Basic Electrical Circuits & Machines (EE-107) Course Teacher Shaheena Noor Assistant Professor Computer Engineering Department Sir Syed University of Engineering & Technology.

Useful Circuit analysis Techniques The basic goals of this is learning methods of simplifying the analysis of more complicated circuits. We are interested only in the detailed performance of an isolated portion of a complex circuit; a method of replacing the remainder of the circuit by a greatly simplified equivalent is then very desirable.

Superposition The superposition Principle It states that “ the response (a desired current or voltage) in a linear circuit having more than one independent source can be obtained by adding the responses caused by the separate independent sources acting alone”

Superposition A voltage source set to zero acts like a short circuit. A current source set to zero acts like an open circuit.

Example 5.1 (page 104) Use superposition to write an expression for the unknown branch current i x. ixix

Source Transformations A real voltage source can be converted to an equivalent real current source and vice versa. For Example:   iLiL

Compute the current through the 4.7kΩ resistor after transforming the 9mA source into an equivalent voltage source. Example 5.4 (page 113) I

For the circuit, compute the voltage V across 1MΩ resistor using repeated source transformations. Drill Problem 5.4 (page 115)

Thevenin’s Theorem It states that “ any linear circuit is equivalent to a single voltage source in series with a single resistance.”

Procedure: 1.Open circuit the terminals with respect to which Thevenin equivalent circuit is desired. 2.The Thevenin equivalent resistance R TH is the total resistance at the open circuited terminals when all voltage sources are replaced by short circuits and all current are replaced by open circuits. 3.The Thevenin equivalent voltage V TH (or E TH ) is the voltage across the open circuited terminals. 4.Replace the original circuitry by its Thevenin equivalent circuit with the Thevenin terminals occupying the same position as the original terminal. Thevenin’s Theorem

Find the Thevenin equivalent to the left of terminal x – y Thevenin’s Theorem (Example)

Drill Problem 5.6 (page 119) Use Thevenin’s theorem to find the current through 2Ω resistor. I2ΩI2Ω

Norton’s Theorem It states that “any linear circuit is equivalent to a real current source at a selected set of terminals.” Procedure: First find the Thevenin’s equivalent circuit and then convert it to an equivalent current source.

Example Find the Norton equivalent current source at terminals x - y

Drill Problem 5.5 (page 118) Determine the Norton equivalent of the high lighted network. RLRL

Maximum Power Transfer An independent voltage source in series with a resistance R S, or an independent current source in parallel with a resistance R S, delivers a maximum power to that load resistance R L for which R L = R S. RLRL VSVS RSRS +VL-+VL- iLiL A voltage source connected to a load resistor R L

Delta-Wye (∆ -Y) Conversion Some electrical circuits have no components in series and in parallel. So they can not be reduced to simpler circuits containing equivalent resistance of series or parallel combination. However in many cases it is possible to transform a portion of the circuit in such a way that the resulting configuration does contain series and parallel connected components.

Delta-Wye (∆ -Y) Conversion The transformation produces an equivalent circuit in the sense that voltages and current in the other (untransformed) components remain the same. Therefore, once the circuit has been transformed, voltages and current in the unaffected components can be determined using series-parallel analysis methods.

Delta-Wye (∆ -Y) Conversion RBRB RCRC RARA RBRB RCRC RARA a c d b (a) ∏ network consisting of three resistors and three unique connections (b) Same network drawn as a Δ network (c) A T network consisting of three resistors (d) Same network drawn as a Y network R1R1 R3R3 R2R2 R3R3 R2R2 R1R1

To convert from a Y network to a ∆ network, the new resistor values are calculated using the following relations: R A = R 1 R 2 + R 2 R 3 + R 3 R 1 R 2 R B = R 1 R 2 + R 2 R 3 + R 3 R 1 R 3 R C = R 1 R 2 + R 2 R 3 + R 3 R 1 R 1 Delta-Wye (∆ -Y) Conversion

To convert from a ∆ network to a Y network. R 1 = R A R B R A + R B + R C R 2 = R B R C R A + R B + R C R 3 = R C R A R A + R B + R C Delta-Wye (∆ -Y) Conversion

Drill Problem (page 129) Use the technique of Y- Δ conversion to find the Thevenin equivalent resistance of the circuit given below. Each R is 10 Ω

Example (page 128) Use the technique of Δ-Y conversion to find the Thevenin equivalent resistance of the circuit given below. 1Ω1Ω 4Ω4Ω 3Ω3Ω 2Ω2Ω 5Ω5Ω