**HIGHLIGHTS**

1. Kirchhoff’s laws :

First law : 1: currents entering = 1: currents leaving

Second law : 1: potential rises = 1: potential drops.

2. Applications of Kirchhoffs laws include the following:

(i) Branch-current method

(ii) Maxwell’s loop (or mesh) current method

(iii) Nodal voltage method.

3. The solutions of the networks involve the use of the following theorems:

- Superposition theorem

- Thevenin’s theorem

- Nortan’s theorem

- Maximum power transfer theorem

- Delta star transformation

- Compensation theorem

- Reciprocity theorem

- Millman’s theorem.

**OBJECTIVE TYPE QUESTIONS
**

**Choose the Correct Answer**

1. Kirchhoff’s current law states that

(a) net current flow at the junction is positive

(b) algebraic sum of the currents meeting at the junction is zero

(c) no current can leave the junction without some current entering it

(d) total sum of currents meeting at the junction is zero.

2. According to Kirchhoff’s voltage law, the algebraic sum of all IR drops and e.m.fs. in any closed loop of a network is always

(a) negative (b) positive

(c) determined by battery e.m.fs. (d) zero.

3. Kirchhoff’s current law is applicable to only

(a) junction in a network (b) closed loops in a network

(c) electric circuits (d) electronic circuits.

4. Kirchhoff’s voltage law is related to

(a) junction voltages (b) battery e.m.fs. (c) IR drops (d) both (a) and (h)

(e) none of the above.

5. Superposition theorem can be applied only to circuits having

(a) resistive elements (b) passive elements

(c) non-linear elements (d) linear bilateral elements.

6. The concept on which superposition theorem is based is

(a) reciprocity (b) duality (c) non-linearity (d) linearity.

7. Thevenin resistance R1h is found

(a) by removing voltage sources along with their intemal resistances

(b) by short-circuiting the given two terminals

(c) between any two ‘open’ terminals

(d) between same open terminals as for E_{th}·

8. An ideal voltage source should have

(a) large value of e.m.f. (b) small value of e.m.f.

(c) zero source resistance (d) infinite source resistance.

9. For a voltage source

(a) terminal voltage is always 1ower than source e.m.f.

(b) terminal voltage cannot be higher than source e.m.f.

(c) the source e.m.f. and terminal voltage are equal.

10. To determine the polarity of the voltage drop across a resistor, it is necessary to know

(a) value of current through the resistor (b) direction of current through the resistor

(c) value of resistor (d) e.m.fs. in the circuit.** **

** **

** **

**THEORETICAL QUESTIONS**

- Define the following terms :

Circuit, Electrical network, Active network, Node and Branch.

- What are the limitations of Ohm’s law?
- State and explain Kirchhoff’s laws.
- Discuss briefly application of Kirchhoff s laws.
- Explain the nodal voltage method for solving networks. How are the nodal equations written ?
- Explain Cramer’s rule used for solving equations by determinants.
- State and explain superposition theorem.
- State Norton’s theorem. List the steps for finding the current in a branch of a network with the help of this theorem.
- State Thevenin’s theorem.
- State the maximum power transfer theorem and explain its importance.
- State the compensation theorem and discuss its application. ,
- State Millman’s theorem.

** **

**UNSOLVED EXAMPLES**

1. Determine the magnitude and direction of the current in each of the batteries L, M and N shown in the Fig. 128.

2. Two batteries are connected in parallel. The e.m.f. and internal resistance of one are 110 V and 6 n respectively and the corresponding values for other are 130 V and 4 Q respectively. A resistance of 20 n is connected across the parallel combination. Calculate :

(i) The value and direction of the current in each battery.

(ii) The terminal voltage.

[Hint. Terminal voltage= (I_{1}+ I

_{2})R] [Ans. (i ) 0.1786 A, 5.2678 A (ii) 108.928 V]

3. Two cells A and Bare connected in parallel, unlike poles being joined together. The terminals of the cells are then joined by two resistors of 4 Q and 2 Q in parallel. The e.m.f. of A is 2 V, its internal resistance is 1 n; the e.m.f. of B is 1 V, its internal resistance is 2 Q. Find the current in each of the four branches of the circuit.

[Ans. 4/3 A, 5/6 A, 116 A, 1/3 A]

4. What is the equivalent resistance of the network shown in Fig. 129?

5. Determine the currents in the three batteries (L, M and N) in the network shown in Fig. 130 and show their values and direction of flow on the diagram. Neglect the internal resistance of batteries

[Ans. 2 A, 1 A, 0 AI

6. Determine the magnitude and direction of flow of current in the branch CD for the circuit shown in

(Ans. 71755 A from C to D]

7. For the lattice-type network shown in Fig. 132, calculate the current in each branch of the network.

8. Two storage batteries, A and B, are connected in parallel to supply a load the resistance of which is 1.2 n. Calculate :

(i) The current in this load.

(ii) The current supplied by each battery if the open-circuit e.m.f. of A is 12.5 V and that of B is 12.8 V, the internal resistance of A being 0.05 n and that of B 0.08 n.

[Ans. (i) 10.25 A (ii) 4 A (A), 6.25 A (B)]

9. A load having a resistance of0.1 n is fed by two storage batteries connected in parallel. The open circuit e.m.f. of one battery (A) is 12.1 V and that of the other battery (B) is 11.8 V. The internal resistances are 0.03 n and 0.04 n respectively. Calculate:

(i) The current supplied to the load

(ii) The current in each battery

(iii) The terminal voltage of each battery.

[Ans. (i) 102.2 A, (i~) 62.7 A (A), 39.5 A (B), (iii) 10.22 V]

10. A battery having an e.m.f. of 110 V and an internal resistance of 0.2 n is connected in parallel with another battery having an e.m.f. of 100 V and internal resistance 0.25 n. The two batteries in parallel are placed in series with a regulating resistance of 5 n and connected across 200 V mains. Calculate :

(i) The magnitude and direction of the current in each battery.

(ii) The total current taken from the supply mains.

[Ans. (i) 11.96 A (discharge) ; 30.43 A (charge) (ii ) 18.4 7 A]

11. A wheatstone bridge ABCD is arranged as follows: AB =50 ohms, BC = 100 ohms and CD= 101 ohms. A galvanometer of l000 ohms resistance is connected between Band D. A 2-volt battery having negligible resistance is connected across A and C. Estimate the current flowing through the galvanometer.

[Ans. 4.141-lA from D to B]

12. Determine the current through the galvanometer G in the wheatstone bridge network of the given Fig. 133.

[Ans. 0.52 mA]

13. Determine the currents through various resistors of the circuit shown in above Fig. 134

[Ans . 42 / 11 A(L to M) : 6 / 11 A(N to M) ; 48 / 11 (M to P)]

14. Find I_{1}, 1_{2} and I_{3} in the network shown in Fig. 135, using-loop-current method.

[Ans. 1 A, 2 A, :l AI

15. Find the currents in the branches of network of Fig. 136 using nodal voltage method.

[Ans./1 = 1.42 A, / 2 = 1.68 A, / 3 = 0.26 A, 14 = 1.1 A, /5 = 1.36 A, Is= 0.32 A]

16. Using superposition theorem find the currents in the different branches of the network shown in Fig. 137.

[Ans./1 = 0.352 A, 12 = 1.082 A,I= 1.432 A]

17. By using superposition theorem, find the current in 1 Q resistance in Fig. 138. Internal resistances of the cells are negligible.

[Ans. 2.066 A]

18. Using superposition theorem find the current in 20 Q resistor of the circuit shown in Fig. 139.

[Ans. 0.443 A]

19. Find the current in the 30 Q resistor of the circuit shown in Fig. 140, using superposition theorem.

[Ans. 0.662 A]

20. With reference to the network of Fig. 141, by applying Thevenin’s theorem find the following: (i) The equivalent e.m.f. of the network when viewed from terminals A and B (ii) The equivalent resistance of the network when looked into from terminals A and B, and (iii) Current in the load resistance R1 of 15 Q.

21. Find the current through 50 n resistance in the circuit shown in Fig. 142. Use Thevenin’s theorem.

[Ans. 0.086 A]

22. Calculate the current in the 8 n resistor of Fig. 143 by using Thevenin’s theorem.

[Ans. 0.8 A]

23. Use Norton’s theorem to calculate current flowing through 10 n resistor of Fig. 144.

24. Fig. 145 shows network with the value of load resistance RL =50 n. Develop Norton’s equivalent circuit and determine the current and power delivered to R1,.

[Ans. 0.114 A ; 0.65 W]

25. Fig. 146 shows a number of resistances connected in delta and star. Find the resistance across the terminals A and B. Use star/delta conversion method

26. In the given circuit, find the resistance between the points Band C (Fig. 147).

27. By using delta/star transformation for Fig. 148, find the current I supplied by the battery.

[Ans. 0.95 A]

28. Find the current in the galvanometer arm in the network shown in the Fig. 149, using delta/ star transformation.

[Ans. 0.236 A]

29. A bridge network ABCD has arms AB, BC, CD and DA of resistances 2, 2, 4 and 2 ohms respectively. If the detector AC has a resistance of 2 ohms, determine by star/delta transformation, the network resistance as viewed from the battery terminals BD.