Electrical Network Analysis Instructor: Dr Alina Mirza Lab Engr: Amir Ali Credit: 3+1 1 Books Textbooks: Fundamentals of Electric Circuits, 5th Edition, by Charles K. Alexander & MatthewN.O.Sadiku, McGraw Hill 2003. Reference Books: Electric Circuits Analysis, 7th Edition, by Hayat Jr(late, Steven Durbin Basic Engineering Circuit Analysis, 7th Edition, by J.David Irwin 2 Topics 1. Topics covered in the Course and Level of Coverage Sinusoidal Input and Steady State Analysis 6hrs AC Power Circuit Analysis 6hrs Poly Phase System 5 hrs Single Phase Three Wire System 5hrs Magnetically Coupled Circuits Complex Frequency & Frequency Response Circuit Analysis in S-Domain Two port Network 6 hrs 6 hrs 6 hrs 6 hrs 3 Grading Policy 1. Course Assessment Exam: 2 OHT and 1 Final Examination Home work: 3 Assignments Lab reports: 14Reports Quizzes: 3 Quizzes Grading: Quizzes: 10% Assignments: 10% 2 One Hour Tests 30% (15% each) (OHTs): Final Exam: Semester Project/Lab: 50% 15% out marks of lab 4 Course Learning Outcome (CLOs) 1. Course Outcomes and their Relation to Program Outcomes (Mapping CLO to PLO) Course Learning Outcome (CLOs) PLOs Learning Level Identify the use of phasor technique and sinusoidal steady state CLO 1 analysis for linear circuits. 1 C1 CLO 2 Perform AC power analysis and describe three phase circuits. Illustrate RLC circuits and their frequency response. Analyze CLO 3 them using Laplace transform. 4 C2 4 C3 CLO 4 Observing the working of different circuits in lab. An ability to work effectively as an individual or in a team in lab CLO 5 or in any assigned group or individual task. 5 P3 9 A3 5 Alexander-Sadiku Chapter 9 Sinusoidal Steady-State Analysis 6 Sinusoids and Phasor Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Motivation Sinusoids’ features Phasors Phasor relationships for circuit elements Impedance and admittance Kirchhoff’s laws in the frequency domain Impedance combinations 7 9.1 Motivation (1) How to determine v(t) and i(t)? vs(t) = 10V How can we apply what we have learned before to determine i(t) and v(t) ? 8 9.2 Sinusoids (1) • A sinusoid is a signal that has the form of the sine or cosine function. • A general expression for the sinusoid, v(t ) = Vm sin(t + ) where Vm = the amplitude of the sinusoid ω = the angular frequency in radians/s Ф = the phase 9 9.2 Sinusoids (2) A periodic function is one that satisfies v(t) = v(t + nT), for all t and for all integers n. T= 2 f = 1 Hz T = 2f • 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. 10 Trigonometric Identities Example 12 9.2 Sinusoids (3) Example 1 Given a sinusoid, 5 sin( 4t − 60 o ), calculate its amplitude, phase, angular frequency, period, and frequency. Solution: Amplitude = 5, phase = –60o, angular frequency = 4 rad/s, Period = 0.5 s, frequency = 2 Hz. 13 9.2 Sinusoids (4) Example 2 Find the phase angle between i1 = −4 sin( 377 t + 25 o ) and i2 = 5 cos(377 t − 40 o ), does i1 lead or lag i2? Solution: Since sin(ωt+90o) = cos ωt i2 = 5 sin(377 t − 40 o + 90 o ) = 5 sin(377 t + 50 o ) i1 = −4 sin(377 t + 25 o ) = 4 sin(377 t + 180 o + 25 o ) = 4 sin(377 t + 205 o ) therefore, i1 leads i2 155o. 14 Example 15 Example Method 1: Convert into Cosine form 16 Figure 9.5 17 9.3 Phasor (1) • 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: a. Rectangular z = x + jy = r (cos + j sin ) b. Polar z = r j z = re c. Exponential where r= x2 + y2 = tan −1 y x 18 9.3 Phasor (3) Mathematic operation of complex number: 1. Addition z1 + z 2 = ( x1 + x 2 ) + j ( y1 + y 2 ) 2. Subtraction z1 − z 2 = ( x1 − x2 ) + j ( y1 − y2 ) 3. Multiplication z1 z 2 = r1r2 1 + 2 4. Division z1 r1 = 1 − 2 z 2 r2 5. Reciprocal 1 1 = − z r 6. Square root z = r 2 7. Complex conjugate z = x − jy = r − = re − j 8. Euler’s identity e j = cos j sin 19 9.3 Phasor (2) Example 3 • Evaluate the following complex numbers: a. [(5 + j2)( −1 + j4) − 5 60 o ] b. 10 + j5 + 340o + 10 30o − 3 + j4 Solution: a. –15.5 + j13.67 b. 8.293 + j2.2 20 9.3 Phasor (4) • Transform a sinusoid to and from the time domain to the phasor domain: v(t ) = Vm cos(t + ) (time domain) V = Vm (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. 21 9.3 Phasor (5) Example 4 Transform the following sinusoids to phasors: i = 6cos(50t – 40o) A v = –4sin(30t + 50o) V Solution: a. I = 6 − 40 A b. Since –sin(A) = cos(A+90o); v(t) = 4cos (30t+50o+90o) = 4cos(30t+140o) V Transform to phasor => V = 4140 V 22 9.3 Phasor (6) Example 5: Transform the sinusoids corresponding to phasors: a. V = − 10 30 V b. I = j(5 − j12) A Solution: a) v(t) = 10cos(t + 210o) V 5 ) = 13 22.62 12 b) Since I = 12 + j5 = 122 + 52 tan−1 ( i(t) = 13cos(t + 22.62o) A 23 9.3 Phasor (7) 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. 24 9.3 Phasor (8) Relationship between differential, integral operation in phasor listed as follow: v(t ) V = V dv dt j V vdt V j 25 9.3 Phasor (9) Example 6 Use phasor approach, determine the current i(t) in a circuit described by the integro-differential equation. di 4i + 8 idt − 3 = 50 cos(2t + 75) dt Answer: i(t) = 4.642cos(2t + 143.2o) A 26 9.3 Phasor (11) The answer is YES! Instead of first deriving the differential equation and then transforming it into phasor to solve for Vo, we can transform all the RLC components into phasor first, then apply the KCL laws and other theorems to set up a phasor equation involving Vo directly. 27 9.4 Phasor Relationships for Circuit Elements (1) Resistor: Inductor: Capacitor: 28 9.4 Phasor Relationships for Circuit Elements (2) Summary of voltage-current relationship Element Time domain R v = Ri L v=L C dv i=C dt di dt Frequency domain V = RI V = jLI V= I jC 29 9.4 Phasor Relationships for Circuit Elements (3) Example 7 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 + 60o) mA 30 9.5 Impedance and Admittance (1) • The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms Ω. V Z = = R + jX I where R = Re, Z is the resistance and X = Im, Z is the reactance. Positive X is for L and negative X is for C. • The admittance Y is the reciprocal of impedance, measured in siemens (S). 1 I Y= = Z V 31 9.5 Impedance and Admittance (2) Impedances and admittances of passive elements Element R Impedance Z=R L Z = j L C 1 Z = jC Admittance 1 Y= R Y= 1 j L Y = jC 32 9.5 Impedance and Admittance (3) = 0; Z = 0 Z = j L → ; Z → = 0; Z → Z= 1 jC → ; Z = 0 33 9.5 Impedance and Admittance (4) After we know how to convert RLC components from time to phasor domain, we can transform a time domain circuit into a phasor/frequency domain circuit. Hence, we can apply the KCL laws and other theorems to directly set up phasor equations involving our target variable(s) for solving. 34 9.5 Impedance and Admittance (5) Example 8 Refer to Figure below, determine v(t) and i(t). vs = 5 cos(10t ) Answers: i(t) = 1.118cos(10t – 26.56o) A; v(t) = 2.236cos(10t + 63.43o) V 35 9.6 Kirchhoff’s Laws in the Frequency Domain (1) • Both KVL and KCL are hold in the phasor domain or more commonly called frequency domain. • Moreover, the variables to be handled are phasors, which are complex numbers. • All the mathematical operations involved are now in complex domain. 36 9.7 Impedance Combinations (1) • The following principles used for DC circuit analysis all apply to AC circuit. • For example: a. voltage division b. current division c. circuit reduction d. impedance equivalence e. Y-Δ transformation 37 9.7 Impedance Combinations (2) Example 9 Determine the input impedance of the circuit in figure below at ω =10 rad/s. Answer: Zin = 32.38 – j73.76 38