Unit 3A – Resistors in Series and in Parallel; Voltage ...



Chapter 9 – Sinusoidal Steady-State Analysis

Study Guide

Objectives:

1. Understand phasor concepts and be able to perform a phasor transform and an inverse phasor transform.

2. Be able to transform a circuit with a sinusoidal source into the frequency domain using phasor concepts.

3. Know how to use the following circuit-analysis techniques to solve a circuit in the frequency domain:

• Kirchhoff’s laws;

• Series, parallel, and delta-to-wye simplifications;

• Voltage and current division;

• Thevenin and Norton equivalents;

• Node-voltage method; and

• Mesh-current method.

4. Be able to analyze circuits containing linear transformers using phasor methods.

5. Understand the ideal-transformer constraints and be able to analyze circuits containing ideal transformers using phasor methods.

Mastering the Objectives:

1. Read the Introduction, Section 9.1, and Section 9.2.

a) What three quantities completely specify a sinusoidal signal?

b) What is the relationship between frequency in rad/s and frequency in hertz?

c) What is the relationship between period and frequency in hertz?

d) Suppose the phase angle ( in the sinusoid described by Eq. 9.1 is increased by 30(. Does this shift the sinusoid left or right on the time axis? How many time units is the sinusoid shifted?

e) If v(t) =Vmcos((t+(), what are the typical units of

• Vm?

• (?

• t?

• (?

What is the relationship between the units of (t and the units of (? What must you do to find the value of cos((t+(), given the values of (, t, and ( in their typical units?

f) Suppose you are given a function f(t) that is the sum of several time-dependent components. How can you tell if a component is transient? How can you tell if a component remains in the steady state?

g) If the input to a circuit is a sinusoid, what two features of the steady state output of the circuit are unchanged from the input? What two features of the steady state output of the circuit are different from the input?

2. Read Section 9.3.

a) If you need to review complex numbers, do so now by reading Appendix B. Make sure you know how to

i. Find the real and imaginary parts of a complex number in polar or rectangular form;

ii. Find the magnitude and phase angle of a complex number in polar or rectangular form;

iii. Convert from rectangular to polar form, and from polar to rectangular form;

iv. Locate a complex number in polar or rectangular form in the complex plane;

v. Find the complex conjugate of a complex number in polar or rectangular form;

vi. Add, subtract, multiply, and divide complex numbers using the most convenient form of the complex numbers;

vii. Enter and perform calculations with complex numbers on your calculator.

b) In this section, how can you tell whether the symbol representing a voltage or current is a phasor? How is that symbol different from the one used to represent a time-domain voltage or current?

c) How do you transform a sinusoid to a phasor?

d) How do you transform a phasor to a sinusoid?

e) Solve Assessment Problems 9.1 and 9.2.

3. Read Section 9.4.

a) State Ohm’s law for phasors.

b) Summarize the impedances of the resistor, the inductor, and the capacitor.

c) The impedance of a component is a complex number – is it also a phasor? Why or why not?

d) 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? 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” 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 man” to remember the relationship between voltage and current in inductors and capacitors.

e) Solve Assessment Problems 9.3 and 9.4.

4. Read Sections 9.5 and 9.6.

a) Once we have the equivalent of Ohm’s law, KVL, and KCL for phasor-transformed circuits, we can use all of the circuit-analysis tools developed for dc circuits in Chapters 3 and 4. Go back and review those chapters and make a list of all of the circuit-analysis techniques introduced there. Use this list as a reminder when deciding how to approach the solution of a particular problem in Chapter 9.

b) Impedances combine in series and in parallel just like resistors. Practice combining impedances to simplify a circuit by solving Assessment Problems 9.7 and 9.8.

5. Read Section 9.7.

a) Find the Thevenin equivalent of the circuit to the left of the (10 – j19)( impedance in Fig. 9.27. Use the Thevenin equivalent to find the voltage phasor Vo.

b) Instead of using the test source method in Example 9.10, calculate the short-circuit current at the output terminals of Fig. 9.30. Verify that you get the same Thevenin resistance using the open-circuit voltage and the short-circuit current.

c) Solve Chapter Problem 9.39 using source transformation; solve Chapter Problem 9.43.

6. Read Sections 9.8 and 9.9.

a) Use the mesh-current method for the circuit in Fig. 9.34 to confirm the answers in Example 9.11.

b) Use the node-voltage method for the circuit in Fig. 9.36 to confirm the answers in Example 9.12.

c) Think about superposition, a circuit-analysis technique presented in Chapter 4. Superposition must be used in the circuits of Chapter 9 when two or more sources in a circuit operate at different frequencies. Consider the circuit in Fig. P9.58. Suppose va = 10 cos(50,000t) V. Show that the current phasor Io( due to just this source is (1-j1) A. Suppose vb = 5 cos(100,000t + 90() V. Show that the current phasor Io(( due just to this source is (–2.5 – j2.5) A. Using superposition, calculate io (t) when both sources are in the circuit?

d) Solve Assessment Problem 9.12. Check your answer using source transformation.

e) Solve Assessment Problem 9.13. Check your answer using the node voltage method.

7. Read Section 9.10.

a) Find Zab for the circuit in Fig. 9.38 if the dot on the secondary coil is at the bottom. Follow the steps in Eqs. 9.57 – 9.64.

b) If the load impedance in Fig. 9.38 is capacitive, the transformer can be used to make the load appear purely resistive to the source. Under what conditions will this happen?

c) Solve Chapter Problem 9.67.

8. Read Section 9.11.

a) What are the three limiting properties that characterize an ideal transformer?

b) If the turns ratio of an ideal transformer is 1:a, what is the relationship between V1 and V2? What is the relationship between I1 and I2

c) Repeat Example 9.14 with the dot on the secondary coil moved to the bottom. Compare your results with those in the example.

d) Simplify the circuit in Fig. 9.47 by replacing the ideal transformer and the load impedance with a single equivalent impedance. This removes V2 and I2 from the circuit but these quantities are easy to calculate from the simplified circuit – how?

e) Simplify the circuit in Fig. 9.47 by replacing everything to the left of the load impedance with the Thevenin equivalent. This removes V1 and I1 from the circuit but these quantities are easy to calculate from the simplified circuit – how?

f) Solve Assessment Problem 9.15 and Chapter Problem 9.71. Use the equivalent circuits you generated in (d) and (e) when solving these problems.

9. Read Section 9.12 and study its examples. You may want to solve Chapter Problem 9.75 to assess your understanding of the phasor-diagram technique.

Assessing Your Mastery:

Review the Objectives for this unit. Once you are satisfied that you have achieved these Objectives, take the Chapter 9 Test.

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