ELECTRICITY & MAGNETISM (Fall 2011)

1. ELECTRICITY & MAGNETISM (Fall 2011) LECTURE # 14 BY MOEEN GHIYAS

2. (Parallel Circuits – Chapter 6) (Series – Parallel Circuits – Chapter 7) ......Basics Only Introductory Circuit Analysis by Boylested (10th Edition) TODAY’S lesson

3. Today’s Lesson Contents • Current Divider Rule • Voltage Sources in Parallel • Open and Short Circuit • Voltmeters: Loading Effect Series – Parallel Circuits (Chapter 7) • Introduction • General Approach • Block Diagram Approach

4. Current Divider Rule • For two parallel elements of equal value, the current will divide equally. • For parallel elements with different values, the smaller the resistance, the greater the share of input current.

5. Current Divider Rule • For parallel elements of different values, the current will split with a ratio equal to the inverse of their resistor values.

6. Current Divider Rule • The input current I = V/RT • Substituting V=IxRx into the above equation, where Ixrefers to the current through a parallel branch of resistance Rx, we have • . For I2,

7. VOLTAGE & CURRENT DIVIDER RULES • Current Divider Rule • Special case (Two Parallel Resisters Only) • In words, for two parallel branches, the current through either branch is equal to the product of the other parallel resistor and the input current divided by the sum (not the total parallel resistance) of the two parallel resistances

8. Current Divider Rule • Example – Determine the magnitude of the currents I1, I2, and I3 for the network of fig • Solution: By the current divider rule, • or by applying KCL, • . And • . or

9. Current Divider Rule • Current seeks the path of least resistance. i.e., • More current passes through the smaller of two parallel resistors. • The current entering any number of parallel resistors divides into these resistors as the inverse ratio of their ohmic values.

10. Current Divider Rule • The current entering any number of parallel resistors divides into these resistors as the inverse ratio of their ohmic values

11. Voltage Sources in Parallel • Voltage sources are placed in parallel only if they have the same voltage rating. • The primary reason for placing two or more batteries in parallel of the same terminal voltage would be to increase the current rating (and, therefore, the power rating) of the source.

12. Voltage Sources in Parallel • If two batteries of different terminal voltages were placed in parallel, both would be left ineffective or damaged. • Rint of btys are only current-limiting elements of resulting series circuit. • The current of 120A far exceeds the continuous drain rating of the larger supply, resulting in a rapid discharge of E1 and a destructive impact on the smaller supply E2.

13. Open and Short Circuits • Open circuits and short circuits can often cause more confusion and difficulty in the analysis of a system • An open circuit is simply two isolated terminals not connected by an element of any kind • An open circuit can have a potential difference (voltage) across its terminals, but the current is always zero amperes.

14. Open and Short Circuits • A short circuit is a very low resistance, direct connection between two terminals of a network. • A short circuit can carry a current of a level determined by the external circuit, but the potential difference (voltage) across its terminals is always zero volts (because V = IR = I(0Ω) = 0 V).

15. Open and Short Circuits • If a short circuit should develop across the 2Ω resistor, the total resistance will be • And current by Ohm’s law • The maximum current is now limited only by the circuit breaker or fuse in series with the source.

16. Open and Short Circuit • Example – Determine the voltages Vab and Vcd for the network of Fig • Solution: • And by KVL

17. Open and Short Circuit • Example – Calculate the current I and the voltage V for the network of fig • Solution:

18. Voltmeters: Loading Effect • The loading of a network by the insertion of meters is not to be taken lightly, especially in research efforts where accuracy is a primary consideration. • Most DMMs have internal resistance levels in excess of 10 MΩ on all voltage scales. • While the internal resistance of VOMs is sensitive to the chosen scale.

19. Voltmeters: Loading Effect • To determine the resistance of scale setting of a VOM in the voltmeter mode, simply multiply the maximum voltage of the scale setting by the ohm/volt (Ω/V) rating of the meter, normally found at the bottom of the meter. • For ohm/volt rating of 20,000, the 2.5-V scale would have internal resistance of

20. Voltmeters: Loading Effect • Example – For the relatively simple network of fig: • What is the open-circuit voltage Vab? • What will a DMM indicate if it has an internal resistance of 11 MΩ? • What will a VOM indicate with an Ω/V rating of 20,000 on the 100-V scale.

21. Voltmeters: Loading Effect • What is the open-circuit voltage Vab? • What will a DMM indicate if it has an internal resistance of 11 MΩ? • Solution:

22. Voltmeters: Loading Effect • What will a VOM indicate with an Ω/V rating of 20,000 on the 100-V scale. • Solution: For the VOM, the internal resistance of the meter is

23. Introduction – Series-Parallel Networks • Series-parallel networks are networks that contain both series and parallel circuit configurations. • A firm understanding of the basic principles is sufficient to begin an investigation of any single-source dc network having a combination of series and parallel elements or branches.

24. General Approach • Take a moment to study the problem “in total” and make a brief mental sketch of the overall approach. • Next examine each region of the network independently before tying them together in series-parallel combinations • Redraw the network as often as possible with the reduced branches towards source keeping unknown quantities undisturbed or have provision for the trip back to unknown quantities from the source.

25. General Approach • Example – For the network of fig, determine the voltages V1 and V2 and the current I. • Solution: • Redraw the circuit • By observation • . By KVL in right loop • . or • . or

26. General Approach • Apply KCL at node a

27. Block Diagram Approach • The block diagram approach is employed to simplify a complex combinations of elements into a simple combination of blocks in series or parallel.

28. Block Diagram Approach • Example - Solve for unknown parameters. • Solution: Working with each region: • Simplifying the circuit

29. Block Diagram Approach • Example - Solve for unknown parameters. • Solution: Working with each region: • Simplifying the circuit as below

30. Block Diagram Approach • Example - Solve for unknown parameters. • Solution: Working with each region: • . Returning to the network

31. Block Diagram Approach • Example - Solve for unknown parameters. • Solution: Working with each region: • . Returning to the network

32. Block Diagram Approach • Example - Solve for unknown parameters. • Solution: Working with each region: • The voltages VA, VB, and VC from either figure are • . Applying KVL

33. Summary / Conclusion • Current Divider Rule • Voltage Sources in Parallel • Open and Short Circuit • Voltmeters: Loading Effect Series – Parallel Circuits (Chapter 7) • Introduction • General Approach • Block Diagram Approach