EE203-SUNYBuffalo-09-Chapter09-07

EE203-SUNYBuffalo-09-Chapter09-07 - 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 Nanobio Sensors & MicroActuators Learning Lab The State University of New York Nanobio Sensors & MicroActuators Learning Lab The State University of New York Ideal Transformer EE 203 Circuit Analysis 2 Lecture 09 Chapter 9.11 Ideal Transformer Kwang W. Oh, Ph.D., Assistant Professor SMALL (Nanobio Sensors & 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 kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 1/9 An An ideal transform consists of two magnetically coupled coils having N1 and N2 turns, respectively, and and exhibiting these three properties: k = 1 (the coefficient of coupling is unity) L1= L2 = ∞ (the self-inductance of each coil is infinite) R1= R2 = 0 (the coil losses, due to parasitic resistance, are negligible) Voltage relationship for an ideal transformer the magnitude of the volts per turn is the same for each coil Current relationship for an ideal transformer the magnitude of the ampere-turns is the same for each coil EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 2/9 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo Nanobio Sensors & MicroActuators Learning Lab The State University of New York Nanobio Sensors & MicroActuators Learning Lab The State University of New York Determining the Polarity of the Voltage and Current Ratios The Ratio of the Turns The The ratio N2/N1 is 2500 to 500 or 5 to 1 to or 1 to 1/5 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 3/9 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 4/9 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo Nanobio Sensors & MicroActuators Learning Lab The State University of New York Nanobio Sensors & MicroActuators Learning Lab The State University of New York Example 9.14 (1) Example 9.14 (2) V1 = 10V2 = 10(0.2375 + j 0.05)I 2 = 100(0.2375 + j 0.05)I1 2500∠0° = (0.25 + j 2)I1 + ( 23.75 + j5)I1 = ( 24 + j 7)I1 2500∠0° 2500∠0° = 7 24 + j 7 242 + 72 ∠ tan −1 ( ) 24 2500∠0° = = 100∠ − 16.26° 25∠16.26° ⇒ i1 = 100 cos(400t − 16.26°) A ∴ I1 = 2500∠0° = (0.25 + j 2)I1 + V1 V2 = (0.2375 + j 0.05)I 2 jωL = j ( 400)(5 × 10 ) = j ( 2000 × 10 ) = j 2 −3 Phasor domain circuit −3 jωL = j ( 400)(125 × 10 ) = j (50000 × 10 ) = j5 × 10 −6 KVL −6 −2 V1 = V2 10 10I1 = I 2 V1 = ( 23.75 + j5)I1 = ( 23.75 + j5)(100∠ − 16.26°) 5 )(100∠ − 16.26°) 23.75 = ( 24.27∠11.89°)(100∠ − 16.26°) = 23.752 + 52 ∠ tan −1 ( = 2427∠(11.89° − 16.26°) = 2427∠ − 4.37°V ⇒ v1 = 2427 cos(400t − 4.37°) V Ideal transformer 4 unknowns & equations 4 equations I 2 = 10I1 ⇒ i2 = 1000 cos(400t − 16.26°) A V2 = 0.1V1 ⇒ v2 = 242.7 cos(400t − 4.37°) V EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 5/9 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 6/9 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo Nanobio Sensors & MicroActuators Learning Lab The State University of New York Nanobio Sensors & MicroActuators Learning Lab The State University of New York Sinusoidal Source and Phasor Transform EE 203 Circuit Analysis 2 Lecture 09 Chapter 9 Summary Kwang W. Oh, Ph.D., Assistant Professor SMALL (Nanobio Sensors & 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 kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu The General Equation for a Sinusoidal Source is v = vm cos(ωt + ø) (voltage source), or i = Im cos(ωt + ø) (current source), where vm (or Im) is the maximum amplitude, ω is the frequency, and ø is the phase angle. The best way to find the steady-state voltages and currents in a circuit driven by sinusoidal sources is to perform the analysis in the frequency domain. The following mathematical transforms allow us to move between the time and frequency domains. The phasor transform: from the time domain to the frequency domain The inverse phasor transform: from the frequency domain to the time domain EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 7/9 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 8/9 SMALL for Big Things University at Buffalo Nanobio Sensors & MicroActuators Learning Lab The State University of New York Impedance, Transformer Element Resistor Capacitor Inductor Impedance (Z) Reactance Admittance (Y) Susceptance R (resistance) G (conductance) — — j(−1/ωC) −1/ωC jωC ωC jωL ωL j(−1/ωL) −1/ωL The two-winding linear transformer is a coupling device made up of two coils wound wound on the same nonmagnetic core. Reflected impedance is the impedance of the secondary circuit as seen from the terminals of the primary circuit or vice versa. The reflected impedance of a linear transformer seen from the primary side is the conjugate of the self-impedance of the secondary circuit scaled by the factor (ωM/|Z22|)2. The two-winding ideal transformer is a linear transformer with the following special properties: perfect coupling (k = 1), infinite self-inductance in each coil infinite (L1 = L2 = ∞), and lossless coils (R1 = R2 = 0). The circuit behavior is governed by the turns ratio a = N2/N1. N1 I 1 = ± N2 I 2 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 09 | Chapter 09 | 7/7 | 9/9 ...
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