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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, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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 selfinductance 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 ampereturns is the same for each coil
EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo Lecture 09  Chapter 09  7/7  3/9 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo Lecture 09  Chapter 09  7/7  5/9 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
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 steadystate 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@SUNYBuffalo Lecture 09  Chapter 09  7/7  7/9 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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 twowinding 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 selfimpedance of the
secondary circuit scaled by the factor (ωM/Z22)2. The twowinding ideal transformer is a linear transformer with the following
special properties: perfect coupling (k = 1), infinite selfinductance 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@SUNYBuffalo Lecture 09  Chapter 09  7/7  9/9 ...
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