1. Sinusodial functions problems (25pts): 1.1. Angle Unit Conversion (4pts) 1.1.1. Convert π 6 rad to degrees. Answer(1pt): 1.1.2. Convert 2π 3 rad to degrees. Answer(1pt): 1.1.3. Convert 270◦ to radians. Answer(1pt): 1.1.4. Convert 30◦ to radians. Answer(1pt): 1.2. Period/Cycle/Time Conversion (5pts): 1.2.1. Given the fundamental frequency of a sinusodial function is equal to 50Hz, find the period T. in seconds. Answer(1pt): 2π 1.2.2. Given the voltage function on time domain: v(t) = 138, 000 cos(1000t + 2π 3 ) + 138, 000 cos(2000t + 3 )V . What is the fundamental frequency in Hertz of v(t)? Answer(2 pts): 1.2.3. How many cycles are in 0.5 seconds of a sinusodial signal with fundamental frequency of 10kHz? Answer(1pt): 1.2.4. How many seconds are in 2 cycles of a sinusodial signal with fundamental frequency of 10,000 rad/sec? Answer(1pt): 1.3. Function Manipulation (5pts): √ 1.3.1. For the function i(t) = −200 2 sin(ωt), rewrite the equivalent function in terms of cosine. Answer(3pt): 2 √ 1.3.2. For the function v(t) = 10 2 cos(1000t + π3 ) what it is the value of the function at 0.5 cycles. Answer(2pt): 1.4. Phasor Representation(6pts): √ √ If v(t) = 100 2 cos(ωt + α◦ ) V and i(t) = 168 2 sin(ωt + β ◦ ) A 1.4.1. Find the maximum value for v(t) and i(t) Answer(1pt): Vmax= , Imax= 1.4.2. Find the rms value for v(t) and i(t) Answer(1pt): Vrms= , Irms= 1.4.3. Find the phasor expression in polar and rectangular form if the current is the reference Answer(4pt): Polar V= , Rectangular V= , Polar I= 3 , Rectangular I= 2. Phasor Domain Operations (25pts): √ 2.1. Supposed two voltage functions in the time domain are given by v1 (t) = 69 2 cos(ωt) V and v2 (t) = √ − → − → 69 2 cos(ωt − 120◦ ) V. Use the phasor polar form analysis to find the phasor polar form of V1 − V2 , and the simplified time domain expression. Answer(5pt): − → − → Polar representation of V1 − V2 = , Time Domain v1 (t) − v2 (t) = → − → − 2.2. Given the phasor in rectangular form Z = 10 + j10 Ω. Find the conjugate denoted by Z ∗ in rectangular and polar form. Answer(5pt): → − Polar representation of Z ∗ = → − , Rectangular form Z ∗ = → − → − → − → − → −→ − 2.3. Given the phasors V = Vr +jVx V, I = Ir +jIx A, and Z = Zr +jZx Ω. Show that if the phasor S = V I ∗ , → − then S can be written as: (Vr )2 + (Vx )2 Zr − jZx hint: Use ohms law. Answer(5pt): − → − → − → 2.4. Consider the following phasors in rectangular form: S1 = 1 + j1, S2 = −1 + j1, and S3 = −0.5 − j0.2. Find − → − → − → S1 + S2 + S3 in rectangular and polar form. Answer(5pt): Polar representation= , Rectangular form = − → − → −→ − →− → − → − → 2.5. Consider two impedances Z1 = 1 + j1Ω and Z2 = 2 − j4Ω find the polar form for ZT = Z1 Z2 /(Z1 + Z2 )Ω Answer(5pt): Polar representation= 4 3. Passive Circuit Components (25pts): 3.1. Show that a series L-R circuit can be considered equivalent to a parallel L-R circuit, as shown in the Figure 1, if the relationships between the resistance and reactance quantities are as follows 1 Answer(10pt): RS = RP XP2 2 + X2 Ω RP P XS = 2 RP XP Ω 2 RP + XP2 RS2 + XS2 Ω RS R2 + XS2 XP = S Ω XS RP = RS XS + Vs − + Vs − RP XP Figure 1: Series and Parallel circuit equivalent 3.2. For the following circuit find the current √ IT in the phasor domain, polar form, rectangular form, and also time domain. Given that vs (t) = 110 2 cos(1000t)V Answer(15pt): Polar representation= , Rectangular form = , Time Domain= IL1 IT + Vs − IR1 L1 50 mH R1 10 kΩ IL2 L2 20 mH C1 10 µF Figure 2: AC circuit analysis problem 1 Disclaimer: This problem is copied from the Course Text Book AC Circuits and Power System Practice by Graeme Vertigan (Chapter 2, problem 6), but the labels were changed. 5 4. Resonant Circuits (25pts) 4.1. Solve from Chapter 2 section 2.8 of the Text Book, problem 5 only B, and in C only find the capacitive reactance. Do not plot a phasor diagram. Answer(13pt): 4.2. Solve from Chapter 2 section 2.8 of the Text Book, problem 11 only A and D. Answer(12pt): 6
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