EE203-SUNYBuffalo-05-Chapter09-03

EE203-SUNYBuffalo-05-Chapter09-03 - 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 Nanobio Sensors & MicroActuators Learning Lab The State University of New York Nanobio Sensors & MicroActuators Learning Lab The State University of New York Phasor Current and Phasor Voltage for a Resistor EE 203 Circuit Analysis 2 Lecture 05 Chapter 9.4 R, L, C in Frequency Domain 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 If the current through a resistor R is Then the voltage is given by Ohm’s Law: And the phasor form is which is Th The voltage is in phase with the current No phase shift EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 1/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 2/12 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 A Resistor In the Frequency Domain Phasor Current and Phasor Voltage for an Inductor If If the current through an Inductor L is Then the voltage is given by The phasor diagram for a resistor is: The voltage is in phase with the current for a Resistor Im And the phasor form is V I θi Re The voltage leads the current by 90 degrees EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 3/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 4/12 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 An Inductor in the Frequency Domain Phasor Current and Phasor Voltage for a Capacitor The phasor diagram for an inductor is: 900 Voltage leading current: the voltage leads the current by for an inductor 0 for an inductor Current lagging voltage: the current lags the voltage by 90 Im i=C dv = −ω C Vm sin(ωt + θ v ) dt = −ω C Vm cos(ωt + θ v − 90°) And the phasor form is I = −ω C Vm e j (θ −90°) = −ω C Vm e jθ e − j 90° V I If If the voltage through a Capacitor C is Then the current is given by v θi v = jω C Vm e jθ v = jω C V Re ∴V = V = jωLI jω C I The voltage lags the current The voltage reaches its negative peak exactly 900 before the current reaches its negative peak. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo 1 by 90 degrees Lecture 05 | Chapter 09 | 3/7 | 5/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 6/12 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 “ELI the ICE man” A Capacitor in the Frequency Domain The phasor diagram for a capacitor is: Voltage lagging current: the voltage lags the current by 900 for a capacitor Current lleading voltage: the current leads the voltage by 900 for a capacitor urrent eading voltage: the current leads the voltage by 90 for capacitor Im I V θi Re V= 1 I jωC The voltage reaches its negative peak exactly 900 after the current reaches its negative peak. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 7/12 There There is a simple memory aid for remembering the relationships between sinusoidal voltage and current in inductors and capacitors. Remember that the voltage and current in a resistor are in phase, meaning that the phase angle of the voltage equals the phase angle of the current. Also remember that the voltage and current in inductors and capacitors differ by 90° of phase angle. But does the voltage lead or lag the current? th The key to remembering is the phrase “ELI the ICE man.” The “L” in “ELI” stands for inductor, the “E” stands for voltage (which used to be called “electromotive force, hence the “E”), and the “I” stands for current. Since “E” th “E”) “E” “I” is before “I” in “ELI,” the voltage leads the current in an inductor by 90°. The “C” in “ICE” stands for capacitor, and because the “I” is before the “E” in “ICE,” the current leads the voltage in a capacitor by 90°. Use “ELI the ICE th ICE man” to remember the relationship between voltage and current in inductors and capacitors. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 8/12 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 Impedance and Reactance Impedance and Reactance V = jωL I Impedance V= 1 1 I = j (− )I j ωC ωC Impedance is the ratio of a circuit element’s voltage phasor to its current phasor A phasor is a complex number that shows up as the coefficient of ejwt Z= V Vm ∠φ Vm = = ∠(φ − β ) I I m ∠β I m ∴ magnitude Z = Impedance Z in the frequency domain Vm , phase angle θ = φ − β Im Z = Z ∠θ = Ze jθ Reactance → exponential form = R + jX is the quantity analogous to resistance R, inductance L, and capacitance C in the time domain. → rectangular form where Z = R 2 + X 2 , θ = tan −1 The imaginary part of the impedance Z EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo → polar form Lecture 05 | Chapter 09 | 3/7 | 9/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo X R Lecture 05 | Chapter 09 | 3/7 | 10/12 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 Equivalent Circuits for C and L as a Function of Frequency Admittance and Susceptance Admittance V= 1 I jωC ∴Z = 1 jωC V = jωLI ∴ Z = jωL Admittance is the reciprocal of impedance Conductance Real part G is called conductance Z= 1 → ∞ (open) j (0)C Z = j (0) L → 0 (short ) Z= Susceptance 1 → 0 (short ) j (∞)C Z = j ( ∞) L → ∞ (open) Imaginary part B is called susceptance susceptance V = jωL I V= 1 1 I = j (− )I j ωC ωC EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 11/12 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 05 | Chapter 09 | 3/7 | 12/12 ...
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