STUDY GUIDE

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The two waveforms are the source voltage v(t) in units of volts (V) and the total system current i(t) in units of milliamps (mA).

Figure 1 – Voltage and Current Waveforms

a. (5 points) Determine the period of each waveform.

b. (5 points) What is the frequency of the voltage signal? Show your work.

c. (5 points) What is the phase difference between v(t) and i(t)? Show your work.

2. (15 points) For the series-parallel network shown in Figure 2.

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Figure 2 – Network for Problem 2

a. (5 points) Calculate the total impedance ZT as seen from the voltage source.

b. (5 points) Determine the source current IS.

c. (5 points) State whether, and why, the total system is overall inductive or capacitive.

3. (10 points) Consider the circuit in Figure 3.

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Figure 3 – Network for Problem 3

a. (5 points) Find the load impedance ZL required to deliver maximum power to the load.

b. (5 points) Find the maximum power delivered to the load.

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4. (15 points) For the network shown above in Figure 4, a 208 V power supply with an operating frequency of 60 Hz is connected to an industrial load.

Figure 4 – Network for Problem 4

a. (10 points) Find and draw the system power triangle with appropriate labels and phase angle.

b. (5 points) To correct the system to a unity power factor, what type of component is required, and what is its value? Hint: It’s not a capacitor!

IL

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5. (15 points) For the RLC series circuit shown in Figure 5, a 10 V power supply at a resonant frequency of 5 kHz is connected in series to a resistor, inductor, and capacitor.

Figure 5 – Network for Problem 5

a. (5 points) What is the value of capacitive reactance XC at the 5 kHz resonant frequency

b. (5 points) Calculate the Q of the circuit and the resulting bandwidth.

c. (5 points) Calculate the cutoff (sideband) frequencies, and calculate the power dissipated by the circuit at these frequencies.

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6. (20 points) For the filter shown in Figure 6,

Figure 6 – Filter Circuit for Problem 6

a. (5 points) What type of filter is this?

b. (5 points) Determine the value of the cutoff frequency.

c. (5 points) Determine the approximate gain,

out V

in

V A

V 

in the pass band.

d. (5 points) Determine the approximate gain

out V

in

V A

V 

in the stop band.

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7. (10 points) For the ideal transformer of Figure 7,

Figure 7 – Network for Problem 7

a. (5 points) Find the primary-side current Ip using reflected impedance principles.

b. (5 points) Find the output voltage VL across load resistor RL.

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Extra Credit (10 points) A balanced a-b-c sequenced source with Van = 1200 V is connected to a -connected load with Z = 30  per phase. Find line voltage (Vab), phase voltage (VAB), phase current (IAB), line current (Ia), and total power delivered to the load (PT). Just like the DC Final last semester, you are still required to answer this even though it’s extra credit!

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Standardized Formula Sheet

Voltage, Current, and Resistance V = W/Q (V = J/C) [V] I = Q/t (A = C/sec) [A] Q = 1.602 x 10-19 C/e- R = l/A [Ω] batt. life = cap./disch. rate

Ohm’s Law and Power I = V/R  V = IR  R = V/I P = W/t (W = J/sec) [W] P = VI = I2R = V2/R [W] 1 HP = 746 W  = (Pout/Pin) x 100%

Series and Parallel Resistance RT = R1 + R2 + R3 + ··· + RN (series) RT = 1/(1/R1 + 1/R2 + 1/R3 + ··· + 1/RN) (parallel) RT = R1R2/(R1 + R2) (2 parallel)

KVL and KCL KVL: ∑V = 0, ∑VRISES = ∑VDROPS KCL: ∑IIN = ∑IOUT

Voltage and Current Divider VDR: VX = E(RX/RT) CDR: IX = IT(RT/RX) CDR: IX = IT(Ropposite/(R1 + R2)) (2 elements)

Maximum Power Transfer Pmax = V2TH/4RTH ZLD = Z*TH

Capacitors C = εA/d [F] W = (1/2)CV2 [J] CT = 1/(1/C1 + 1/C2 + 1/C3 + ··· + 1/CN) (series) CT = C1C2/(C1 + C2) (2 series) CT = C1 + C2 + C3 + ··· + CN (parallel) iC = CdvC/dt νC = E(1 – e-t/τ) (charging) νC = Ee-t/τ’ (discharging) τ = RchargeC and τ’ = RdischargeC νC = Vf + (Vi – Vf)e-t/τ

Inductors L = μN2A/l [H] W = (1/2)LI2 [J] LT = L1 +L2 + L3 + ··· + LN (series) LT = 1/(1/L1 + 1/L2 + 1/L3 + ··· + 1/LN) (parallel) LT = L1L2/(L1 + L2) (2 parallel) vL = LdiL/dt iL = I0(1 – e-t/τ) (storage) iL = I0e-t/τ’ (release) τ = L/Rstorage and τ’ = L/Rrelease iL = If + (Ii – If)e-t/τ

Sinusoidal Equations v(t) = VMAX sin(ωt + θ) v(t) = VMAX sin(2πft + θ) v(t) = VMAX sin(2πt/T + θ) ω = 2πf [rad/s] f = 1/T [Hz] Radians = (π/180°) x (degrees) Degrees = (180°/π) x (radians) VRMS = VMAX/ = 0.707VMAX IRMS = IMAX/ = 0.707IMAX Δθ = Δt/T x 360°

AC Impedance XL = ωL = 2πfL XC = 1/ωC = 1/(2πfC) Z = R + jX ZR = R + j0 = R0° ZL = jXL = XL 90° ZC = -jXC = XC -90°

AC Power S = P + jQ S = VRMSIRMS* = VRMSIRMS (θv – θi) S = VRMSIRMS [VA] P = V2/R = I2R = S cos(θ) [W] Q = V2/X = I2X = S sin(θ) [VAR] pf = cos(θ) = P/S θ = (θv – θi)

Transformers a = Npri/Nsec = Vpri/Vsec = Isec/Ipri Vsec = (1/a)Vpri and Isec = aIpri Zpri reflected = a2Zsec

Filters Gp (dB) = 10 log10(Pout/Pin) Gv (dB) = 20 log10(Vout/Vin) RC LP and HP filter: fC = 1/(2πRC) RC LP filter: Vout/Vin = XC/√(R2 + X2C) -tan-1(R/XC) RC HP filter: Vout/Vin = R/√(R2 + X2C) tan-1(XC/R)

Series Resonance ω0 = 1/√(LC) and f0 = 1/(2π√(LC)) ω1 = -(R/2L) + √[(R/2L)2 + 1/LC] and f1 = ω1/2π ω2 = (R/2L) + √[(R/2L)2 + 1/LC] and f2 = ω2/2π BW = ω2 – ω1 = f2 – f1 Q = ω0/BW = ω0L/R = 1/ω0RC ω1 ≈ ω0 – BW/2 and ω2 ≈ ω0 + BW/2 (Q ≥ 10)

Parallel Resonance ω0 = 1/√(LC) and f0 = 1/(2π√(LC)) ω1 = -(1/2RC) + √[(1/2RC)2 + 1/LC] and f1 = ω1/2π ω2 = (1/2RC) + √[(1/2RC)2 + 1/LC] and f2 = ω2/2π BW = ω2 – ω1 = f2 – f1 Q = ω0/BW = ω0RC = R/ω0L ω1 ≈ ω0 – BW/2 and ω2 ≈ ω0 + BW/2 (Q ≥ 10)

3φ Systems PT = √(3) VLIL cosθφ = 3Pφ [W] QT = √(3) VLIL sinθφ = 3Qφ [VAR] ST = √(3) VLIL = 3Sφ [VA] Y source/load: Vab = √(3)30° Van Δ load: Ian = √(3)-30° Iab Δ → Y: ZY = ZΔ/3

Pulse Waveforms f = 1/T Duty cycle = (tpulse width/T) x 100% R-C circuits: νC = Vf + (Vi – Vf)e-t/RC

Nonsinusoidal Circuits Fourier series: f(t) = A0 + A1 sin ωt + A2 sin 2ωt + ··· + AN sin nωt + B1 cos ωt + B2 cos 2ωt + ··· + BN cos nωt