Methods of AnalysisMethods of Analysis

Circuits 1Circuits 1Fall 2005Fall 2005

Harding UniversityHarding University

Jonathan WhiteJonathan White

OutlineOutline Nodal AnalysisNodal Analysis

• Define a symbol for all unknown node voltages.Define a symbol for all unknown node voltages.• Write KCL at each node where variables occurWrite KCL at each node where variables occur• Using Ohm’s Law, solve resulting equations.Using Ohm’s Law, solve resulting equations.

Mesh CurrentsMesh Currents• Set up the currentsSet up the currents• Use KVLUse KVL

Methods to solve linear equationsMethods to solve linear equations• SubstitutionSubstitution• DeterminantsDeterminants• CalculatorCalculator• Method from Numerical MethodsMethod from Numerical Methods

Nodal AnalysisNodal Analysis Steps:Steps:

• Define a voltage at every node in the circuitDefine a voltage at every node in the circuit Note: Some may be known, such as the source and groundNote: Some may be known, such as the source and ground

• Write KCL at the nodes where the unknown voltages Write KCL at the nodes where the unknown voltages existexist

• Now, plug into these KCL equations with the unknown Now, plug into these KCL equations with the unknown voltages, remembering how Ohm’s Law works. In this voltages, remembering how Ohm’s Law works. In this case, I = (Vcase, I = (VHH – V – VLL)/R, because we are writing voltages for )/R, because we are writing voltages for nodes, not just resistors. Since current flows from a nodes, not just resistors. Since current flows from a higher potential to a lower potential, the voltage over a higher potential to a lower potential, the voltage over a resistor that is connected to 2 nodes is just Vresistor that is connected to 2 nodes is just VHH – V – VLL

• Other current and voltage sources must be factored in to Other current and voltage sources must be factored in to either the KCL equations or the unknown voltages. They either the KCL equations or the unknown voltages. They sometimes actually make the equations easier.sometimes actually make the equations easier.

• Solve for the unknown voltages.Solve for the unknown voltages.

Nodal Analysis Example 1Nodal Analysis Example 1Find all voltages and currents.

Nodal Analysis Example 2Nodal Analysis Example 2Find Vo

+ Vo -

Mesh CurrentsMesh Currents Steps:Steps:

• Label each unknown current in each mesh, going Label each unknown current in each mesh, going clockwise.clockwise.

A mesh is a loop which does not contain any other loops A mesh is a loop which does not contain any other loops within it.within it.

Also, write down the polarities of the currents as they go Also, write down the polarities of the currents as they go through each resistor.through each resistor.

• Write KVL equations for each mesh. In this case, use Write KVL equations for each mesh. In this case, use V=I*R. When resistors are in both meshes, I=(IV=I*R. When resistors are in both meshes, I=(I11-I-I22).).

• Use Ohm’s Law to express the voltages in terms of the Use Ohm’s Law to express the voltages in terms of the mesh currents.mesh currents.

• Again, you may need extra equations if there are other Again, you may need extra equations if there are other current/voltage sources.current/voltage sources.

• Solve for the unknown currents.Solve for the unknown currents.

Mesh Current Example - 1Mesh Current Example - 1Calculate the mesh currents.

Mesh Current Example - 2Mesh Current Example - 2Find the current through the 1 ohm R

Methods of Solving Sets of Methods of Solving Sets of EquationsEquations

CalculatorCalculator• rref functionrref function• solve functionsolve function

Linear AlgebraLinear Algebra SubstitutionSubstitution GraphingGraphing Euclid’s MethodEuclid’s Method