**Simple alternating current (A.C.) circuits**

Most present day household and industrial power distribution systems operate with alternating current. Any appliance plugged into a wall outlet uses a.c. and even many battery powered devices such as radio receivers, cordless phones make use of d.c. power supplies derived from a.c. mains source; circuit in modern communication equipment also makes extensive use of a.c.

**(A)Graphical representation**

Graphical illustration of an a.c. circuit

Source: www.commons.wikimedia.org

**(B)Peak and root mean square value of a.c.**

The r.m.s value of an alternating current is defined as that value of steady current which would dissipate heat at the same

rate in a given resistance.

*I*_{r.m.s}* = I*_{p}* / √2 *

* =0.7071*_{p}

Sine representation of a.c. current/voltage.

Source: www.commons.wikimedia.org

**Note:** the average of any sinusoidal current alternating over any number of cycles is zero and half average value is given as *I*_{av}* = 0.637 I*_{p}

**(C) Alternating currents in resistors, inductors and capacitors (R.L.C) **

Variation of current and voltage in a resistor

Source: www.electronics.tutorials.ws

**Circuit elements with alternating current**

Circuit element | Amplitude relation | Circuit Quantity | Phase of V |

Resistor | V_{R} = IR | R | In phase with i |

Inductor | V_{L} = IX_{L} | X_{L} = wL | Leads i by 90° |

Capacitor | V_{c} = IX_{c} | X_{c} = 1/wC | Lags i by 90° |

**(D)The L.R.,R.C.and L.R.C. series circuit**

The overall opposition to the flow of current in the a.c. a circuit by any two or all of the three circuit elements (R, L and C) is called **impedance** and is usually represented by **Z** while **R** stands for **resistor**, **L** stands for **inductor** and **C** stands for **capacitor**.

* Z*_{RL}* = √X*^{2}_{L}* + R*^{2}

*The phase angle between voltage and current is given as:*

* ¢*_{RL}* = tan*^{-1}* (X*_{L}* / R)*

L.R circuit with switch

Source: www.commons.wikimedia.com

R.C series circuit

Source: www.commons.wikimedia.org

L.C series circuit

Source: www.commons.wikimedia.org

**(E)Vector diagram of series RLC. circuit**

RLC circuit

Source: www.commons.wikimedia.org

*Z*_{RLC}* = √(X*_{L}* – X*_{C}*)*^{2}* + R*^{2}

**(F) Resonance and power in RLC circuit**

Graph of current versus frequency

Source: www.researchgate.net

**(G)power in a.c circuit**

Power absorbed in any a.c circuit is given by the equation P=iV

It average power *P*_{av}* = V*_{p}*I*_{p}* cos ¢*

**Past questions**

1.Which of the following conditions occur at resonance in a series RLC circuit? (Wassce 2014)

I.Impedance equals resistance

II.Current is maximum

III.Reactances are maximum

IV. Current and voltage are 90° out of phase.

A. I and II

B. I and III only

C. I, II and III only

D. I,II, III and IV

**Answer:** A

2. A coil of inductance 0.12H and resistance 4 ohms, is connected across a 240V, 50Hz supply. Calculate the current through it. [π=3.14]. (Wassce 2013)

A. 6.3A

B. 33.3A

C. 37.2A

D. 40.0A

**Answer: **A

**Solution:** X_{L} =wL =2πFL =2×3.14×50×0.12= 37.68

V_{L} / X_{L} = 240/37.68 =6.3A

3.An inductor is connected to a 24V, 50Hz mains supply. If the current through the inductor is 1.5A, calculate the inductance of the inductor. (π=22/7). (Wassce 2012)

A. 8.0×10^{2}H

B. 7.5×10^{0}H

C. 1.6×10^{0}H

D. 5.1×10^{-2}H

**Answer:** D

**Solution:** X_{L} = v/I = 2πFL

F=50Hz, I=1.5A, V=24

L=V/2πFl = 24/2×22/7×50×1.5

=5.1×10^{-2}H

4.the unit of capacitance is? (Wassce 2011)

A. Farad

B. Coulomb

C. Henry

D. Ohm

**Answer:** A

5.

In the series a.c circuit shown above, the p.d across the inductor is 8V_{r.m.s} and that across the resistor is 6 V_{r.m.s}. The effective voltage is? (Jamb 2016)

A. 10V

B. 2V

C. 14V

D. 48V

**Answer:** A

**Solution**: VT = √8^{2} + 6^{2}= 10V

6.

The net capacitance in the circuit shown above is? (Jamb 2017)

A. 8.0uF

B. 6.0uF

C. 2.0uF

D. 4.0uF

**Answer:** C

**Solution:** C_{T} = 4×(2+2)/4+2+2 = 16/8 =2uF

7.capacitors are used in the induction coil to? (Jamb 2013)

A. Control circuit

B. Dissipate energy

C. Prevent Electric Sparks

D. Prevent distortion of Electric fields

**Answer:** C

8.In an a.c circuit, the ratio of r.m.s value to peak value of current is? (Jamb 2012)

A. 1/√2

B. √2

C. 2

D. ½

**Answer:** A

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