Methods of Analysis ELEC 202 Electric Circuit Analysis II

Nodal Analysis 1. For an AC circuit, transform the circuit into the phasor domain. 2. Identify every node in the circuit. 3. Label each node with a node voltage. The node with the highest number of branches connected should be labeled as the ground node having zero potential.

Nodal Analysis (Cont’d) 4. At a particular node of interest (except ground), use Ohm’s law to express the current through any branch connected to that node as the difference between the two node voltages at both end of that branch divided by the branch impedance. The voltage at the node of interest is always considered to be at higher potential than the rest of the node voltages.

Nodal Analysis (Cont’d) 5. Apply KCL to sum all currents at that node of interest. The resulting algebraic equation (called nodal equation) has all node voltages as its unknowns. 6. Repeat steps 4 and 5 until all nodes except ground are accounted for. The number of equations must be equal to the number of node voltages.

Nodal Analysis (Cont’d) 7. If a branch not connected to ground contains a voltage source, the two nodes at both ends are collapsed into a single node called a supernode, and the voltage source and any elements connected in parallel with it removed. However, KCL must still be satisfied at a supernode using the old node voltage labels. Also, the removal of the voltage source provides another nodal equation.

Nodal Analysis (Cont’d) 8. If a branch connected to ground contains a voltage source, the non-ground node voltage is equal to the source voltage, and KCL is not applied to this node. 9. Solve the resulting simultaneous nodal equations to obtain the values of the unknown node voltages. 10. Use the values of node voltages above to find voltages and/or currents throughout the rest of the circuit.

Example 1 Find i x in the circuit using nodal analysis.

Example 1 (cont’d) The resulting circuit in the phasor domain.

Example 2 Compute v 1 and v 2 in the circuit.

Example 3 Compute V 1 and V 2 in the circuit.

Example 3 (Cont’d)

Example 4 Compute V 1 and V 2 in the circuit.

Mesh Analysis 1. For an AC circuit, transform the circuit into the phasor domain. 2. Identify every mesh in the circuit. 3. Label each mesh with a mesh current. It is recommended that all mesh currents be labeled in the same direction (either clockwise (CW) or counter- clockwise (CCW)).

Mesh Analysis (Cont’d) 4. Within a particular mesh of interest, use Ohm’s law to express the voltage across any element within that mesh as the difference between the two mesh currents of contiguous meshes shared by the element times the element impedance. The current within the mesh of interest is always considered to be larger than the rest of the mesh currents.

Mesh Analysis (Cont’d) 5. Apply KVL to sum all voltages in that mesh of interest. The resulting algebraic equation (called mesh equation) has all mesh currents as its unknowns. 6. Repeat steps 4 and 5 until all meshes are accounted for. The number of equations must be equal to the number of mesh currents.

Mesh Analysis (Cont’d) 7. If a current source exists between two contiguous meshes, the two meshes are collapsed into a single mesh called a supermesh, and the current source and any elements connected in series with it removed. However, KVL must still be satisfied within a supermesh using the old mesh current labels. Also, the removal of the current source provides another mesh equation.

Mesh Analysis (Cont’d) 8. If a current source exists only in one mesh, the mesh current is equal to the source current, and KVL is not applied to this mesh. 9. Solve the resulting simultaneous mesh equations to obtain the values of the unknown mesh currents. 10. Use the values of mesh currents above to find voltages and/or currents throughout the rest of the circuit.

Example 5 Find I 0 in the circuit using mesh analysis.

Example 6 Find I 0 in the circuit using mesh analysis.

Example 7 Find V 0 in the circuit using mesh analysis.

Example 7 (cont’d)