ELECTRIC CIRCUITS EIGHTH EDITION JAMES W. NILSSON & SUSAN A. RIEDEL

CHAPTER 9 SINUSOIDAL STEADY – STATE ANALYSIS © 2008 Pearson Education

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 © 2008 Pearson Education

CONTENTS 9.5 Kirchhoff ’ s Laws in the Frequency Domain 9.6 Series, Parallel, and Delta-to-Wye Simplifications 9.7 Source Transformations and Thévein- Norton Equivalent Circuits © 2008 Pearson Education

CONTENTS 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 © 2008 Pearson Education

9.1 The Sinusoidal Source A sinusoidal voltage © 2008 Pearson Education

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. © 2008 Pearson Education

  The general equation for a sinusoidal source is (voltage source) or (current source) © 2008 Pearson Education 9.1 The Sinusoidal Source

© 2008 Pearson Education 9.1 The Sinusoidal Source rms value of a sinusoidal voltage source

© 2008 Pearson Education 9.1 The Sinusoidal Source Example: Calculate the rms value of the periodic triangular current shown below. Express your answer in terms of the peak current I p. Periodic triangular current

9.2 The Sinusoidal Response  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. © 2008 Pearson Education

9.3 The Phasor Phasor transform (from the time domain to the frequency domain) © 2008 Pearson Education  The phasor is a complex number that carries the amplitude and phase angle information of a sinusoidal function. P P

9.3 The Phasor The inverse phasor transform (from the frequency domain to the time domain) © 2008 Pearson Education PR

9.4 The Passive Circuit Elements in the Frequency Domain © 2008 Pearson Education Relationship between phasor voltage and phasor current for a resistor The frequency-domain equivalent circuit of a resistor  The V-I Relationship for a Resistor

9.4 The Passive Circuit Elements in the Frequency Domain A plot showing that the voltage and current at the terminals of a resistor are in phase © 2008 Pearson Education

9.4 The Passive Circuit Elements in the Frequency Domain Relationship between phasor voltage and phasor current for an inductor © 2008 Pearson Education  The V-I Relationship for an Inductor

A plot showing the phase relationship between the current and voltage at the terminals of an inductor (θ i = 60°) © 2008 Pearson Education 9.4 The Passive Circuit Elements in the Frequency Domain

© 2008 Pearson Education  The V-I Relationship for a Capacitor 9.4 The Passive Circuit Elements in the Frequency Domain Relationship between phasor voltage and phasor current for a capacitor The frequency domain equivalent circuit of a capacitor

A plot showing the phase relationship between the current and voltage at the terminals of a capacitor (θ i = 60 o ) © 2008 Pearson Education 9.4 The Passive Circuit Elements in the Frequency Domain

  Impedance and Reactance Definition of impedance Z = the impedance of the circuit element © 2008 Pearson Education 9.4 The Passive Circuit Elements in the Frequency Domain

Impedance and reactance values © 2008 Pearson Education 9.4 The Passive Circuit Elements in the Frequency Domain

9.5 Kirchhoff’s Laws in the Frequency Domain   Kirchhoff’s Voltage Law in the Frequency Domain   Kirchhoff’s Current Law in the Frequency Domain © 2008 Pearson Education

9.6 Series, Parallel, and Delta-to-Wye Simplifications Impedances in series © 2008 Pearson Education The equivalent impedance between terminals a and b

9.6 Series, Parallel, and Delta-to-Wye Simplifications Admittance and susceptance values © 2008 Pearson Education

Delta-to-Wye transformations © 2008 Pearson Education 9.6 Series, Parallel, and Delta-to-Wye Simplifications

9.7 Source Transformations and Thévenin-Norton Equivalent Circuits A source transformation in the frequency domain © 2008 Pearson Education

The frequency – domain version of a Thévenin equivalent circuit © 2008 Pearson Education 9.7 Source Transformations and Thévenin-Norton Equivalent Circuits

The frequency – domain version of a Norton equivalent circuit © 2008 Pearson Education 9.7 Source Transformations and Thévenin-Norton Equivalent Circuits

9.8 The Node – Voltage Method Example: Use the node-voltage method to find the branch currents I a, I b, and I c in the circuit shown below. © 2008 Pearson Education

9.9 The Mesh-Current Method Example: Use the mesh-current method to find the voltages V 1, V 2, and V 3 in the circuit shown below. © 2008 Pearson Education

9.10 The Transformer   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. © 2008 Pearson Education

  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 (ω M / |Z 22 |) 2. © 2008 Pearson Education 9.10 The Transformer

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 3 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. © 2008 Pearson Education

9.11 The Ideal Transformer © 2008 Pearson Education  Determining the Voltage and Current Ratios Voltage relationship for an ideal transformer Current relationship for an ideal transformer

Circuits that show the proper algebraic signs for relating the terminal voltages and currents of an ideal transformer © 2008 Pearson Education 9.11 The Ideal Transformer  Determining the Polarity of the Voltage and Current Ratios

Three ways to show that the turns ratio of an ideal transformer is 5 © 2008 Pearson Education 9.11 The Ideal Transformer

Using an ideal transformer to couple a load to a source © 2008 Pearson Education 9.11 The Ideal Transformer  The Use of an Ideal Transformer for Impedance Matching

9.12 Phasor Diagrams A graphic representation of phasors © 2008 Pearson Education

9.12 Phasor Diagrams Example: Using Phasor Diagrams to Analysis a Circuit. © 2008 Pearson Education For the circuit shown at below, 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, i s, by 45° when  = 5 krad/s.

THE END © 2008 Pearson Education