Name: Lab: M8 M2 W8 Th8 Th11 Th2 F8 Trigonometric Identities cos(θ) = cos(θ) sin(θ) = sin(θ) sin(θ) = cos 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 ( θ π ) 2 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 Impedances R Z R = R, L Z L = jωl, C Z C = 1/(jωC) = j/(ωc) Series: R 1 R 2 or Z 1 Z 2, Parallel: R 1 R 2 = R 1R 2 R 1 R 2 or Z 1 Z 2 = Z 1Z 2 Z 1 Z 2 Ohm s Law, Voltage Division, Current Division Ohm s Law Assuming passive sign convention: V = IR or V = IZ R 1 R 2 Voltage Division For R 1 and R 2 in series with V s : V 1 = V s, V 2 = V s R 1 R 2 R 1 R 2 Current Division For R 1 and R 2 in parallel with I s : I 1 = I s, I 2 = I s R 1 R 2 R 1 R 2 Voltage Division For Z 1 and Z 2 in series with V Z 1 s : V1 = Vs, Z 1 Z V Z 2 2 = Vs 2 Z 1 Z 2 Current Division For Z 1 and Z 2 in parallel with Īs: Z 2 Z 1 Ī 1 = Ī s, Z 1 Z Ī2 = Ī s 2 Z 1 Z 2 R 2 R 1
Opamp 1 (5 points) Consider the following circuit where the opamp is ideal. Recall that for an ideal opamp, V = V, I = 0A and I = 0A. Hint: Start by determining V then use KVL, KCL and Ohm s Law. I R2 R 2 =10kΩ V in =1V I R1 R 1 =2kΩ V out (a) Find the current I R1. (b) Find the current I R2. (c) Find the output voltage V out. (d) Find the power dissipation P diss,r1 in resistor R 1. (e) Find the power dissipation P diss,r2 in resistor R 2. Opamp 2 (5 points) Consider the following circuit where the opamp is ideal. Recall that for an ideal opamp, V = V, I = 0A and I = 0A. Hint: Start by determining V then use KVL, KCL and Ohm s Law. V in =1V R 2 =2kΩ V out I R1 R 1 =1kΩ I R2 (a) Find the current I R1. (b) Find the current I R2. (c) Find the output voltage V out. (d) Find the power dissipation P diss,r1 in resistor R 1. (e) Find the power dissipation P diss,r2 in resistor R 2.
Truth Tables and Logic Functions (6 points) Consider the following logic circuit with inputs x and y and output z. (a) (4 points) Determine the truth table for the circuit. (b) (2 points) Write z as a logical function of x and y. x y z Logic Function Implementation (4 points) Given the logical function w = [ xyz (xy) (x z) ], synthesize a logic circuit with inputs x, y, z and output w. Hint: start working with the innermost parentheses first.
Equivalent Resistance Calculation (6 points, 3 points each) Find the equivalent resistance R eq for each of the circuits shown below. 8kΩ 3kΩ 2kΩ R eq 2kΩ 3kΩ 6kΩ R eq 6kΩ 8kΩ Logic Function Implementation (4 points) Given the logical function w = [ x yz x(yz) (x y) z ], synthesize a logic circuit with inputs x, y, z and output w. Hint: start working with the innermost parentheses first and be careful with how you implement x(yz) and (x y) z,
Complex Impedance Calculation (5 points) Suppose you are given a 50Ω resistor and a 0.2H inductor operating at ω = 1000rad/s. (a) Find the impedance Z R of the resistor. (b) Find the impedance Z L of the inductor. (c) Find the parallel combination Z p of the resistor and inductor. (d) Simplify and express Z p in polar form as A θ. (e) Simplify and express Z p in rectangular form as ajb. LED Current Limit Calculation (5 points) In the circuit shown below, the resistor R s is used to limit the current through the LED. When the LED is on (conducting), its voltage is V LED = 2V. 5V V LED I LED R s (a) Determine the value of R s so that the LED current I LED = 2mA when it is on. (b) Find the power dissipation P diss,rs in resistor R s when the LED is on. (c) Find the power dissipation P diss,led in the LED when it is on. (d) Determine the power supplied by the voltage source when the LED is on.
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