8 Circuit Analysis using Series/Parallel Equivalents Begin by locating a combination of resistances that are in series or parallel. Often the place to start is farthest from the source.Redraw the circuit with the equivalent resistance for the combination found in step 1.Repeat steps 1 and 2 until the circuit is reduced as far as possible. Often (but not always) we end up with a single source and a single resistance.Solve for the currents and voltages in the final equivalent circuit.
27 No. of unknown: v1, v2, v3No. of linear equation : 3Setting up nodal equation withKCL at Node 1, Node 2, Node 3
28 No. of unknown: v1, v2, v3No. of linear equation : 3Setting up nodal equation withKCL at Node 1, Node 2, Node 3
29 Problem with node 3, it is rather hard to set the nodal equation at node 3, but still solvable. Why? As there is no way to determine the current through the voltage source, but v3=Vsv3Problem with node 3, it is rather hard to set the nodal equation at node 3 but still solvable.Same as before.
31 Circuits with Voltage Sources We obtain dependent equations if we use all of the nodes in a network to write KCL equations.Any branch with a voltage source:define SUPERNODE, sum all current either in or out at the supernode with KCLuse KVL to set up dependent equation involving the voltage source.
32 (a) The circuit of Example 4 (a) The circuit of Example 4.2 with a 22-V source in place of the 7-W resistor. (b) Expanded view of the region defined as a supernode; KCL requires that all currents flowing into the region must sum to zero, or we would pile up or run out of electrons.At node 1:At the “supernode:”
33 ABSumming all the current outfrom the supernode AFor supernode A,EXCLUDE THE SOURCEWhy?As the current via the 10V source is equal to the current viaR4 plus the current via R3
34 BFor supernode B,EXCLUDE THE SOURCESumming all the current into the supernode B
39 Node-Voltage Analysis with a Dependent Source First, we write KCL equations at each node, including the current of the controlled source just as if it were an ordinary current source.Next, we find an expression for the controlling variable ix in terms of the node voltages.
44 Node-Voltage Analysis 1. Select a reference node and assign variables for the unknown node voltages. If the reference node is chosen at one end of an independent voltage source, one node voltage is known at the start, and fewer need to be computed.2. Write network equations. First, use KCL to write current equations for nodes and supernodes. Write as many current equations as you can without using all of the nodes. Then if you do not have enough equations because of voltage sources connected between nodes, use KVL to write additional equations.3. If the circuit contains dependent sources, find expressions for the controlling variables in terms of the node voltages. Substitute into the network equations, and obtain equations having only the node voltages as unknowns.4. Put the equations into standard form and solve for the node voltages.5. Use the values found for the node voltages to calculate any other currents or voltages of interest.
51 Writing Equations to Solve for Mesh Currents If a network contains only resistors and independent voltage sources, we can write the required equations by following each current around its mesh and applying KVL.
52 For mesh 1, we haveFor mesh 2, we obtainFor mesh 3, we have
53 Determine the two mesh currents, i1 and i2, in the circuit below. For the left-hand mesh,i ( i1 - i2 ) = 0For the right-hand mesh,3 ( i2 - i1 ) i = 0Solving, we find that i1 = 6 A and i2 = 4 A.(The current flowing downward through the 3-W resistor is therefore i1 - i2 = 2 A. )
54 Mesh Currents in Circuits Containing Current Sources *A common mistake is to assume the voltages across current sources are zero. Therefore, loop equation cannot be set up at mesh one due to the voltage across the current source is unknownAnyway, the problem isstill solvable.
55 It is the supermesh. Mesh 3: As the current source common to two mesh, combine meshes 1 and 2 into a supermesh. In other words, we write a KVL equation around the periphery of meshes 1 and 2 combined.It is the supermesh.Mesh 3:Three linear equationsand three unknown
56 Find the three mesh currents in the circuit below. Creating a “supermesh” from meshes 1 and 3:( i1 - i2 ) ( i3 - i2 ) i3 = 0 Around mesh 2:1 ( i2 - i1 ) i ( i2 - i3 ) = 0 Finally, we relate the currents in meshes 1 and 3:i1 - i3 = Rearranging,i i i3 = 7 -i i i3 = 0 i i3 = 7 Solving,i1 = 9 A, i2 = 2.5 A, and i3 = 2 A.
57 supermesh of mesh1 and mesh2 branch currentcurrent source
59 Mesh-Current Analysis 1. If necessary, redraw the network without crossing conductors or elements. Then define the mesh currents flowing around each of the open areas defined by the network. For consistency, we usually select a clockwise direction for each of the mesh currents, but this is not a requirement.2. Write network equations, stopping after the number of equations is equal to the number of mesh currents. First, use KVL to write voltage equations for meshes that do not contain current sources. Next, if any current sources are present, write expressions for their currents in terms of the mesh currents. Finally, if a current source is common to two meshes, write a KVL equation for the supermesh.3. If the circuit contains dependent sources, find expressions for the controlling variables in terms of the mesh currents. Substitute into the network equations, and obtain equations having only the mesh currents as unknowns.4. Put the equations into standard form. Solve for the mesh currents by use of determinants or other means.5. Use the values found for the mesh currents to calculate any other currents or voltages of interest.
60 SuperpositionSuperposition Theorem – the response of a circuit to more than one source can be determined by analyzing the circuit’s response to each source (alone) and then combining the resultsInsert Figure 7.2
80 Finding the Thévenin Resistance Directly When zeroing a voltage source, it becomes a short circuit. When zeroing a current source, it becomes an open circuit.We can find the Thévenin resistance by zeroing the sources in the original network and then computing the resistance between the terminals.
90 Applications of Thevenin’s Theorem Load Voltage Ranges – Thevenin’s theorem is most commonly used to predict the change in load voltage that will result from a change in load resistance
91 Applications of Thevenin’s Theorem Maximum Power TransferMaximum power transfer from a circuit to a variable load occurs when the load resistance equals the source resistanceFor a series-parallel circuit, maximum power occurs when RL = RTH
92 Applications of Thevenin’s Theorem Multiload CircuitsInsert Figure 7.30
93 Norton’s TheoremNorton’s Theorem – any resistive circuit or network, no matter how complex, can be represented as a current source in parallel with a source resistance
94 Norton’s TheoremNorton Current (IN) – the current through the shorted load terminalsInsert Figure 7.35
96 Norton’s TheoremNorton Resistance (RN) – the resistance measured across the open load terminals (measured and calculated exactly like RTH)
97 Norton’s Theorem Norton-to-Thevenin and Thevenin-to-Norton Conversions Insert Figure 7.39
98 Step-by-step Thévenin/Norton-Equivalent-Circuit Analysis 1. Perform two of these:a. Determine the open-circuit voltage Vt = voc.b. Determine the short-circuit current In = isc.c. Zero the sources and find the Thévenin resistance Rt looking back into the terminals.
99 2. Use the equation Vt = Rt In to compute the remaining value. 3. The Thévenin equivalent consists of a voltage source Vt in series with Rt .4. The Norton equivalent consists of a current source In in parallel with Rt .