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The Transformer - Lecture Slides - Elements of Electrical Engineering | ES A309, Study notes of Engineering

University of Alaska AnchorageEngineering
Prof. Jeffrey A. Miller

Material Type: Notes; Professor: Miller; Class: Elements of Electrical Engineering; Subject: Engineering Science ; University: University of Alaska - Anchorage; Term: Fall 2007;

Typology: Study notes

2009/2010

Uploaded on 03/28/2010

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ES309
Elements of Electrical Engineering
Lecture #25
Jeffrey Miller, Ph.D.
Outline
• Final Exam Review
• Chapter 9.10
Final Exam Details
• 9 Questions
• Closed book
• Calculators are allowed
• The formula sheet posted on the class web
site will be provided with the exam
• 2 hours will be allotted for the exam, from
the beginning of the exam time period
Final Exam Details
1) Find the current, voltage, power, and energy in a purely resistive circuit
with a voltage source (Chapter 2 – 4%)
2) Find the equivalent resistance in a purely resistive circuit (Chapter 3 – 3%)
3) Find the voltage and current in a purely resistive circuit with a current
source (Chapter 3 – 4%)
4) Using the node-voltage method to find the voltage and power in a purely
resistive circuit with multiple voltage or current sources (Chapter 4 – 4%)
5) Find the Thevenin equivalent in a purely resistive circuit with multiple
voltage or current sources (Chapter 4 – 4%)
6) Determine the output voltage and source voltage to ensure linear operation
in an operation amplifier circuit (Chapter 5 – 4%)
7) Find the current and voltage in an RL circuit (Chapter 7 – 4%)
8) Find the current and voltage in an RLC circuit (Chapter 8 – 4%)
9) Determine the impedence and phase information given a sinusoidal voltage
and current circuit (Chapter 9 – 4%)
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Partial preview of the text

Download The Transformer - Lecture Slides - Elements of Electrical Engineering | ES A309 and more Study notes Engineering in PDF only on Docsity!

ES

Elements of Electrical Engineering

Lecture

Jeffrey Miller, Ph.D.

Outline

• Final Exam Review

• Chapter 9.

Final Exam Details

• 9 Questions

• Closed book

• Calculators are allowed

• The formula sheet posted on the class web

site will be provided with the exam

• 2 hours will be allotted for the exam, from

the beginning of the exam time period

Final Exam Details

1) Find the current, voltage, power, and energy in a purely resistive circuit

with a voltage source (Chapter 2 – 4%)

2) Find the equivalent resistance in a purely resistive circuit (Chapter 3 – 3%)

3) Find the voltage and current in a purely resistive circuit with a current

source (Chapter 3 – 4%)

4) Using the node-voltage method to find the voltage and power in a purely

resistive circuit with multiple voltage or current sources (Chapter 4 – 4%)

5) Find the Thevenin equivalent in a purely resistive circuit with multiple

voltage or current sources (Chapter 4 – 4%)

6) Determine the output voltage and source voltage to ensure linear operation

in an operation amplifier circuit (Chapter 5 – 4%)

7) Find the current and voltage in an RL circuit (Chapter 7 – 4%)

8) Find the current and voltage in an RLC circuit (Chapter 8 – 4%)

9) Determine the impedence and phase information given a sinusoidal voltage

and current circuit (Chapter 9 – 4%)

The Transformer

• Transformers are based on magnetic coupling and

are used in communication and power circuits

– In communication circuits, a linear transformer

matches impedances and eliminates DC signals from

parts of the system

– In power circuits, a ferromagnetic transformer

(which we’ll model as an ideal transformer) is used to

establish AC voltage levels that facilitate the

transmission, distribution, and consumption of

electrical power

The Transformer

• A simple linear transformer is formed when two

coils are wound on a single core to ensure magnetic

coupling

• The winding connected to the source is known as

the primary winding and the winding connected to

the load is the secondary winding

Transformer Circuit Parameters

• R 1 - the resistance of the primary winding

• R

- the resistance of the secondary winding

• L

– the self-inductance of the primary winding

• L 2 – the self-inductance of the secondary winding

• M – the mutual inductance

• V

s

– the sinusoidal voltage source

• Zs – the internal impedance of the voltage source

• Z

L

– the load connected to the secondary winding

• i

, i

– the phasor currents associated with the primary and secondary

currents of the transformer

Transformer Circuit Analysis

• To analyze the transformer circuit below, we need to

find i

and i

as functions of the circuit parameters V

s

Z

s

, R

, L

, L

, R

, M, Z

L

, and ω

• To find i

and i

, we can write two mesh current

equations

V

s

= (Z

s

+ R

+ jωL

) i

– jωM i

0 = -jωM i

+ (R

+ jωL

+ Z

L

) i

Linear Transformer Example

• The parameters of a certain linear transformer are

R

1

=200Ω, R

2

=100Ω, L

1

=9H, L

2

=4H, and k=0.

(coefficient of coupling). The transformer couples an

impedance consisting of an 800Ω resistor in series with a

1μF capacitor to a sinusoidal voltage source. The 300V

source has an internal impedance of 500 + j100Ω and a

frequency of 400 rad/s.

– Construct a frequency-domain equivalent circuit of the system.

– Calculate the self-impedance of the primary circuit.

– Calculate the self-impedance of the secondary circuit.

– Calculate the impedance reflected into the primary winding.

– Calculate the impedance seen looking into the primary terminals of

the transformer.

Linear Transformer Example

• Construct a frequency-domain equivalent circuit of the system.

– Given

ω=400 rad/s, R 1 =200Ω, R 2 =100Ω, L 1 =9H, L 2 =4H, k=0.5,

Zs=500 + j100Ω, Vs = 300V, ZL=800Ω + ZC

jωL

= j (400rad/s)(9H) = 3600jΩ

jωL 2 = j (400rad/s)(4H) = 1600jΩ

M = k √(L

* L

) = 0.5 √(4H * 9H) = 3H

jωM = j (400rad/s)(3H) = 1200jΩ

ZC = 1 / (jωC) = 1 / (j * 400rad/s * 1x

F) = -2500jΩ

Linear Transformer Example

• Calculate the self-impedance of the primary

circuit

Z

= Z

s

+ R

+ jωL

= 500Ω + 100jΩ + 200Ω + 3600jΩ

= 700 + 3700j Ω

Linear Transformer Example

• Calculate the self-impedance of the secondary

circuit

Z

= Z

L

+ R

+ jωL

= 800Ω – 2500jΩ + 100Ω + 1600jΩ

= 900 - 900j Ω

Linear Transformer Example

• Calculate the impedance reflected into the primary winding.

Zr = ω

M

/ (R 2 + jωL 2 + ZL)

/ (100 + 1600j + 800 – 2500j)

= 1440000 / (900 – 900j)

= 1440000 (900 – 900j) / (900 – 900j)^2

= 1440000 (900 – 900j) / (810000 – 1620000j + 810000j^2 )

= 1440000 (900 – 900j) / -1620000j

= (1296000000 – 1296000000j) / -1620000j

= -800/j + 800

= 800j + 800 Ω

Linear Transformer Example

• Calculate the impedance seen looking into the primary terminals of the transformer.

Z

ab

= Z

M

/ Z

– Z

s

= 700 + 3700j + (400)

/ (900 – 900j) – 500 – 100j

= 200 + 3600j + 1440000 / (900 – 900j)

= 200 + 3600j + 1440000 (900 – 900j) / (900 – 900j)^2

= 200 + 3600j + 1440000 (900 – 900j) / (810000 – 1620000j + 810000j^2 )

= 200 + 3600j + 1440000 (900 – 900j) / -1620000j

= 200 + 3600j + (1296000000 – 1296000000j) / -1620000j

= 200 + 3600j + -800/j + 800

= 200 + 3600j + 800j + 800

= 1000 + 4400j Ω

Concluding Remarks

• I hope you enjoyed the class

• I hope you learned a lot, not only about

electric circuits

• Good luck on all of your final exams!

Homework

• Final exam next week!