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1 Lecture 2 Dr Kelvin Tan Electrical Systems 100

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2 Contents Series-Parallel Circuits Wye-Delta Conversion Ladder Networks Current Sources Source Conversion Current Sources in Parallel (Series?) Mesh Analysis Nodal Analysis

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3 Series-Parallel Circuits Series-parallel circuits are networks where there are both series and parallel elements

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4 Reduce and Return Approach Step 1: Take an overall mental look at the problem Step 2: Examine each section of the network independently before tying them together Step 3: Redraw the network as often as you will need to arrive at reduced branches. Maintain the original unknown quantities to be found for clarity where applicable Step 4: Take the trip back to the original network to find detailed solution (Some time it may help to draw branche/s as blocks and work out blockwise)

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5 An Example of Reduced and Return Approach Finding V 4 ?

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6 An Example of Reduced and Return Approach

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7 Wye-Delta Conversion Often we encounter a different kind of network which appears to be not in series or parallel in relation to the rest of the network. Under these circumstances it is necessary to convert this portion of circuit from one form to the other to find appropriate branch connection which then appears clearly in series or parallel with rest of the network.

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8 Y- Conversion The purpose is then to be able to convert Y to or to Y.

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9 -Y Conversion

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10 Y- Conversion Similarly, for converting Wye quantities to Delta are given as:

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11 -Y /Y- Conversion If all resistors in the or Y are the same (R A = R B = R C ): From -Y eq: This shows that for a Y of three equal resistors the value of each resistor equivalent is 3 times the Y resistor.

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12 Ladder Networks A Ladder Network is one where a series-parallel section of a network occur repeatedly within the network. An example of such network is a Low Pass Filter circuit. Figure below shows a three section Ladder Network. To solve a ladder network follow the steps: Calculate the total resistance Calculate the source current or total current drawn from source Work back through the ladder until desired current or voltage is obtained

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13 Ladder Networks Combining parallel and series elements to reduce the circuit we get :

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14 Ladder Networks Using current divider I 6 can be found. Finally, (Current divider)

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15 Current Sources A battery supplies fixed voltage and the source current may vary according to load. Similarly, a current source is one where it supplies constant current to the branch where it is connected and the voltage and polarity of voltage across it may vary according to the network condition.

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16 Source Conversion A Voltage source can be converted to a current source and vice versa. In reality, Voltage sources has an internal resistance Rs and current sources has a shunt resistance Rsh. In ideal cases, Rs equal to 0 and Rsh equal to .

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17 Source Conversion For us to be able to convert sources, the voltage source must have a series resistance and current source must have some shunt resistance. Eg.

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18 Current Sources in Parallel If two or more sources are in parallel, then the equivalent source is obtained by summing the currents with direction taken into account and the new shunt resistance is the parallel combination of all the individual resistances. Note that current sources can not be connected in series!

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19 Method of Circuit Analysis (Mesh Analysis and Nodal Analysis) Mesh is a closed loop which does not contain any other loops within it. In most circumstances, a mesh will contain one or more voltage sources and one or more type of circuit elements. In dc circuit theory, these elements are limited to resistances only in steady state analysis. Mesh analysis determines the mesh or loop currents in the circuit. i1i1 i2i2 I 1 = i 1 I 2 = i 2 I 3 = i 1 -i 2 By solving i 1 and i 2

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20 Mesh Analysis Steps to determine Mesh Currents: Identify the “n” number meshes in the circuit Assign mesh currents, in clockwise directions Apply KVL to each of the “n” meshes. Use Ohm’s law to express the voltages in terms of the mesh currents. Take appropriate voltage drop polarity (+ve clockwise and –ve anticlockwise) into consideration in writing these equations. Solve the “n” simultaneous equations to get the “n” mesh currents.

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21 Mesh Analysis-An Example Find I 1, I 2 and I 3. i1i1 i2i2 (Loop i 1 ) (Loop i 2 ) Method 1: Using method of substitution

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22 Mesh Analysis-An Example Method 2: Using Cramer’s Rule (Also known as Format Approach): From previous 2 loops : We obtain the determinant Δ as Thus:

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23 Mesh Analysis-Super Mesh Sometime, there may be a current source in one of the mesh or a current source in common between two meshes. If the current source involves only one of the meshes, then the analysis is easier as we have 1 less equation to solve as the current is defined by the current source in one of the mesh already. If however, the current source is common between two meshes, then we need to form a super mesh. This is necessary as we need to apply KVL in solving mesh equations and we do not know the voltage across the current source in advance.

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24 Mesh Analysis-Super Mesh Case 1: In this example, i1i1 i2i2 Applying KVL in Mesh 1

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25 i1i1 i2i2 Mesh Analysis-Super Mesh Case 2: we create a super-mesh by excluding the current source and any element connected in series with it as shown. i1i1 i2i2 In super-mesh super-mesh

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26 Nodal Analysis A Node is defined as the junction of two or more branches. In a “n” node circuit, 1 node is taken as the reference (usually the ground is taken as reference) and we need to solve node voltages using KCL. By solving node v 1 and v 2 We can solve :

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27 Nodal Analysis Steps to determine Nodal voltages: Select a node as reference node Assign voltages to the remaining nodes Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express currents in terms of node voltages. Solve the n-1 simultaneous equations to get the unknown node voltages.

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28 Nodal Analysis Applying KCL to each nodes Example in node v 1 node v 2

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29 Nodal Analysis- An Example At node 1, applying KCL, v1v1 v2v2 At node 2, applying KCL, v2v2 v1v1

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30 Nodal Analysis- An Example Using Cramer’s Rule or Format Approach:

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31 Nodal Analysis-Supernode Sometime, there may be a voltage source connected between a reference and nonreference node. If this is the case, the voltage of the nonreference node is simply set equal to the voltage source and we have 1 less equation to solve. If however, the voltage source is common between two or more unknown nodes, then we need to form a super node. This is necessary as we need to apply KCL in solving node equations and we do not know the current through the voltage source in advance. A Supernode is formed by enclosing the voltage source between two nonreference nodes and any branches in parallel with it.

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32 Nodal Analysis-Supernode Consider the following circuit. Nodes 2 and 3 form a supernode. At Supernode we get, v1v1 v3v3 v2v2 v1v1 v2v2 v3v3 At Supernode, applying KCL,

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33 Nodal Analysis-Supernode Applying KVL to supernode we get, v2v2 v3v3

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34 Nodal Analysis-Supernode We note the following properties of a supernode, The voltage source inside the supernode provides a constraint equation to solve for the node voltages A supernode do not have a voltage of its own A supernode requires the application of both the KCL and KVL

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