EE203-SUNYBuffalo-30-Chapter14-03

EE203-SUNYBuffalo-30-Chapter14-03 - SMALL for Big Things...

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Unformatted text preview: SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York EE 203 Circuit Analysis 2 Lecture 30 Chapter 14.3 High High-Pass Filters (continue…) Kwang W. Oh, Ph.D., Assistant Professor SMALL (nanobioSensors and MicroActuators Learning Lab) Department of Electrical Engineering University at Buffalo, The State University of New York 215E Bonner Hall, SUNY-Buffalo, Buffalo, NY 14260-1920 Tel: (716) 645-3115 Ext. 1149, Fax: (716) 645-3656 E-mail: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu Recall Lecture 5 Equivalent Circuits for C and L as a Function of Frequency 1 I jωC V= ∴Z = 1 jωC V = jωLI ∴ Z = jωL Z= 1 → ∞ (open) j (0)C Z = j (0) L → 0 (short ) Z= 1 → 0 (short ) j ( ∞)C Z = j ( ∞) L → ∞ (open) EE 203 Circuit Analysis 2 | Spring 2007 | Prof. Kwang W. Oh | EE@SUNY-Buffalo EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 1/13 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 13/17 Lecture 30 | Chapter 14 | 3/5 | 2/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Low-Pass VS High-Pass (a) A series RL Low-pass filter ∴Z = (b) at ω = 0 High-Pass (b) at ω = 0 (c) at ω =∞ VS (a) A series RC high-pass filter ∴ Z = jωL Low-Pass (c) at ω =∞ EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo 1 jωC Lecture 30 | Chapter 14 | 3/5 | 3/13 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 4/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Band-Pass Filter (1) EE 203 Circuit Analysis 2 Lecture 30 Chapter 14.4 Band-Pass Filters Kwang W. Oh, Ph.D., Assistant Professor SMALL (nanobioSensors and MicroActuators Learning Lab) Department of Electrical Engineering University at Buffalo, The State University of New York 215E Bonner Hall, SUNY-Buffalo, Buffalo, NY 14260-1920 Tel: (716) 645-3115 Ext. 1149, Fax: (716) 645-3656 E-mail: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 5/13 The The next filters we examine are those that pass voltages within a band of frequencies to the output while filtering out voltages at frequencies outside this band. These filters are somewhat more complicated than the low-pass and highpass filters of the previous sections. Ideal bandpass filters have two cutoff frequencies, ωc1 and ωc2, which identify the passband. For realistic bandpass filters, these cutoff frequencies are again defined as the frequencies for which the magnitude of the transfer function equals (1/√2)Hmax. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 6/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Band-Pass Filter (2) Series RLC Circuits – Qualitative Analysis At ω = 0 Center Frequency, ωo the frequency for which a circuit's transfer function is purely real. Another name for the center frequency is the resonant frequency. The center frequency is the geometric center of the passband, that is, ωo = ωc1ωc 2 For bandpass filters, the magnitude of the transfer function is a maximum at the center frequency (Hmax = |H(jωo)|) Bandwidth, β the width of the passband. Quality Factor, Q the ratio of the center frequency to the bandwidth. The quality factor gives a measure of the width of the passband, independent of independent its location on the frequency axis. It also describes the shape of the magnitude plot, independent of frequency. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 7/13 Capacitor behaves like an open circuit Inductor behaves like a short circuit At ω = ∞ Capacitor behaves like a short circuit Inductor behaves like an open circuit Between ω = 0 and ω = ∞ and Remember that Zcapacitor is negative, whereas Zinductor is positive. Th Thus, at some frequency (ωo), Zcapacitor and Zinductor have equal magnitudes and opposite signs. The two impedances cancel out, causing v0 = vi. On either side of ωo, the output voltage is less than the source voltage. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 8/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Series RLC Circuits – Quantitative Analysis (1) Series RLC Circuits – Quantitative Analysis (2) Transfer Function R ( R / L) s =2 sL + 1 / sC + R s + ( R / L) s + (1 / LC ) ω ( R / L) H ( jω ) = [(1 / LC ) − ω 2 ]2 + [ω ( R / L)]2 H (s) = R ( R / L) s = sL + 1 / sC + R s 2 + ( R / L) s + (1 / LC ) ω ( R / L) H ( jω ) = [(1 / LC ) − ω 2 ]2 + [ω ( R / L)]2 H ( s) = Center Frequency The frequency for which the circuit’s transfer function is purely real jω o L + 1 1 1 1 2 = 0 ⇒ jω o L − j = 0 ⇒ ωo L = ⇒ ωo = ⇒∴ ωo = jω o C ωo C ωo C LC 1 LC The frequency for which the circuit’s transfer function has the maximum H max = H ( jω0 ) = ω0 ( R / L ) [(1 / LC ) − ω0 ]2 + [ω0 ( R / L)]2 2 ∴[(1 / LC ) − ω0 ]2 = 0 ⇒ ω0 = 2 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 9/13 =1 1 LC EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 10/13 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Series RLC Circuits – Quantitative Analysis (3) Series RLC Circuits – Quantitative Analysis (4) Cutoff Frequencies Center Frequency & Cutoff Frequencies at the ωc (ω1 or ω2) 1 = [( 1 / LC ωc ( R / L) ωo = ωc1 ⋅ ωc 2 |H(jωc)|=(1/√2)Hmax ωc ( R / L) 1 1 H max = = = 2 2 2 [(1 / LC ) − ωc ]2 + [ωc ( R / L)]2 ωc2 + 1 ωc ( R / L ) 2 )− = 1 1 2 {[(1 / LC ) − ωc ]2 + ωc ( R / L)]2 } ωc 2 ( R / L ) 2 2 1 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo 2 1 ⎛R⎞ ⎛R⎞ = −⎜ ⎟ +⎜ ⎟ + = 2 L ⎠ ⎝ 2 L ⎠ CL ⎝ 1 L [ − ω c ]2 + 1 ωc RC R 1 L 2 ∴ − ωc = ±1 ⇒ ωc L m ωc R − 1 / C = 0 ωc RC R 2 R 1 ⎛R⎞ ωc > 0 ωc1 = − + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 2 R 1 ⎛R⎞ 2 + ⎜ ⎟+ ∴ ωc = m R 1 ⎛R⎞ 2L ⎝ 2 L ⎠ CL ωc 2 = + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 2 2 ⎡R 1 ⎤⎡ R R⎞ 1⎤ ⎛R⎞ ⎥ ⎥⎢ + ⎛ ⎟ + = ⎢− + ⎜ ⎟+ ⎜ ⎢ 2L ⎝ 2 L ⎠ CL ⎥ ⎢ 2 L ⎝ 2 L ⎠ CL ⎥ ⎣ ⎦ ⎦⎣ Bandwidth 2 ωc1 = − R 1 ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 2 ωc 2 = R 1 ⎛R⎞ + ⎜ ⎟+ 2L 2 L ⎠ CL ⎝ 1 LC β = ωc 2 − ωc1 2 2 ⎡R 1⎤ ⎡ R 1⎤ R ⎛R⎞ ⎛R⎞ ⎥ − ⎢− ⎥= =⎢ + ⎜ ⎟ + + ⎜ ⎟+ ⎢ 2L ⎝ 2 L ⎠ CL ⎥ ⎢ 2 L ⎝ 2 L ⎠ CL ⎥ L ⎣ ⎦⎣ ⎦ ax + bx + c = 0 2 2 ⇒ x= b ⎛b⎞ c ± ⎜ ⎟− 2a ⎝ 2a ⎠ a Lecture 30 | Chapter 14 | 3/5 | 11/13 Quality Factor Q= (1 / LC ) o L = = β R/L CR 2 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 30 | Chapter 14 | 3/5 | 12/13 SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York Series RLC Circuits – Quantitative Analysis (5) ωc1 = − 2 β ⎛β ⎞ 2 + ⎜ ⎟ + ωo 2 ⎝2⎠ 2 β ⎛β ⎞ ωc 2 = + ⎜ ⎟ + ωo 2 2 ⎝2⎠ ⎡ 1 + ⎢ 2Q ⎣ ωc1 = ωo ⎢− ⎡ 1 ωc 2 = ωo ⎢ + ⎢ 2Q ⎣ 2 ⎤ ⎛1⎞ ⎟ + 1⎥ ⎜ ⎟ ⎜ ⎥ ⎝ 2Q ⎠ ⎦ 2 ⎤ ⎛1⎞ ⎜ ⎟ + 1⎥ ⎜ 2Q ⎟ ⎥ ⎝ ⎠ ⎦ EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo 2 ωc1 = − R 1 ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 2 ωc 2 = ωo = β= Q= R 1 ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 1 LC R L L CR 2 Lecture 30 | Chapter 14 | 3/5 | 13/13 ...
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This note was uploaded on 10/19/2011 for the course EE 203 taught by Professor Staff during the Spring '08 term at SUNY Buffalo.

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