04/03/2015 Hafiz Zaheer Hussain

04/03/2015 Hafiz Zaheer Hussain Circuit Analysis-II Spring-2015 EE -1112 Instructor: Hafiz Zaheer Hussain Email: zaheer.hussain@ee.uol.edu.pk www.hafizzaheer.pbworks.com Department of Electrical Engineering The University of Lahore Week 1 & 2 04/03/2015 Hafiz Zaheer Hussain

Fundamentals of Electric Circuits by Alexander-Sadiku Chapter 9 Sinusoidal Steady-State Analysis 04/03/2015 Hafiz Zaheer Hussain

Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors 9.4 Phasor relationships for circuit elements 9.5 Impedance and admittance 9.6 Kirchhoff’s laws in the frequency domain 9.7 Impedance combinations 04/03/2015 Hafiz Zaheer Hussain

9.1 Introduction We now begin the analysis of circuits in which the source voltage or current is time-varying. In this chapter, we are particularly interested in sinusoidally time-varying excitation, or simply, excitation by a sinusoid A sinusoid is a signal that has the form of the sine or cosine function. A sinusoidal current is usually referred to as alternating current (AC). Such a current reverses at regular time intervals and has alternately positive and negative values. Circuits driven by sinusoidal current or voltage sources are called ac circuits 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ features We are interested in sinusoids for a number of reasons. nature itself is characteristically sinusoidal. The motion of a pendulum The vibration of a string The ripples on the ocean surface etc. Sinusoidal signal is easy to generate and transmit. Sinusoidal play an important role in the analysis of periodic signals. Sinusoid is easy to handle mathematically. The derivative and integral of a sinusoid are themselves sinusoids. For these the sinusoid is an extremely important function in circuit analysis. 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ (2) where Vm = the amplitude of the sinusoid A sinusoid is a signal that has the form of the sine or cosine function. A general expression for the sinusoid, where Vm = the amplitude of the sinusoid ω = the angular frequency in radians/s Ф = the phase 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ (3) A periodic function is one that satisfies v(t) = v(t + nT), for all t and for all integers n. If Only two sinusoidal values with the same frequency can be compared by their amplitude and phase difference. If phase difference is zero( ) , they are in phase; If phase difference is not zero( ) , they are out of phase. 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ (4) 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ (5) Q 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ (6) 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ Example 9.1 Given a sinusoid, , calculate its amplitude, phase, angular frequency, period, and frequency. 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ PP 9.2 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ (7) Trigonometric Identities (9.9) (9.9) 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ (8) A graphical approach may be used to relate or compare sinusoids as an alternative to using the trigonometric identities in Eqs. (9.9) and (9.10). Consider the set of axes shown in Fig. 9.3(a). The horizontal axis represents the magnitude of cosine, while the vertical axis (pointing down) denotes the magnitude of sine. Angles are measured positively counterclockwise from the horizontal, as usual in polar coordinates. This graphical technique can be used to relate two sinusoids. 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ (9) We obtain 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ Example Given v(t) = Vm sin (ωt +10o). Transform to Cosine Solution v(t) = Vm sin (ωt +10o) v(t) = Vm cos (ωt + 10o - 90o) v(t) = Vm cos (ωt – 80o) 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ Example 9.2 Solution: Let us calculate the phase in three ways. The first two methods use trigonometric identities, while the third method uses the graphical approach. ■ METHOD 1 In order to compare v1 and v2 we must express them in the same form. If we express them in cosine form with positive amplitudes, 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ Example 9.2(count..) 04/03/2015 Hafiz Zaheer Hussain

9.2 Sinusoids’ PP 9.2 04/03/2015 Hafiz Zaheer Hussain

Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors 9.4 Phasor relationships for circuit elements 9.5 Impedance and admittance 9.6 Kirchhoff’s laws in the frequency domain 9.7 Impedance combinations 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors A phasor is a complex number that represents the amplitude and phase of a sinusoid. It can be represented in one of the following three forms: Rectangular Polar Exponential where 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , Example 9.3 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , Example 9.3 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , Example 9.3 (count..) 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , PP 9.3 Evaluate the following complex numbers: a. b. Solution: a. –15.5 + j13.67 b. 8.293 + j2.2 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors (2) Mathematic operation of complex number: Addition Subtraction Multiplication Division Reciprocal Square root Complex conjugate Euler’s identity 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors (3) (time domain) (phasor domain) Transform a sinusoid to and from the time domain to the phasor domain: (time domain) (phasor domain) Amplitude and phase difference are two principal concerns in the study of voltage and current sinusoids. Phasor will be defined from the cosine function in all our proceeding study. If a voltage or current expression is in the form of a sine, it will be changed to a cosine by subtracting from the phase. 04/03/2015 Hafiz Zaheer Hussain

Table 9.1 Sinusoid-phasor transformation 9.3 Phasors (3) Table 9.1 Sinusoid-phasor transformation Time domain representation Phasor domain representation 04/03/2015 Hafiz Zaheer Hussain

Lead or Lag seen Via Phasors Such a graphical representation of phasors is known as a phasor diagram. 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , Example 9.4 Transform the following sinusoids to phasors: i = 6cos(50t – 40o) A v = –4sin(30t + 50o) V Solution: I A b. v(t) = -4 sin (30t +50o)V v(t) = 4 cos (30t + 50 + 90o) v(t) = 4 cos (ωt + 140o) 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , Example 9.5 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , Example 9.5 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , PP 9.5 Transform the sinusoids corresponding to phasors: Solution: v(t) = 10cos(wt + 210o) V Since i(t) = 13cos(wt + 22.62o) A 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , Example 9.6 Answer 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , PP 9.6 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors (4) The differences between v(t) and V: v(t) is instantaneous or time-domain representation V is the frequency or phasor-domain representation. v(t) is time dependent, V is not. v(t) is always real with no complex term, V is generally complex. Note: Phasor analysis applies only when frequency is constant; when it is applied to two or more sinusoid signals only if they have the same frequency. 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors (5) Relationship between differential, integral operation in phasor listed as follow: 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , Example 9.7 Use phasor approach, determine the current i(t) in a circuit described by the integrodifferential equation. 04/03/2015 Hafiz Zaheer Hussain

9.3 Phasors , PP 9.7 04/03/2015 Hafiz Zaheer Hussain

Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors 9.4 Phasor relationships for circuit elements 9.5 Impedance and admittance 9.6 Kirchhoff’s laws in the frequency domain 9.7 Impedance combinations 04/03/2015 Hafiz Zaheer Hussain

9.4 Phasor relationships for circuit elements Transform the voltage-current relationship from the time domain to the frequency domain for each element. we will assume the passive sign convention. Voltage and Current are in phase in resistance 04/03/2015 Hafiz Zaheer Hussain

9.4 Phasor relationships for circuit elements 04/03/2015 Hafiz Zaheer Hussain

9.4 Phasor relationships for circuit elements 04/03/2015 Hafiz Zaheer Hussain

9.4 Phasor relationships for circuit elements 04/03/2015 Hafiz Zaheer Hussain

The V-I Relationships for Capacitor 9.4 Phasor relationships for circuit elements The V-I Relationships for Capacitor 04/03/2015 Hafiz Zaheer Hussain

9.4 Phasor relationships for circuit elements 04/03/2015 Hafiz Zaheer Hussain

9.4 Phasor relationships for circuit elements 04/03/2015 Hafiz Zaheer Hussain

Summary of voltage-current relationship 9.4 Phasor relationships for circuit elements Summary of voltage-current relationship Resistor: Inductor: Capacitor: 04/03/2015 Hafiz Zaheer Hussain

Summary of voltage-current relationship 9.4 Phasor relationships for circuit elements Summary of voltage-current relationship Element Time domain Frequency domain R L C 04/03/2015 Hafiz Zaheer Hussain

9.4 Phasor relationships for circuit elements Example 9.8 04/03/2015 Hafiz Zaheer Hussain

9.4 Phasor relationships for circuit elements PP 9.8 If voltage v(t) = 6cos(100t + 30o) is applied to a 50 μF capacitor, calculate the current, i(t), through the capacitor. Answer: i(t) = 30 cos(100t + 120o) mA 04/03/2015 Hafiz Zaheer Hussain

04/03/2015 Hafiz Zaheer Hussain