Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson...

transcript

Electronic Circuits, Tenth EditionJames W. Nilsson | Susan A. Riedel

CHAPTER 9

SinusoidalSteady-State Analysis

CHAPTER CONTENTS

• 9.1 The Sinusoidal Source • 9.2 The Sinusoidal Response • 9.3 The Phasor • 9.4 The Passive Circuit Elements in the Frequency Domain • 9.5 Kirchhoff’s Laws in the Frequency Domain • 9.6 Series, Parallel, and Delta-to-Wye Simplifications • 9.7 Source Transformations and Thévenin-Norton Equivalent

Circuits • 9.8 The Node-Voltage Method • 9.9 The Mesh-Current Method• 9.10 The Transformer • 9.11 The Ideal Transformer • 9.12 Phasor Diagrams

CHAPTER 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; Thévenin 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.

9.1 The Sinusoidal Source

• A sinusoidal voltage source (independent or dependent) produces a voltage that varies sinusoidally with time.

• A sinusoidal current source (independent or dependent) produces a current that varies sinusoidally with time.

Figure 9.1 A sinusoidal voltage.

• The sinusoidal function repeats at regular intervals is called periodic.

• The length of time is referred to as the period of the function and is denoted T.

• The angular frequency of the sinusoidal function

• The angle f is know as the phase angle of the sinusoidal voltage.

Figure 9.2 The sinusoidal voltage from Fig. 9.1 shifted to the right when ϕ = 0.

Root mean square (rms)

• The rms value of a periodic function is defined as the square root of the mean value of the squared function.

• rms value of a sinusoidal voltage source

Example 9.1

• A sinusoidal current has a maximum amplitude of 20 A.The current passes through one complete cycle in 1 ms. The magnitude of the current at zero time is 10 A.

a) What is the frequency of the current in hertz?

b) What is the frequency in radians per second?

c) Write the expression for i(t) using the cosine function. Express f in degrees.

d) What is the rms value of the current?

Example 9.1

Example 9.2

• A sinusoidal voltage is given by the expression v = 300 cos (120pt + 30°).

a) What is the period of the voltage in milliseconds?

b) What is the frequency in hertz?

c) What is the magnitude of at t = 2.778 ms?

d) What is the rms value of v?

Example 9.2

Example 9.3

• We can translate the sine function to the cosine function by subtracting 90° (p/2 rad) from the argument of the sine function.

a) Verify this translation by showing that sin (ωt + θ) = cos (ωt + θ – 90°).

b) Use the result in (a) to express sin (ωt + 30°) as a cosine function.

Example 9.3

Example 9.4

• Calculate the rms value of the periodic triangular current shown in Fig. 9.3. Express your answer in terms of the peak current Ip.

Figure 9.3 Periodic triangular current.

Example 9.4

Figure 9.4 i2 versus t.

Example 9.4

9.2 The Sinusoidal Response

• Kirchhoff’s voltage law

Figure 9.5 An RL circuit excited by a sinusoidal voltage source.

transient component steady-state component

Transient & Steady-state component

1. The steady-state solution is a sinusoidal function.

2. The frequency of the response signal is identical to the frequency of the source signal. If R, L, and C, are constant.

3. The maximum amplitude of the response signal is

and the maximum amplitude of the signal source is Vm.

4. The phase angle of the current is f – q and that of the voltage source is .f

2 2 2/mV R L

9.3 The Phasor

• The phasor is a complex number that carries the amplitude and phase angle information of a sinusoidal function.1 The phasor concept is rooted in Euler’s identity, which relates the exponential function to the trigonometric function:

• The phasor transform transfers the sinusoidal function from the time domain to the complex-number domain, which is also called the frequency domain.

• Phasor transform

the phasor transform of Vm cos (ωt + θ)

Inverse Phasor Transform

• finding the inverse phasor transform

the inverse phasor transform of

• Equation 9.17 indicates that to find the inverse phasor transform, we multiply the phasor by and then extract the real part of the product.

.jmV e

• The transient component vanishes as time elapses, so the steady state component of the solution must also satisfy the differential equation.

• In a linear circuit driven by sinusoidal sources, the steady-state response also is sinusoidal, and the frequency of the sinusoidal response is the same as the frequency of the sinusoidal source.

• Using the notation introduced in Eq. 9.11, we can postulate that the steady-state solution is of the form where A is the maximum amplitude of the response and is the phase angle of the response.

• When we substitute the postulated steady-state solution into the differential equation, the exponential term cancels out, leaving the solution for A and b in the domain of complex numbers.

Example 9.5

• If y1 = 20 cos (ωt – 30°) and y2 = 40 cos (ωt + 60°) express y = y1 + y2 as a single sinusoidal function.

a) Solve by using trigonometric identities.

b) Solve by using the phasor concept.

Example 9.5

Figure 9.6 A right triangle used in the solution for y.

Example 9.5

Phasor as Rotating Vectors

( ) cos( )

( ) Re

( ) Re ( )

Rotating Phasor

v t V t

v t V e

v t V j t

Phasor Diagrams

cos( )

Domain Re presentation Phasor Domain Re

sin( ) 0

Time Domain Versus Phasor Domain

9.4 The Passive Circuit Elements in the Frequency Domain• The V-I Relationship for a Resistor• If i = Im cos (ωt + qi),

• The phasor transform of this voltage is

• Relationship between phasor voltage and phasor current for a resistor

Figure 9.7 A resistive element carrying a sinusoidal current.

• The signals are said to be in phase because they both reach corresponding values on their respective curves at the same time.

Figure 9.8 The frequency-domain equivalent circuit of a resistor. Figure 9.9 A plot showing that the

voltage and current at the terminals of a resistor are in phase.

• The V-I Relationship for an Inductor

• Relationship between phasor voltage and phasor current for an inductor.

• voltage leading current or current lagging voltage

Figure 9.10 The frequency-domain equivalent circuit for an inductor.

Figure 9.11 A plot showing the phase relationship between the current and voltage at the terminals of an inductor (θi = 60°).

The V-I Relationship for a Capacitor

• Relationship between phasor voltage and phasor current for a capacitor.

Figure 9.12 The frequency domain equivalent circuit of a capacitor.

• The current leads the voltage by 90°.

Figure 9.13 A plot showing the phase relationship between the current and voltage at the terminals of a capacitor (θi = 60°).

Impedance and Reactance

• Definition of impedance

where Z represents the impedance of the circuit element.

Phasor Diagrams The SINOR

Rotates on a circle of radius Vm at an angular velocity of ω in the counterclockwise direction

j te V

9.5 Kirchhoff’s Laws in the Frequency Domain• Kirchhoff’s Voltage Law in the Frequency Domain

• Euler’s identity

• KVL in the frequency domain

Kirchhoff’s Current Law in the Frequency Domain

• KCL in the frequency domain

9.6 Series, Parallel, and Delta-to-WyeSimplifications• Combining Impedances in Series and Parallel

Figure 9.14 Impedances in series.

Example 9.6

• A resistor, a 32 mH inductor, and a 5 mF capacitor are connected in series across the terminals of a sinusoidal voltage source, as shown in Fig. 9.15.The steady-state expression for the source voltage vs is 750 cos (5000t + 30°) V.

a) Construct the frequency-domain equivalent circuit.

b) Calculate the steady-state current i by the phasor method.

Figure 9.15 The circuit for Example 9.6.

Example 9.6

Figure 9.16 The frequency-domain equivalent circuit of the circuit shown in Fig. 9.15.

Example 9.6

Impedances connected in parallel

Figure 9.17 Impedances in parallel.

• Just two impedances in parallel

• Admittance, defined as the reciprocal of impedance and denoted Y

G, is called conductance and whose imaginary part, B, is called susceptance.

Example 9.7

• The sinusoidal current source in the circuit shown in Fig. 9.18 produces the current is = 8 cos 200,000t A.

a) Construct the frequency-domain equivalent circuit.

b) Find the steady-state expressions for v, i1, i2, and i3.

Example 9.7

Figure 9.19 The frequency-domain equivalent circuit.

Example 9.7

Delta-to-Wye Transformations

Figure 9.20 The delta-to-wye transformation.

Example 9.8

• Use a D-to-Y impedance transformation to find I0, I1, I2, I3, I4, I5, V1 and V2 in the circuit in Fig. 9.21.

Example 9.8

Figure 9.22 The circuit shown in Fig. 9.21, with the lower delta replaced by its equivalent wye.

Example 9.8

Figure 9.23 A simplified version of the circuit shown in Fig. 9.22.

Example 9.8

9.7 Source Transformations andThévenin-Norton Equivalent Circuits• The source transformations and the Thévenin-• Norton equivalent circuits can be applied to

frequency-domain circuits

Figure 9.24 A source transformation in the frequency domain.

Figure 9.25 The frequency-domain version of a Thévenin equivalent circuit.

Figure 9.26 The frequency-domain version of a Norton equivalent circuit.

Example 9.9

• Use the concept of source transformation to find the phasor voltage V0 in the circuit shown in Fig. 9.27.

Example 9.9

Figure 9.28 The first step in reducing the circuit shown in Fig. 9.27.

Example 9.9

Figure 9.29 The second step in reducing the circuit shown in Fig. 9.27.

Example 9.10

• Find the Thévenin equivalent circuit with respect to terminals a,b for the circuit shown in Fig. 9.30.

Example 9.10

Figure 9.31 A simplified version of the circuit shown in Fig. 9.30.

Example 9.10

Figure 9.32 A circuit for calculating the Thévenin equivalent impedance.

Example 9.10

Figure 9.33 The Thévenin equivalent for the circuit shown in Fig. 9.30.

9.8 The Node-Voltage Method

• Example 9.11• Use the node-voltage method to find the branch

currents Ia, Ib, and Ic in the circuit shown in Fig. 9.34.

Example 9.11

Figure 9.35 The circuit shown in Fig. 9.34, with the node voltages defined.

Example 9.11

9.9 The Mesh-Current Method

• Example 9.12• Use the mesh-current method to find the voltages V1,

V2, and V3 in the circuit shown in Fig. 9.36.

Example 9.12

Figure 9.37 Mesh currents used to solve the circuit shown in Fig. 9.36.

Example 9.12

9.10 The Transformer

• A transformer is a device that is based on magnetic coupling.

• In communication circuits, the transformer is used to match impedances and eliminate dc signals from portions of the system.

• In power circuits, transformers are used to establish ac voltage levels that facilitate the transmission, distribution, and consumption of electrical power.

The Analysis of a Linear Transformer Circuit• Linear transformer is found primarily in

communication circuits.• Primary winding: the winding connected to the

source• Secondary winding: the winding connected to the

Figure 9.38 The frequency domain circuit model for a transformer used to connect a load to a source.

Reflected Impedance

• The third term in Eq. 9.64 is called the reflected impedance (Zr), because it is the equivalent impedance of the secondary coil and load impedance transmitted, or reflected, to the primary side of the transformer.

• Note that the reflected impedance is due solely to the existence of mutual inductance; that is, if the two coils are decoupled, M becomes zero, Zr becomes zero, and Zab reduces to the self-impedance of the primary coil.

• The load impedance in rectangular form:

• The reflected impedance in rectangular form:

Example 9.13• The parameters of a certain linear transformer are R1 = 200 Ω, R2 =

100 Ω, L1 = 9 H, L2 = 4 H and k = 0.5. The transformer couples an impedance consisting of an 800 Ω resistor in series with a 1 mF capacitor to a sinusoidal voltage source. The 300 V (rms) source has an internal impedance of 500 + j100 Ω and a frequency of 400 rad/s.

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

b) Calculate the self-impedance of the primary circuit.

c) Calculate the self-impedance of the secondary circuit.

d) Calculate the impedance reflected into the primary winding.

e) Calculate the scaling factor for the reflected impedance.

f) Calculate the impedance seen looking into the primary terminals of the transformer.

g) Calculate the Thévenin equivalent with respect to the terminals c,d.

Example 9.13

Figure 9.39 The frequency-domain equivalent circuit for Example 9.13.

Example 9.13

Figure 9.40 The Thévenin equivalent circuit for Example 9.13.

9.11 The Ideal Transformer

• An ideal transformer consists of two magnetically coupled coils having N1 and N2 turns, respectively, and exhibiting these three properties:1. The coefficient of coupling is unity (k = 1).

2. The self-inductance of each coil is infinite (L1 = L2 = ∞).

3. The coil losses, due to parasitic resistance, are negligible.

Exploring Limiting Values

Figure 9.41 The circuits used to verify the volts-per-turn and ampere-turn relationships for an ideal transformer.

Determining the Voltage and Current Ratios

• For unity coupling, the mutual inductance equals 1 2L L

• Voltage relationship for an ideal transformer

for k = 1,

• Current relationship for an ideal transformer

• Figure 9.42 shows the graphic symbol for an ideal transformer.

Figure 9.42 The graphic symbol for an ideal transformer.

Determining the Polarity of the Voltageand Current Ratios

• Dot convention for ideal transformers

Figure 9.43 Circuits that show the proper algebraic signs for relating the terminal voltages and currents of an ideal transformer.

Ratio of the turns

Figure 9.44 Three ways to show that the turns ratio of an ideal transformer is 5.

Example 9.14

• The load impedance connected to the secondary winding of the ideal transformer in Fig. 9.45 consists of a 237.5 mΩ resistor in series with a 125 mH inductor.

• If the sinusoidal voltage source (vg) is generating the voltage 2500 cos 400t V, find the steady-state expressions for: (a) i1; (b) v1; (c) i2; and (d) v2.

Example 9.14

Figure 9.46 Phasor domain circuit for Example 9.14.

Example 9.14

The Use of an Ideal Transformer for Impedance Matching

Figure 9.47 Using an ideal transformer to couple a load to a source.

9.12 Phasor Diagrams

• Constructing phasor diagrams of circuit quantities generally involves both currents and voltages.

Figure 9.48 A graphic representation of phasors.

Figure 9.49 The complex number −7 − j3 = 7.62 /–156.80°.

Example 9.15

• For the circuit in Fig. 9.50, use a phasor diagram to find the value of R that will cause the current through that resistor, iR, to lag the source current. is, by 45° when w = 5 krad/s.

Example 9.15

Figure 9.51 The phasor diagram for the currents in Fig. 9.50.

Example 9.16

• The circuit in Fig. 9.52 has a load consisting of the parallel combination of the resistor and inductor. Use phasor diagrams to explore the effect of adding a capacitor across the terminals of the load on the amplitude of Vs if we adjust Vs so that the amplitude of VL remains constant. Utility companies use this technique to control the voltage drop on their lines.

Example 9.16

Figure 9.53 The frequency-domain equivalent of the circuit in Fig. 9.52.

Example 9.16

Figure 9.54 The step-by-step evolution of the phasor diagram for the circuit in Fig. 9.53.

Example 9.16

Figure 9.55 The addition of a capacitor to the circuit shown in Fig. 9.53.

Figure 9.56 The effect of the capacitor current Ic on the line current I.

Example 9.16

Figure 9.57 The effect of adding a load-shunting capacitor to the circuit shown in Fig. 9.53 if VL is held constant.

Summary

• The general equation for a sinusoidal source is

where Vm (or Im) is the maximum amplitude, w is the frequency, f and is the phase angle.

• The frequency, w, of a sinusoidal response is the same as the frequency of the sinusoidal source driving the circuit. The amplitude and phase angle of the response are usually different from those of the source.

Summary

• The best way to find the steady-state 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):

Summary

• When working with sinusoidally varying signals, remember that voltage leads current by 90° at the terminals of an inductor, and current leads voltage by 90° at the terminals of a capacitor.

Summary

• Impedance (Z) plays the same role in the frequency domain as resistance, inductance, and capacitance play in the time domain. Specifically, the relationship between phasor current and phasor voltage for resistors, inductors, and capacitors is

where the reference direction for I obeys the passive sign convention. The reciprocal of impedance is admittance (Y), so another way to express the currentvoltage relationship for resistors, inductors, and capacitors in the frequency domain is

Summary

• All of the circuit analysis techniques developed in Chapters 2–4 for resistive circuits also apply to sinusoidal steady-state circuits in the frequency domain. These techniques include KVL, KCL, series, and parallel combinations of impedances, voltage and current division, node voltage and mesh current methods, source transformations and Thévenin and Norton equivalents.

Summary

• The two-winding linear transformer is a coupling device made up of two coils 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 self-impedance of the secondary circuit scaled by the factor (wM/|Z22|)2.

Summary

• The two-winding ideal transformer is a linear transformer with the following special properties: perfect coupling (k = 1), infinite self-inductance in each coil (L1 = L2 = ∞), and lossless coils (R1 = R2 = 0). The circuit behavior is governed by the turns ratio a = N2/N1.In particular, the volts per turn is the same for each winding, or

and the ampere turns are the same for each winding, or

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson...

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