EE203-SUNYBuffalo-31-Chapter14-04

EE203-SUNYBuffalo-31-Chapter14-04 - 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 Series RLC Circuits – Quantitative Analysis (5) EE 203 Circuit Analysis 2 Lecture 31 Chapter 14.4 Band-Pass Filters (continue…) ωc1 = − Lecture 31 | Chapter 14 | 4/5 | 1/12 ⎛β ⎞ 2 + ⎜ ⎟ + ωo 2 ⎝2⎠ 2 ωc1 = − ⎛β ⎞ ωc 2 = + ⎜ ⎟ + ωo 2 2 ⎝2⎠ ⎡ 1 + ⎢ 2Q ⎣ ωc1 = ωo ⎢− ⎡ 1 ωc 2 = ωo ⎢ + ⎢ 2Q ⎣ R 1 ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 2 2 β 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 2 β ωc 2 = R 1 ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 1 LC ωo = 2 ⎤ ⎛1⎞ ⎟ + 1⎥ ⎜ ⎟ ⎜ ⎥ ⎝ 2Q ⎠ ⎦ β= 2 ⎤ ⎛1⎞ ⎜ ⎟ + 1⎥ ⎜ 2Q ⎟ ⎥ ⎝ ⎠ ⎦ R L L CR 2 Q= EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 31 | Chapter 14 | 4/5 | 2/12 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 Example 14.5 Designing a Bandpass Filter (2) Example 14.5 Designing a Bandpass Filter (1) Q: Q: A graphic equalizer is an audio amplifier that allows you to select different levels of amplification within different frequency regions. Using the series RLC circuit in Fig. 14.19(a), choose values for R, L, and C that yield a bandpass circuit able to select inputs within the 1-10 kHz frequency band. Such a circuit might be used in a graphic equalizer to select this frequency band from the larger audio band (generally 0–20 kHz) prior to amplification. Choose C = 1 uF. Solution f c1 = 1000 Hz, f c 2 = 10000 Hz ω = 2πf , ωo = ωc1ωc 2 ⇒ f o = ωo = Q= Q= f c1 f c 2 = (1000)(10000) = 3162.28 Hz 1 1 1 ⇒L= 2 = = 2.533 mH LC ωo C (2π 3162.28) 2 (10 −6 ) ωo ωo fo 3162.28 = = = = 0.3514 β ωc 2 − ωc1 f c 2 − f c1 10000 − 1000 0.0025 L L ⇒R= = = 143.24 Ω 2 2 −6 CR CQ (10 )(0.3514) 2 We need to compute values for R, L, and C that produce a bandpass filter with cutoff frequencies of 1 kHz and 10 kHz. cutoff frequencies of and EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 31 | Chapter 14 | 4/5 | 3/12 ωo = Q= f c1 f c 2 = (1000)(10000) = 3162.28 Hz 28 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo R 1 ⎛R⎞ + ⎜ ⎟+ 2L 2 L ⎠ CL ⎝ 2 ωc 2 = β= f c1 = 1000 Hz, f c 2 = 10000 Hz ω = 2πf , ωo = ωc1ωc 2 ⇒ f o = 2 ωc1 = − R 1 ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 1 LC R L L CR 2 Lecture 31 | Chapter 14 | 4/5 | 4/12 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 Loaded RLC Bandpass Filter Two RLC Bandpass Filters 2 ωc1 = − R 1 ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL H (s) = 2 ωc 2 = ωo = β= 1 R ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL 1 LC 2 ωc1 = − R L ωo = β= Q= Lecture 31 | Chapter 14 | 4/5 | 5/12 1 1 ⎛1⎞ +⎜ ⎟+ 2 RC ⎝ 2 RC ⎠ CL 2 ωc 2 = L Q= CR 2 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo βs 2 s + β s + ωo 2 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo 1 1 ⎛1⎞ +⎜ ⎟+ 2 RC 2 RC ⎠ CL ⎝ 1 LC 1 RC R 2C L Lecture 31 | Chapter 14 | 4/5 | 6/12 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 – Qualitative Analysis EE 203 Circuit Analysis 2 Lecture 31 Chapter 14.5 Band-Reject 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 31 | Chapter 14 | 4/5 | 7/12 short ω=0 open open ω=∞ short Band-Reject Filter Band-Pass Filter EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 31 | Chapter 14 | 4/5 | 8/12 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 Cutoff Frequencies sL + 1 / sC s 2 + 1 / LC H ( s) = =2 R + sL + 1 / sC s + ( R / L) s + (1 / LC ) H ( jω ) = 1 / LC − ω at the ωc (ω1 or ω2) [(1 / LC ) − ω 2 ]2 + [ω ( R / L)]2 1 1 = 2 1 2 ⎡ ω ( R / L) ⎤ {[(1 / LC ) − ωc ]2 + ωc ( R / L)]2 } 22 1+ ⎢ c 2⎥ (1 / LC − ωc ) ⎣1 / LC − ωc ⎦ = The frequency for which the circuit’s transfer function is purely real 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 2 ⎡ ω ( R / L) ⎤ ⎡ ω ( R / L) ⎤ 2 ∴⎢ c = ±1 ⇒ ω c L ± ω c R − 1 / C = 0 =1 ⇒ ⎢ c 2⎥ 2⎥ ⎣1 / LC − ωc ⎦ ⎣1 / LC − ωc ⎦ The frequency for which the circuit’s transfer function has the minimum H min = H ( jωo ) = 1 / LC − ωo 2 [(1 / LC ) − ωo ] + [ωo ( R / L)] 22 ∴ 1 / LC − ωo = 0 ⇒ ωo = 2 2 1 1 H max = = 2 2 2 [(1 / LC ) − ωc ]2 + [ωc ( R / L)]2 2 Center Frequency jω o L + |H(jωc)|=(1/√2)Hmax 1 / LC − ωc 2 =0 ωc1 = − 2 ∴ ωc = ± 1 LC EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo 2 ωc > 0 Lecture 31 | Chapter 14 | 4/5 | 9/12 R 1 ⎡R⎤ + ⎢ ⎥+ 2L ⎣ 2 L ⎦ CL R 1 ⎡R⎤ + ⎢ ⎥+ 2L ⎣ 2 L ⎦ CL 2 ωc 2 = R 1 ⎡R⎤ + ⎢ ⎥+ 2L ⎣ 2 L ⎦ CL EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo ax 2 + bx + c = 0 2 ⇒ x= b ⎡b⎤ c ± ⎢ ⎥− 2a ⎣ 2a ⎦ a Lecture 31 | Chapter 14 | 4/5 | 10/12 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 (4) Series RLC Circuits – Quantitative Analysis (3) 2 Center Frequency & Cutoff Frequencies ωo = ωc1 ⋅ ωc 2 2 2 ⎡R 1 ⎤⎡ R R⎞ 1⎤ ⎛R⎞ ⎥ ⎥⎢ + ⎛ ⎟ + = ⎢− + ⎜ ⎟+ ⎜ ⎢ 2L ⎝ 2 L ⎠ CL ⎥ ⎢ 2 L ⎝ 2 L ⎠ CL ⎥ ⎣ ⎦ ⎦⎣ 2 2 1 ⎛R⎞ ⎛R⎞ = −⎜ ⎟ +⎜ ⎟ + = 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 ωc 2 = β β 2 ⎛β ⎞ 2 + ⎜ ⎟ + ωo 2 ⎝2⎠ 2 ⎛β ⎞ 2 + ⎜ ⎟ + ωo ⎝2⎠ 2 ⎡ ⎤ ⎞ ⎢− 1 + ⎛ 1 ⎟ + 1⎥ ⎜ ωc1 = ωo ⎜ 2Q ⎟ ⎥ ⎢ 2Q ⎝ ⎠ ⎣ ⎦ β = ωc 2 − ωc1 ⎡ 1 ωc 2 = ωo ⎢ + ⎢ 2Q ⎣ 2 2 ⎡R 1⎤ ⎡ R 1⎤ R ⎛R⎞ ⎛R⎞ ⎥ − ⎢− ⎥= =⎢ + ⎜ ⎟ + + ⎜ ⎟+ ⎢ 2L ⎝ 2 L ⎠ CL ⎥ ⎢ 2 L ⎝ 2 L ⎠ CL ⎥ L ⎣ ⎦⎣ ⎦ Quality Factor Q= ωc1 = − 2 2 ⎛1⎞ ⎜ ⎟ + 1⎥ ⎜ 2Q ⎟ ⎥ ⎝ ⎠ ⎦ ωc1 = − R 1 ⎛R⎞ + ⎜ ⎟+ 2L 2 L ⎠ CL ⎝ 2 ωc 2 = 1 LC ωo = β= Q= R 1 ⎛R⎞ + ⎜ ⎟+ 2L ⎝ 2 L ⎠ CL R L L CR 2 (1 / LC ) ωo L = = β R/L CR 2 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 31 | Chapter 14 | 4/5 | 11/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 31 | Chapter 14 | 4/5 | 12/12 ...
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