Transcript Chapter 10 Sinusoidal steady

Chapter 10
Sinusoidal steady-state
analysis
SJTU
1
Steps to analyze ac circuit
1. Transform the circuit to the phasor or
frequency domain
2. Solve the problem using circuit
techniques(nodal analysis, mesh analysis,
superposition,etc)
3. Transform the resulting phasor to the time
domain
SJTU
2
Nodal analysis
Fig. 8-28: An example node
SJTU
3
Mesh analysis
planar circuits:
Circuits that can be drawn on a flat
surface with no crossovers
the sum of voltages around mesh
A is
Fig. 8-29: An example mesh
SJTU
4
EXAMPLE 8-21
Use node analysis to find the current IX in Fig. 8-31.
SOLUTION:
Node C :VC  750
or
Fig. 8-31
SJTU
5
SJTU
6
EXAMPLE 8-24
The circuit in Fig. 8-32 is an equivalent circuit of an ac induction
motor. The current IS is called the stator current, IR the rotor
current, and IM the magnetizing current. Use the mesh-current
method to solve for the branch currents IS, IR and IM.
SJTU
7
SJTU
8
EXAMPLE 8-25
Use the mesh-current method to solve for output voltage V2 and
input impedance ZIN of the circuit below.
SOLUTION:
SJTU
9
SJTU
10
Example
Frequency domain
equivalent of the circuit
SJTU
11
Example
SJTU
12
Find Vo/Vi, Zi
See F page417
SJTU
13
Circuit Theorems with Phasors
PROPORTIONALITY
The proportionality property states that phasor output responses
are proportional to the input phasor
where X is the input phasor, Y is the output phasor, and K is the
proportionality constant.
SJTU
14
EXAMPLE 8-13
Use the unit output method to find the input impedance, current
I1, output voltage VC, and current I3 of the circuit in Fig. 8-20
for Vs= 10∠0°
SOLUTION: 1.Assume a unit output voltage
2.By Ohm's law,
.
3.By KVL,
4.By Ohm's law,
5.By KCL,
6.By KCL,
SJTU
.
15
Given K and ZIN, we can now calculate the required responses
for an input
SJTU
16
SUPERPOSITION
Two cases:
1. With same frequency sources.
2. With different frequency sources
EXAMPLE 8-14
Use superposition to find the steady
- state voltage vR (t) in Fig. 8 - 21
for R=20 , L1 = 2mH, L2 = 6mH,
C = 20 F, V s1= 100cos 5000t V ,
and Vs2=120cos (5000t +30 )V.
SJTU
17
SOLUTION:
Fig. 8-22
SJTU
18
SJTU
19
EXAMPLE 8-15
Fig. 8-23
Use superposition to find the steady-state current i(t) in Fig. 823 for R=10k , L=200mH, vS1=24cos20000t V, and
vS2=8cos(60000t+30 ° ).
SOLUTION:
With source no. 2 off and no.1 on
SJTU
20
With source no.1 off and no.2 on
The two input sources operate at different frequencies, so
that phasors responses I1 and I2 cannot be added to obtain
the overall response. In this case the overall response is
obtained by adding the corresponding time-domain
functions.
SJTU
21
More examples
See F page403
SJTU
22
THEVENIN AND NORTON EQUIVALENT CIRCUITS
The thevenin and Norton circuits
are equivalent to each other, so
their circuit parameters are related
as follows:
SJTU
23
Source transformation
SJTU
24
EXAMPLE 8-17
Both sources in Fig. 8-25(a) operate at a frequency of =5000 rad/s.
Find the steady-state voltage vR(t) using source transformations.
SOLUTION:
+
SJTU
25
EXAMPLE 8-18
Use Thevenin's theorem to find the current Ix in the bridge
circuit shown in Fig. 8-26.
Fig. 8-26
SJTU
26
SOLUTION:
SJTU
27
SJTU
28
SJTU
29