05 Dec Use the Delta-Wye transformation to find the resistance as seen at ?the input. Label each Delta/Wye you identify on the circuit diagram and ?discuss your approach an
- Use the Delta-Wye transformation to find the resistance as seen at the input. Label each Delta/Wye you identify on the circuit diagram and discuss your approach and your reasoning for the solution.
- Why do industrial plants use power factor correction? Here is a good resource to read. Find other examples and list your resources.
Delta to wye transformation
Eet310 circuit analysis
∆-to-Y and Y-to-∆ conversion (1 of 2)
The ∆-to-Y and Y-to-∆ conversion formulas allow a three terminal resistive network to be replaced with an equivalent network.
For the ∆-to-Y conversion, each resistor in the Y is equal to the product of the resistors in the two adjacent ∆ branches divided by the sum of all three ∆ resistors.
For example,
2
∆-to-Y and Y-to-∆ conversion (2 of 2)
The ∆-to-Y and Y-to-∆ conversion formulas allow a three terminal resistive network to be replaced with an equivalent network.
For the Y-to-∆ conversion, each resistor in the ∆ is equal to the sum of all products of Y resistors, taken two at a time divided by the opposite Y resistor.
For example,
3
Wye – Delta Conversions
4
Wye – Delta Conversions
5
Wye – Delta Conversions
Determine the total resistance of the circuit shown below.
6
Wye – Delta Conversions
Convert the delta configuration to wye
7
Wye – Delta Conversions
Determine the equivalent resistance of the following circuit
8
Wye – Delta Conversions
Determine Rab for the following circuit
9
Wye – Delta Conversions
Determine the total resistance Rab for the circuit shown below. All the resistors are equal
10
image1.jpeg
image2.wmf
oleObject1.bin
image3.png
image4.wmf
image5.png
oleObject2.bin
image6.png
image7.png
image8.png
image9.png
image10.png
image11.png
image12.png
image13.png
image14.png
image15.jpeg
image16.jpeg
image17.png
image18.png
image19.png
,
Analysis of ac circuits
EET310 Circuit analysis
Objective of the Lecture
After successful completion of the lecture you will be able to:
Determine the reactance of both a capacitor and an inductor.
Analyze an AC circuit using either NVA or LCA.
Determine the frequency response an AC circuit
2
Introduction
3
A large
complex AC
circuits
Simplify
circuit analysis
Circuit Theorems
‧Thevenin’s theorem ‧ Norton theorem
‧Circuit linearity ‧ Superposition
‧source transformation ‧ max. power transfer
Superposition theorem
The superposition theorem that you studied in dc circuits can be applied to ac circuits by using complex numbers. Recall that it is applied to circuits with multiple independent sources to solve for the current in any element.
In a linear circuit with multiple independent sources, the current in any element is the algebraic sum of the currents produced by each source acting alone.
One way to summarize the superposition theorem is
4
Summary
Superposition theorem
Leave one of the sources in the circuit, and replace all others with their internal impedance. For ideal voltage sources, the internal impedance is zero. For ideal current sources, the internal impedance is infinite. We will call this procedure zeroing the source.
Find the current in the branch of interest produced by the one remaining source.
Repeat Steps 1 and 2 for each source in turn. When complete, you will have a number of current values equal to the number of sources in the circuit.
Add the individual current values as phasor quantities.
Steps for applying the superposition theorem:
5
Superposition theorem
Steps from the text are applied in the following example. To simplify this example, reactances are given for the capacitors.
What is the current in C2? The internal source resistances are zero.
Example
-j10 kW
-j20 kW
6.8 kW
C1
C2
R
VS1
VS2
Step 1.
Replace VS2 with its internal impedance (zero) and find IC2 due to VS1 acting alone.
-j10 kW
-j20 kW
6.8 kW
C1
C2
R
VS1
VS2
zeroed
Note: To simplify math equations, units are not shown on the solution. All impedances are in kW, currents in mA, and voltages in V.
continued…
6
Superposition theorem
Step 1. (continued)
-j10 kW
-j20 kW
6.8 kW
C1
C2
R
VS1
VS2
zeroed
Looking from VS1, the total impedance (in kW) is
The total current from VS1 is
Apply current divider to find IC2 due to VS1:
continued…
7
Superposition theorem
Step 2.
-j10 kW
-j20 kW
6.8 kW
C1
C2
R
VS1
zeroed
Find IC2 due to VS2 by zeroing VS1. The total impedance (in kW) is
The total current from VS2 is
Notice that IS2 = IC2 for this source and is opposite to IC2 due to VS1.
continued…
VS2
8
Superposition theorem
Step 3.
Find IC2 due to both sources by combining the results from steps 1 and 2.
In polar form, the total current in IC2 is
-j10 kW
-j20 kW
6.8 kW
C1
C2
R
VS1
VS2
IC2 due to VS1 is
IC2 due to VS2 is
IC2 due to both sources is
Subtract because currents oppose.
The result is in the 3rd quadrant, meaning that the actual direction is reversed from the assumed positive direction. It is due mainly to VS2.
Problems like this example can be easily solved by Multisim, a popular computer simulation. The Multisim circuit is shown on the following slide.
9
Superposition theorem
Voltage sources are specified in Multisim as peak voltages.
10
Superposition Theorem
Using the superposition theorem, find the current I through the 4ohm reactance (XL2)
11
Superposition Theorem
12
=
+
Determine Io using superposition theorem
Superposition Theorem
13
Superposition Theorem
Determine Vo using superposition theorem
14
Superposition Theorem
Due to the 5 V DC source
15
Superposition Theorem
Due to the 10 V Peak Voltage source
16
Superposition Theorem
Due to the AC current source
17
Source Transformation
18
Source Transformation
Find Vx by applying source transformation
19
Thevenin’s theorem
Thevenin’s theorem can be applied to ac circuits. As applied to ac circuits, Thevenin’s theorem can be stated as:
The Thevenin equivalent circuit is:
Vth
Zth
Any two-terminal linear ac circuit can be reduced to an equivalent circuit that consists of an ac voltage source in series with an equivalent impedance.
20
Thevenin’s theorem
Vth
Zth
ac circuit
Equivalency means that when the same value of load is connected to both the original circuit and Thevenin’s equivalent circuit, the load voltages and currents are the same for both.
R
R
For example, if the same R is connected to both circuits, the voltage across R will be identical for each circuit.
21
Thevenin’s theorem
Vth
Zth
Thevenin’s equivalent voltage (Vth) is the open-circuit voltage between two specified terminals in a circuit.
With load removed, this is the Thevenin voltage.
ac circuit
With load removed, this is the Thevenin voltage.
RL
RL
22
Thevenin’s theorem
Thevenin’s equivalent impedance (Zth) is the total impedance appearing between two specified terminals in a given circuit with all sources replaced by their internal impedances.
Vth
Zth
With sources replaced with their internal impedance, you could measure Zth here.
ac circuit
With sources replaced with their internal impedance, you could measure Zth here.
Replace with internal resistance
23
Thevenin’s theorem
Steps for applying Thevenin’s theorem:
Open the two terminals between which you want to find the Thevenin circuit. This is done by removing the component from which the circuit is to be viewed.
Determine the voltage across the two open terminals.
Determine the impedance viewed from the two open terminals with ideal voltage sources replaced with shorts and ideal current sources replaced with opens.
Connect Vth and Zth in series to produce the complete Thevenin equivalent circuit.
24
Norton’s theorem
Norton’s theorem is also an equivalent circuit that provides a way to reduce complicated circuits to a simpler form. As applied to ac circuits, Norton’s theorem can be stated as:
The Norton equivalent circuit is:
Any two-terminal linear ac circuit can be reduced to an equivalent circuit that consists of an ac current source in parallel with an equivalent impedance.
In
Zn
25
Norton’s theorem
With the load replaced with a short, the current in the short is Norton’s current.
ac circuit
Norton’s equivalent current (In) is the short-circuit current (ISL) between two specified terminals in a circuit.
In
Zn
With the load replaced with a short, the current in the short is Norton’s current.
RL
RL
26
Norton’s theorem
Norton’s equivalent impedance (Zn) is the total impedance appearing between two specified terminals in a given circuit with all sources replaced by their internal impedances.
An ideal current source is replaced with an open circuit.
Notice that Zn = Zth.
With sources replaced with their internal impedance, you could measure Zth here.
ac circuit
With sources replaced with their internal impedance, you could measure Zth here.
In
Zn
27
Norton’s theorem
Steps for applying Norton’s theorem:
Replace the load connected to the two terminals between which the Norton circuit is to be determined with a short.
Determine the current through the short. This is In.
Open the terminals and determine the impedance between the two open terminals with all sources replaced with their internal impedances. This is Zn.
Connect In and Zn in parallel.
28
Maximum power transfer theorem
Recall that in resistive circuits, maximum power is transferred from a given source to a load if RL = RS.
Impedance consists of a resistive and reactive part. To transfer maximum power in reactive circuits, the load resistance should still match the source resistance but it should cancel the source reactance by matching the opposite reactance as illustrated.
RS
RL
RS ± jX
29
Maximum power transfer theorem
The impedance that cancels the reactive portion of the source impedance is called the complex conjugate. The complex conjugate of R – jXC is R + jXL. Notice that each is the complex conjugate of the other.
RS ± jX
In effect the reactance portion of the load forms a series resonant circuit with the reactive portion of the source at the resonant frequency. Thus the matching is frequency dependent.
30
Thevenin and Norton Theorems
31
Thevenin and Norton Theorems
Determine the Thevenin voltage and resistance in terminal ab
32
Thevenin and Norton Theorems
Determine the Thevenin equivalent circuit
33
Thevenin and Norton Theorems
34
Thevenin and Norton Theorems
Determine Io using Norton Theorem
35
Thevenin and Norton Theorems
36
Nodal Analysis in AC Circuits
Find ix in the circuit shown below
37
Nodal Analysis in AC Circuits
38
Nodal Analysis in AC Circuits
Determine the voltages V1 and V2 in the circuit below
39
Applying KCL for the supernode gives 1 equations.
Applying KVL at the supernode gives 1 equations.
2 variables solved by 2 equations.
Nodal Analysis in AC Circuits
40
Mesh Analysis in AC Circuits
Using Mesh current analysis determine Io
41
Mesh Analysis in AC Circuits
Using Mesh current analysis determine Vo
42
Applying KVL for mesh 1 & 2 gives 2 equations.
Applying KVL for the supermesh gives 1 equations.
Applying KCL at node A gives 1 equations.
4 variables solved by 4 equations
Mesh Analysis in AC Circuits
43
Superposition theorem
Equivalent circuit
Thevenin’s theorem
A method for the analysis of circuits with more than one source.
A circuit that produces the same voltage and current to a given load as the original circuit that it replaces.
Key Terms
A method for simplifying a two-terminal circuit to an equivalent circuit with only a voltage source in series with an impedance.
44
Norton’s theorem
Complex conjugate
A method for simplifying a two-terminal circuit to an equivalent circuit with only a current source in parallel with an impedance.
A complex number having the same real part and oppositely signed imaginary part; an impedance containing the same resistance and a reactance opposite in phase but equal in magnitude to that of a given impedance.
Key Terms
45
Quiz
Compared to the original circuit that it replaces, an equivalent circuit will always have
a. the same power dissipation.
b. the same number of components
c. the same voltage and current for a specific load
d. all of the above
46
Quiz
2. To apply the superposition theorem, all sources but the one of interest are replaced with
a. a short
b. an open
c. their internal impedance
d. an impedance equal to the load
47
Quiz
3. Compared to the Thevenin impedance for a given circuit, the Norton impedance is
a. the same
b. smaller
c. larger
d. equal to the complex conjugate
48
Quiz
4. If you put a short on a Thevenin circuit, the current in the short is
a. zero
b. smaller than the Norton current
c. larger than the Norton current
d. the same as the Norton current
49
Quiz
5. A Thevenin circuit is a
a. voltage source in parallel with an impedance
b. voltage source in series with an impedance
c. current source in parallel with an impedance
d. current source in parallel with an impedance
50
Quiz
6. A Norton circuit is a
a. voltage source in parallel with an impedance
b. voltage source in series with an impedance
c. current source in parallel with an impedance
d. current source in parallel with an impedance
51
Quiz
A theorem which is algebraically adds the current from several sources acting independently to find the total current is
a. superposition theorem
b. Thevenin’s theorem
c. Norton’s theorem
d. maximum power theorem