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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, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
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@SUNYBuffalo Lecture 05  Chapter 09  3/7  1/12 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo Lecture 05  Chapter 09  3/7  3/12 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo 1 by 90 degrees
Lecture 05  Chapter 09  3/7  5/12 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo 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@SUNYBuffalo 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@SUNYBuffalo → polar form Lecture 05  Chapter 09  3/7  9/12 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo Lecture 05  Chapter 09  3/7  11/12 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 05  Chapter 09  3/7  12/12 ...
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