Alternating Current

Alternating current, whose magnitude and direction change periodically, is the form in which electricity reaches our homes, because it can be stepped up or down easily with transformers and transmitted over long distances with small losses. In this chapter you will study an AC voltage v = vm sin(omega t), the meaning of root mean square values, and why the 220 V of household supply is an rms value with a peak of about 311 V. You will use phasors to represent alternating voltages and currents and learn how current behaves in a pure resistor, a pure inductor and a pure capacitor, including inductive and capacitive reactance. You will then analyse a series LCR circuit using a phasor diagram, find its impedance and phase angle, and study resonance, sharpness of resonance and the quality factor, as used in radio tuning. The chapter explains power in AC circuits, the power factor, wattless current and why capacitor banks improve the power factor. Finally, you will learn the principle, working and energy losses of a transformer.

Detailed Worksheet: Alternating Current Section A - Definitions (10 marks) 1. Define the rms value of an alternating current. How is it related to the peak value? (2 marks) 2. Define inductive reactance and capacitive reactance. How does each depend on frequency? (2 marks) 3. What is impedance in a series LCR circuit? Write its formula. (2 marks) 4. Define power factor. What is its value for a pure resistor and for a pure inductor? (2 marks) 5. What is wattless current? (2 marks) Section B - Calculations and Applications (15 marks) 6. A 100 ohm resistor is connected to a 220 V, 50 Hz supply. Find (i) the peak voltage, (ii) the rms current and (iii) the average power consumed. (3 marks) 7. A pure inductor of 44 mH is connected to a 220 V, 50 Hz supply. Find the inductive reactance and the rms current in the circuit. (3 marks) 8. A 60 microF capacitor is connected to a 110 V, 60 Hz supply. Find the capacitive reactance and the rms current. (3 marks) 9. A 200 ohm resistor and a 15.0 microF capacitor are connected in series to a 220 V, 50 Hz supply. Calculate the capacitive reactance, impedance and rms current in the circuit. (3 marks) 10. A series LCR circuit with L = 5.0 H, C = 80 microF and R = 40 ohm is connected to a 230 V variable frequency source. Find the resonant angular frequency, the rms current at resonance and the quality factor. (3 marks) Section C - Diagrams (10 marks) 11. Draw the phasor diagram for a series LCR circuit with XL > XC and show the phase angle between voltage and current. Derive the expression for impedance from it. (4 marks) 12. Draw graphs of voltage and current against omega t for (i) a pure inductor and (ii) a pure capacitor, showing the phase relation in each case. (3 marks) 13. Draw a labelled diagram of a step-up transformer showing the primary and secondary coils and the laminated soft iron core. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A step-down transformer converts 2,200 V to 220 V. The primary has 3,000 turns. Find the number of turns in the secondary. If the output power is 4.4 kW, find the secondary and primary currents for an ideal transformer. Explain four causes of energy loss in real transformers and how each is reduced. (5 marks) 15. A series LCR circuit with R = 3 ohm, L = 25.48 mH and C = 796 microF is connected to an AC source of peak voltage 283 V and frequency 50 Hz. Find (i) the impedance, (ii) the peak current, (iii) the power factor and phase angle, and (iv) the average power dissipated. (5 marks) 16. Explain why a choke coil is preferred over a resistor to control current in a fluorescent tube circuit. Why do industries install capacitor banks? Explain the idea of wattless current using a phasor diagram. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Use pi = 3.14 where needed and show units at every step. Draw neat phasor diagrams.