fourier series as a tool for computation of Fourier spectrum. • Analyze first and second-order electrical filters by determining their filtering properties. • Computation of frequency response and its graphical representation as Bode plot. Këpuska 2005**Sinusoidal Frequency Response**• Provides a circuit response to a sinusoidal input of arbitrary frequency. • The frequency response of a circuit is a measure of voltage or current (magnitude and phase) as a function of the frequency of excitation (source) signal. Këpuska 2005**Methods to Compute Frequency Response**• Thevenin equivalent source circuit: + Zs Z1 ZT VL ~ ~ VT VT Z2 - Këpuska 2005**Load Voltage VT**ZT ZL ~ VT Këpuska 2005**Frequency Response**• From definition: • VL(j) is a phase-shifted and amplitude-scaled version of VS(j) ⇨ Këpuska 2005**Frequency Response (cont)**• Phasor form of the load voltage: Këpuska 2005**Example 6.1**• Compute the frequency response Hv(j) of the circuit for R1= 1k, C=10F; and RL= 10k. Këpuska 2005**Magnitude & Phaze**Këpuska 2005**Fourier Analysis**• Let x(t) be a periodic signal with period T. • x(t) = x(t+nT) for n=1,2,3,… Këpuska 2005**Fourier Series**• A signal x(t) can be expressed as an infinite summation of sinusoidal components know as Fourier Series: • Sine-cosine (quadrature) representation • Magnitude and Phase form: • Fundamental Frequency and Period T: Këpuska 2005**Fourier Series**• It can be shown • Or similarly Këpuska 2005**Fourier Series Aproximation**• Infinite summation practically not possible • Replaced by finite summation that leads to approximation. • Higher order coefficients; n, are associated with higher frequencies; (2/T)n. ⇒ • Better approximations require larger bandwidths. Këpuska 2005**Odd and Even Functions**Fourier Series Këpuska 2005**Frequency Spectrum**Këpuska 2005**Computation of Fourier Series Coefficients**Këpuska 2005**Example of Fourier Series Approximation**• Square wave and its representation by a Fourier series. (a) Square wave (even function); (b) first three terms; (c) sum of first three terms Këpuska 2005**Example 6.3 Computation of Fourier Series Coefficients**• Problem: Compute the complete Fourier Spectrum of the sawtooth function shown in the Figure below for T=1 and A=1: Këpuska 2005**Solution**• x(t) is an odd function. • Evaluate the integral in equation Këpuska 2005**Solution (cont)**• Spectrum computation: Këpuska 2005**Matlab Simulation**• Components of the sawtooth wave function: Këpuska 2005**Matlab Simulation**• Fourier Series approximation of sawtooth wave function Këpuska 2005**Example 6.4**• Problem: Compute the complete Fourier series expansion of the pulse waveform shown in the Figure for /T=0.2 • Plot the spectrum of the signal Këpuska 2005**Solution**• Expression for x(t) • Evaluate Integral Equations: Këpuska 2005**Solution (cont)**Këpuska 2005**Spectrum Computation**• Magnitude: • Phase: Këpuska 2005**Graphical Representation**Këpuska 2005**Matlab Simulation**Këpuska 2005**Matlab Simulation**Këpuska 2005**Linear Systems Response to Periodic Inputs**• Any periodic signal x(t) can be represented as a sum of finite number of pure periodic terms: Këpuska 2005**General Input-Output Representation of a System**Këpuska 2005**Linear Systems**• For Linear Systems - by definition Principle of superposition applies: T{ax1(t) + bx2(t)} = aT{x1(t)} + bT{x2(t)} a x1 ax1[n] + bx2[n] T{} y= T{ax1(t) + bx2(t)} x2 b a aT{x1[n]} T{} x1 y= aT{x1(n)}+bT{x2(n)} T{} x2 bT{x2[n]} b Këpuska 2005**Linear System View of a Circuit**• Output of a circuit y(t) as a function of the input x(t): Këpuska 2005**Example 6.6 Response of Linear System to Periodic Input**• Problem: • Linear system: • Input: sawtooth waveform approximated with only first two Fourier components of the input waveform. Këpuska 2005**Solution**• Approximation of the sawtooth function with first two terms of Fourier Series: • Spectrum Computation: Këpuska 2005**Frequency Response**• Magnitude and Phase • Computation of Frequency Response for two frequency values of 1 = 8 and 2 = 16: Këpuska 2005**Frequency Response (cont)**• Computation of steady-state periodic output of the system: Këpuska 2005**Matlab Simulation**Këpuska 2005**Matlab Simulation**Këpuska 2005**Filters**• Low-Pass Filters Simple RC Filter Këpuska 2005**Low-Pass Filter**Këpuska 2005**Low-Pass Filter**• =0 • H(j)=1 ⇨ Vo(j)=Vi(j) • >0 Këpuska 2005**Low-Pass Filter**Cutoff Frequency Këpuska 2005**Example 6.7**• Compute the response of the RC filter to sinusoidal inputs at the frequencies of 60 and 10,000 Hz. • R=1k, C=0.47F, vi(t)=5cos(t) V • 0=1/RC=2,128 rad/sec • = 120 rad/sec ⇒ /0 = 0.177 • = 20,000 rad/sec ⇒ /0 = 29.5 Këpuska 2005**Solution**Këpuska 2005**High-Pass Filters**Këpuska 2005**High-Pass Filter**• The expression in previous slide can be written in magnitude-and-phase form: Këpuska 2005**High-Pass Filter Response**Këpuska 2005**Band-Pass Filters**Këpuska 2005**Frequency Response of Band-Pass Filter**Këpuska 2005