Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. CHAPTER 9 Sinusoidal.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. The angle  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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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  in degrees. d) What is the rms value of the current?

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.2 A sinusoidal voltage is given by the expression v = 300 cos (120  t + 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?

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.3 We can translate the sine function to the cosine function by subtracting 90° (  /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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 I p. Figure 9.3 Periodic triangular current.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 9.2 The Sinusoidal Response Kirchhoff’s voltage law Figure 9.5 An RL circuit excited by a sinusoidal voltage source. transient componentsteady-state component

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 V m. 4. The phase angle of the current is  –  and that of the voltage source is 

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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:

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. The phasor transform transfers the sinusoidal function from the time domain to the complex-number domain, which is also called the frequency domain.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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  in the domain of complex numbers.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.5 If y 1 = 20 cos (ωt – 30°) and y 2 = 40 cos (ωt + 60°) express y = y 1 + y 2 as a single sinusoidal function. a) Solve by using trigonometric identities. b) Solve by using the phasor concept.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.5 Figure 9.6 A right triangle used in the solution for y.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 9.4 The Passive Circuit Elements in the Frequency Domain The V-I Relationship for a Resistor If i = I m cos (ωt +  i ), 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. The V-I Relationship for an Inductor Relationship between phasor voltage and phasor current for an inductor.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 ( θ i = 60°).

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 ( θ i = 60°).

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Impedance and Reactance Definition of impedance where Z represents the impedance of the circuit element.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Phasor Diagrams  The SINOR Rotates on a circle of radius V m at an angular velocity of ω in the counterclockwise direction

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 9.5 Kirchhoff’s Laws in the Frequency Domain Kirchhoff’s Voltage Law in the Frequency Domain Euler’s identity

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Kirchhoff’s Current Law in the Frequency Domain KCL in the frequency domain

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 9.6 Series, Parallel, and Delta-to-Wye Simplifications Combining Impedances in Series and Parallel Figure 9.14 Impedances in series.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 v s 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.6 Figure 9.16 The frequency-domain equivalent circuit of the circuit shown in Fig. 9.15.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Impedances connected in parallel Figure 9.17 Impedances in parallel.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.7 The sinusoidal current source in the circuit shown in Fig. 9.18 produces the current i s = 8 cos 200,000t A. a) Construct the frequency-domain equivalent circuit. b) Find the steady-state expressions for v, i 1, i 2, and i 3. Figure 9.18 The circuit for Example 9.7.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.7 Figure 9.19 The frequency- domain equivalent circuit.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Delta-to-Wye Transformations Figure 9.20 The delta-to-wye transformation.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.8 Use a  -to-Y impedance transformation to find I 0, I 1, I 2, I 3, I 4, I 5, V 1 and V 2 in the circuit in Fig. 9.21. Figure 9.21 The circuit for Example 9.8.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.8 Figure 9.22 The circuit shown in Fig. 9.21, with the lower delta replaced by its equivalent wye.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.8 Figure 9.23 A simplified version of the circuit shown in Fig. 9.22.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 9.7 Source Transformations and Thévenin-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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Figure 9.25 The frequency-domain version of a Thévenin equivalent circuit. Figure 9.26 The frequency-domain version of a Norton equivalent circuit.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.9 Use the concept of source transformation to find the phasor voltage V 0 in the circuit shown in Fig. 9.27. Figure 9.27 The circuit for Example 9.9.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.9 Figure 9.28 The first step in reducing the circuit shown in Fig. 9.27.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.9 Figure 9.29 The second step in reducing the circuit shown in Fig. 9.27.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.10 Find the Thévenin equivalent circuit with respect to terminals a,b for the circuit shown in Fig. 9.30. Figure 9.30 The circuit for Example 9.10.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.10 Figure 9.31 A simplified version of the circuit shown in Fig. 9.30.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.10 Figure 9.32 A circuit for calculating the Thévenin equivalent impedance.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.10 Figure 9.33 The Thévenin equivalent for the circuit shown in Fig. 9.30.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 9.8 The Node-Voltage Method Example 9.11 Use the node-voltage method to find the branch currents I a, I b, and I c in the circuit shown in Fig. 9.34. Figure 9.34 The circuit for Example 9.11.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.11 Figure 9.35 The circuit shown in Fig. 9.34, with the node voltages defined.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 9.9 The Mesh-Current Method Example 9.12 Use the mesh-current method to find the voltages V 1, V 2, and V 3 in the circuit shown in Fig. 9.36. Figure 9.36 The circuit for Example 9.12.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.12 Figure 9.37 Mesh currents used to solve the circuit shown in Fig. 9.36.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 load

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Figure 9.38 The frequency domain circuit model for a transformer used to connect a load to a source.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Reflected Impedance The third term in Eq. 9.64 is called the reflected impedance (Z r ), 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, Z r becomes zero, and Z ab reduces to the self-impedance of the primary coil.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. The load impedance in rectangular form: The reflected impedance in rectangular form:

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.13 The parameters of a certain linear transformer are R 1 = 200 Ω, R 2 = 100 Ω, L 1 = 9 H, L 2 = 4 H and k = 0.5. The transformer couples an impedance consisting of an 800 Ω resistor in series with a 1  F 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.13 Figure 9.39 The frequency-domain equivalent circuit for Example 9.13.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.13 Figure 9.40 The Thévenin equivalent circuit for Example 9.13.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 9.11 The Ideal Transformer An ideal transformer consists of two magnetically coupled coils having N 1 and N 2 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 (L 1 = L 2 = ∞). 3.The coil losses, due to parasitic resistance, are negligible.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Figure 9.41 The circuits used to verify the volts-per-turn and ampere-turn relationships for an ideal transformer.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Determining the Voltage and Current Ratios For unity coupling, the mutual inductance equals

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Voltage relationship for an ideal transformer for k = 1, Current relationship for an ideal transformer

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Figure 9.42 shows the graphic symbol for an ideal transformer. Figure 9.42 The graphic symbol for an ideal transformer.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Determining the Polarity of the Voltage and Current Ratios Dot convention for ideal transformers

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Figure 9.43 Circuits that show the proper algebraic signs for relating the terminal voltages and currents of an ideal transformer.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Ratio of the turns Figure 9.44 Three ways to show that the turns ratio of an ideal transformer is 5.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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  H inductor. If the sinusoidal voltage source (v g ) is generating the voltage 2500 cos 400t V, find the steady-state expressions for: (a) i 1 ; (b) v 1 ; (c) i 2 ; and (d) v 2. Figure 9.45 The circuit for Example 9.14.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. The Use of an Ideal Transformer for Impedance Matching Figure 9.47 Using an ideal transformer to couple a load to a source.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 − j 3 = 7.62 /–156.80°.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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, i R, to lag the source current. is, by 45° when  = 5 krad/s. Figure 9.50 The circuit for Example 9.15.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.15 Figure 9.51 The phasor diagram for the currents in Fig. 9.50.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.16 Figure 9.52 The circuit for 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 V s if we adjust V s so that the amplitude of V L remains constant. Utility companies use this technique to control the voltage drop on their lines.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.16 Figure 9.53 The frequency-domain equivalent of the circuit in Fig. 9.52.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.16 Figure 9.54 The step- by-step evolution of the phasor diagram for the circuit in Fig. 9.53.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 I c on the line current I.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Example 9.16 Figure 9.57 The effect of adding a load-shunting capacitor to the circuit shown in Fig. 9.53 if V L is held constant.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. Summary The general equation for a sinusoidal source is or where V m (or I m ) is the maximum amplitude,  is the frequency,  and is the phase angle. The frequency, , 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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):

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 (  M/|Z 22 |) 2.

Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. 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 (L 1 = L 2 = ∞), and lossless coils (R 1 = R 2 = 0). The circuit behavior is governed by the turns ratio a = N 2 /N 1.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 Higher Education. All rights reserved. CHAPTER 9 Sinusoidal.

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Electronic Circuits, Tenth Edition James W. Nilsson | Susan A. Riedel Copyright ©2015 by Pearson Higher Education. All rights reserved. CHAPTER 9 Sinusoidal.

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