EECE310_Chapter_7_Lecture_Notes__F05_

EECE310_Chapter_7_Lecture_Notes__F05_ - College of...

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Unformatted text preview: College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7 Network Frequency Characteristics EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics This chapter explores the frequency characteristics of an electrical network. Particularly how the frequencies at the source affect the circuit. The frequency response (i.e., the sinusoidal frequency response) defines the behavior of the network as a function of the frequency. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Elucidating the frequency response of a electrical circuit will provide a better understanding of its performance. Electrical networks are designed to operate at a single frequency. In general, schematics will respond differently to sinusoidal signals of various frequencies. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Achieving the proper frequency response in certain circuits is extremely important in the circuit's functionality. To examine the sinusoidal frequency response in a network, we need to examine the ratio of the output to the input (ergo the system's transfer function). EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics A network's transfer function, H(j), may be expressed as the ratio of two polynomials, which both are a function of j: N ( j ) a m ( j ) m + a m - 1 ( j ) m - 1 + K + a1 ( j ) 1 + a 0 H ( j ) = = D ( j ) bn ( j ) n + bn -1 ( j ) n -1 + K + b1 ( j )1 + b0 N(j) represents the numerator polynomial (output) of degree m. M(j) represents the denominator polynomial (input) of degree n. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The transfer function, H(j), may be re-written as: ( j - z1 )( j - z 2 ) K ( j - z m ) H ( j ) = K 0 ( j - p1 )( j - p 2 ) K ( j - p n ) K0 represents a constant coefficient. z1, ..., zm are the roots of N(j) and are formally called the zeros of the function, H(j). p1, ..., pm are the roots of D(j) and are formally called the poles of the function, H(j). EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The transfer function, H(j) (continued): ( j - z1 )( j - z 2 ) K ( j - z m ) H ( j ) = K 0 ( j - p1 )( j - p 2 ) K ( j - p n ) When examining a resistor-inductor-capacitor (RLC) circuit, the values of the poles of the function, H(j), determine the type of response. The zeros and/or poles of the function, H(j), may be complex. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Often to achieve a specific frequency response, it is necessary to discard other responses. This can be accomplished by the use of electrical circuits known as filters. Filters are two-port networks. When a signal is applied to the input port, then the filter only allows a certain range of frequencies to appear at the output port. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics There are four (4) basic filters to be examined: 1) Low Pass Filter (LPF) designed to pass low frequencies and reject high frequencies. 2) High Pass Filter (HPF) designed to pass high frequencies and reject low frequencies. 3) Band-Pass Filter (BPF) designed to pass a particular range of frequencies and reject all frequencies outside that range. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The four basic filters to be examined (continued): 4) Band-Rejection Filter (BRF) designed to reject a specific range of frequencies and pass all other frequencies. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Illustrations of the four examined filters: LPF: BPF: Chapter 7-Network Frequency Characteristics HPF: BRF: EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Important Note about the Four Profiled Filters (LPF, HPF, BPF, and BRF): The overall configuration of the circuit's real (R) and reactive (jX) elements establishes the filter's capability, not any of the passive circuit elements' values. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Determine the voltage transfer function H(j) for a r r r LPF: VIN VIN IT r = ; IT = 1 Z eq j C R+ j C r r r VIN V0 1 1 V0 = ; r ( j ) = j C R + 1 j RC + 1 VIN j C EECE 310 Preston D. Frazier, Ph.D., P.E., PMP r V0 = College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The voltage transfer function, H(j), for a LPF may be expressed in polar form, H(j) = M()exp(j()): 2 2 1 +0 1 M ( ) = H V ( j ) = = 2 2 [1 + ( ) 2 ] 0 .5 1 + ( ) -1 0 tan ( ) 1 = 0 - tan -1 ( ) = - tan -1 ( ) ( ) = -1 tan ( ) 1 EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Determine the voltage transfer function H(j) for a r r HPF: r r V0 R j RC R ; r ( j ) = V0 = = j RC + 1 1 j RC + 1 VIN R+ j C j C EECE 310 Preston D. Frazier, Ph.D., P.E., PMP r r r VIN VIN = V0 = IT R ; IT = 1 Z eq R+ j C r VIN College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The voltage transfer function, H(j), for a HPF may be expressed in polar form, H(j) = M()exp(j()): 0 2 + ( ) 2 ( ) M ( ) = H V ( j ) = = 2 0 .5 2 2 [1 + ( ) ] 1 + ( ) -1 tan ( ) 0 = 90 - tan -1 ( ) ( ) = -1 tan ( ) 1 EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics An important frequency to analysis for the LPF and the HPF is the break frequency (also known as the half-power frequency). In general, the break frequency occurs when = -1. At the half-power frequency, the magnitude of the voltage or current is 70.7% of its maximum value. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The magnitude of the voltage and the phase angle of the voltage at the break frequency ( = 1/) for a LPF are: -1 0 tan ( ) 1 = 0 - tan -1 (1) = - 45 ( ) = -1 1 tan ( ) 1 EECE 310 Preston D. Frazier, Ph.D., P.E., PMP M ( = 1 ) = H ( j ) = 12 + 0 2 1 = 2 12 + 12 College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The magnitude of the voltage and the phase angle of the voltage at the break frequency ( = 1/) for a HPF are: -1 1 tan ( ) 0 = 90 - tan -1 (1) = 45 ( ) = -1 1 tan ( ) 1 EECE 310 Preston D. Frazier, Ph.D., P.E., PMP M ( = 1 ) = H ( j ) = 1 = 2 12 + 12 0 2 + 12 College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics When studying BPFs or BRFs, there are four (4) frequencies, which must be accounted. 1) Center Frequency (0) where the maximum (BPF) or minimum (BRF) amplitude occurs. 2) Lower-Break Frequency (LO) the lower frequency where the amplitude is 70.7% of its peak value. 3) Upper-Break Frequency (HI) the upper frequency where the amplitude is 70.7% of its peak value. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The four frequencies for the BPF and the BRF (continued): 4) Bandwidth (BW) the width of the pass or rejection band (i.e., BW = HI - LO). Bandwidth may be measured in Hertz (f) or radians/second (). EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Another frequency response to elucidate is when the frequency is at resonance. Resonance is a phenomenon which exists in a wide variety of engineering systems. For electric networks at Resonance when a sinusoidal source of the proper frequency is applied, then voltages much larger than the source voltage can appear in the circuit. Let's examine RLC circuits at resonance. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics To analyze the phenomenon of resonance, look at the total impedance for the two basic RLC configurations. r Series: Z( j ) = R + 1 r 1 Parallel: Y ( j ) = G + j C + (**) j L Note: Y = 1/Z and G = 1/R EECE 310 Preston D. Frazier, Ph.D., P.E., PMP j C + j L (*) College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The imaginary terms in both (*) and (**) will equal to zero if: 1 L = C Therefore either circuit configuration will be at resonance if the resonant frequency (0) is: 0 = 1 LC EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics At resonance, the voltage and current are in phase, and therefore the phase angle is zero and the power factor is at unity (PF = 1). In a series RLC circuit, the current plays a significant role because it is identical in all three elements. In a parallel RLC schematic, the same holds true for the voltage. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Important Note about Series RLC Resonant Circuits: The impedance is at minimum and the current is at maximum. At low frequencies, the system's impedance is dominated by the capacitive term. At high frequencies, the inductive term subjugates the network. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Important Note about Parallel RLC Resonant Circuits: The admittance is at minimum and the voltage is at maximum. At low frequencies, the admittance of the system is dictated by the inductive term. At high frequencies, the admittance of the network is commanded by the capacitive term. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics There are three important determinants in the study of resonant circuits. Two factors have already been discussed resonant frequency (0) and bandwidth (BW). The third is called the quality factor (Q). The quality factor for a series RLC network is defined to be the ratio of the reactance of the inductance at 0 to the resistance. Q= 0L R = ( 0 CR ) -1 EECE 310 Preston D. Frazier, Ph.D., P.E., PMP L = R C College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics The quality factor for a parallel RLC network is defined to be the ratio of the resistance to the reactance of the inductance at 0. The quality factor (for a series or parallel RLC 0 networks) is the quotient of the resonant Q= frequency and the bandwidth. BW EECE 310 Preston D. Frazier, Ph.D., P.E., PMP R R C Q= = 0 CR = 0L L College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 7-Network Frequency Characteristics Important Note about the Quality Factor: The frequency selectivity of the system is determined by the value of Q. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP ...
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