Kevin D. Donohue, University of Kentucky 1Combining Circuit ElementsSeries, Parallel, Delta, and Wye Connected Elements, Simplification, and Problem Solving Strategies.Kevin D. Donohue, University of Kentucky 2Series and Parallel ElementsØ Two elements are connected in series if and only if they have a common node to which no other elements are attached.Ø Elements are connected in parallel if and only if they have common nodes at both terminals. R1R2R3Is R1R2 R3VsKevin D. Donohue, University of Kentucky 3Equivalent ResistanceØ Series and parallel resistors can be combined into one element with an equivalent resistance.Ø Equivalent resistance refers to a single resistance replacing a multiple resistor subcircuit such that it draws the same current and produces the same voltage drop.Ø Apply Ohm’s law to derived formula for finding equivalent resistance. Ø Note the same current flows through all elements in series and the same voltage drop occurs over all elements in parallel.Kevin D. Donohue, University of Kentucky 4Series and Parallel EquivalentsR1R2RNMReqRNLReqR1R2NeqRRRR+++=L21NeqRRRR111121+++=LKevin D. Donohue, University of Kentucky 5Series and Parallel SourcesØ Voltage sources in series can be combined into a single voltage source:Ø Current sources in parallel can be combined into a single current source:Ia + Ib + IcVa+Vb+Vc VaVbVcIa Ib IcKevin D. Donohue, University of Kentucky 6ExamplesSimplify complex circuits containing independent sources and resistors to a minimum number of equivalent elements.Kevin D. Donohue, University of Kentucky 7 Delta and Wye ElementsØ Three elements are ∆ (delta) connected if and only if any pair has only one common node to which other elements are attached.Ø Three elements are Y (Wye) connected if and only if they have a single common nodes to which no other elements are connected and their other nodes are connected to other elements.R1R2 R3RaRbRcabcabcKevin D. Donohue, University of Kentucky 8 Delta and Wye ConversionsØ Conversion formulas between ∆ and Y connection can be derived by equating equivalent resistances between each terminal pair (while leaving the other terminal open) and solving the set of equations for each resistor set in terms of the other.Ø The following formulas will result:R1R2 R3RaRbRcabcabc321313213232121RRRRRRRRRRRRRRRRRRcba++=++=++=acacbbaccacbbabcacbbaRRRRRRRRRRRRRRRRRRRRRRRR++=++=++=321Kevin D. Donohue, University of Kentucky 9ExamplesØ Find voltages and currents in branches of circuits containing resistors and independent sources through simplifying and expanding the circuit.Ø Voltages and currents in the original circuit can be determined byØ Finding currents and voltages of a simplified circuitØ Expanding back to the original configuration while applying current and voltage division to determine current and voltage values on the expanded elements.Kevin D. Donohue, University of Kentucky 10 Single Loop Circuits with Dependent SourcesSince current is the same throughout the entire loop, express all unknowns in terms of the loop current. Find Vo in circuit below:6 V+Vo-2 Ω120 Ω3 Ω+ Vd -100VdKevin D. Donohue, University of Kentucky 11Single Node-Pair Circuits with Dependent SourcesSince voltage is the the same across every element, express all unknowns in terms of the node voltage difference. Find Io in circuit below:6 AIo2 Ω3