Name (print): Lab (circle): W8 Th8 Th11 Th2 F8 Trigonometric Identities ( cos(θ) = cos(θ) sin(θ) = sin(θ) sin(θ) = cos θ π ) 2 Cosines and Sines of common angles Euler s Formula θ (radians) θ (degrees) cos θ sin θ 0 0 1 0 π/6 30 3/2 1/2 π/4 45 2/2 2/2 π/3 60 1/2 3/2 π/2 90 0 1 Complex Numbers e jθ = cosθ jsinθ e jθ = cosθ jsinθ rectangular form z = ajb a is the real part of z, b is the imaginary part of z polar form z = Ae jθ A is the magnitude of z, θ is the angle of z rectangular polar z = ajb a 2 b 2 e j tan1 (b/a) = Ae jθ polar rectangular z = Ae jθ Acosθ jasinθ = ajb complex conjugate z = ajb z = ajb complex conjugate z = Ae jθ z = Ae jθ Complex Number Properties j = 1 = e jπ/2 = e j3π/2 j 2 = 1 = e jπ = e jπ j 3 = j = e j3π/2 = e jπ/2 j 4 = 1 = e j0 = e j2π = e j2π (see note #1 below) Notes 1. Remember to draw the complex number in the real-imaginary plane to determine the angle θ since the four-quadrant version of the tan 1 function must be used. For example, the phase angle of complex numbers 2j2 and 2j2 are not equal. 2. Complex numbers are easier to multiply and divide when in polar form. 3. Complex numbers are easier to add and subtract when in rectangular form. Prof. Vahe Caliskan 1 of 6 Posted: December 9, 2014
Problem 1 (2 points) Suppose we have entered two row vectors x and y given by x = [1 2 3 4] and y = [4 3 2 1] into Matlab. Determine the Matlab answers to the following operations: (a) x.* y (b) x * y Problem 2 (3 points) We are given the voltage signal v(t) = 5sin(100tπ/3) Volts. (a) What is the frequency f of v(t) in Hertz (Hz)? (b) What is the period T of v(t) in seconds (s)? (c) If we would like to plot 5 cycles of v(t), what time range should we use? Problem 3 (2 points) You are given the transformer circuit shown below where the number of primary turns is N p = 200 and the the number of secondary turns is N s = 400. Recall that v p (t)i p (t) = v s (t)i s (t). i p (t) N p : N s i s (t) v p (t) v s (t) R = 10Ω (a) Determine the secondary voltage v s (t) if the primary voltage v p (t) = 20sin(120πt). (b) Determine the secondary current i s (t) if the primary voltage v p (t) = 20sin(120πt). Problem 4 (2 points) Given the charge q(t) = 2sin(10πt) (in Coulombs), determine the current i(t) and sketch it for two cycles. Prof. Vahe Caliskan 2 of 6 Posted: December 9, 2014
Problem 4 (3 points) For a = 1, b = 0 and c = 0, give the results of each of the following logical operations: (a) f = (a & c) (b & a) (b) g = (a & b) (c & b) (c) q = a & or( b, c) Problem 5 (3 points) You are given the circuit shown below with the dc (constant) voltage source V in = 8V and resistors R 1 = 15Ω and R 2 = 5Ω. Given this information, answer the following questions (using the correct units): I 1 V 1 V in = 8V R 1 =15Ω R 2 =5Ω I 2 V 2 (a) Find I 1. (b) Find V 2. (c) What is the power dissipated in resistor R 2? Problem 6 (3 points) You are given the following matrices: 2 1 2 A = 2 1 0, B = 1 1 2 2 1 1 1 0 3, C = [ 3 2 1 1 2 4 ], x = 2 3 1 (a) Matlab is used to find D = A * B * C. What is the dimension of D? (b) What does Matlab return as an answer to g = max(x)? (c) What does Matlab return as an answer to F = x * x? Prof. Vahe Caliskan 3 of 6 Posted: December 9, 2014
Problem 7 (3 points) Design a logic circuit with three inputs a, b and c and a single output y that implements the following logic expression y = (a bc) (ab c). You are allowed to use all standard gates (AND, OR, NOT, NAND, NOR). Problem 8 (4 points) Show all work for the following problems. (a) Find the magnitude of 2j5. (b) Find the angle (in degrees) of 5j5 (c) Given the complex numbers z 1 = 4j3 and z 1 = 2j5, find the product z 1 z 2 (d) Convert 2j2 2j2 to polar form. Prof. Vahe Caliskan 4 of 6 Posted: December 9, 2014
Problem 9 (4 points) Determine the truth table for the following digital circuit where a and b are inputs and z is the output. You should also include the values of x and y as intermediate outputs in determining z. Show all work. a b x a b x y z z y Problem 10 (4 points) Suppose we are given a circuit that has a 1mH inductor, 20µF capacitor and a 10Ω resistor and is driven by a sinusoidal source with a frequency of 5000 rad/s. (a) What is the impedance of the inductor? (b) What is the impedance of the capacitor? (c) What is the impedance of the resistor? (d) If the three components are in series, what is the total impedance? Prof. Vahe Caliskan 5 of 6 Posted: December 9, 2014
Problem 11 (4 points) Show all work for the following problems. (a) Find the real part of j3e jπ/3. (b) Find the imaginary part of j2e jπ/6. (c) Draw the complex product e jπ/2 (2j3) as a vector in the complex xy (real-imaginary) plane. (d) Draw the complex product e jπ/2 (2j3) as a vector in the complex xy (real-imaginary) plane. Problem 12 (3 points) Consider the circuit shown below with the two current source I s1 and I s2. Write three node equations in terms of node voltages V 1, V 2 and V 3 and put them in standard form. R 4 R 1 V 2 R 3 V 1 V 3 I s1 R 2 I s2 Prof. Vahe Caliskan 6 of 6 Posted: December 9, 2014