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

(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 = √X²_L + R²

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 + R²

(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_pI_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. 8.0×10²H

B. 7.5×10⁰H

C. 1.6×10⁰H

D. 5.1×10^-2H

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^-2H

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 = √8² + 6²= 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