Methods of Analysis Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Contents Introduction Nodal Analysis Nodal Analysis with Voltage Sources Mesh Analysis Mesh Analysis with Current Sources Nodal Analysis vs. Mesh Analysis Applications

Introduction Nodal Analysis –Based on KCL Mesh Analysis –Based on KVL Linear algebra is applied to solve the resulting simultaneous equations. –Ax=B, x=A -1 B

Nodal Analysis Circuit variables = node voltages Steps to determine node voltages –Select a reference node, assign voltages v 1, v 2,…, v n-1 for the remaining n-1 nodes –Use Ohm’s law to express currents of resistors –Apply KCL to each of the n-1 nodes –Solve the resulting equations

Symbols for Reference Node (Ground) Used in this course

Case Study Assign v n

Nodal Analysis with Voltage Sources If a voltage source is connected between a nonreference node and the reference node (or ground) –The node voltage is defined by the voltage source –Number of variables is reduced –Simplified analysis

Continued If a voltage source is connected between two nonreference nodes –The two nodes form a supernode –Apply KCL to the supernode (similar to a closed boundary) –Apply KVL to derive the relationship between the two nodes Supernode

Case Study with Supernode

Example 1

Example 2

What is a mesh? A mesh is a loop that does not contain any other loop within it.

Mesh Analysis Circuit variables = mesh currents Steps to determine mesh currents –Assign mesh currents i 1, i 2,…, i n –Use Ohm’s law to express voltages of resistors –Apply KVL to each of the n meshes –Solve the resulting equations

Continued Applicable only for planar circuits An example for nonplanar circuits is shown below

Case Study

Mesh Analysis with Current Sources If a current source exists only in one mesh –The mesh current is defined by the current source –Number of variables is reduced –Simplified analysis

Continued Supermesh Excluded If a current source exists between two meshes –A supermesh is resulted –Apply KVL to the supermesh –Apply KCL to derive the relationship between the two mesh currents

Example 1

Example 2 Supermesh

Example 3 Supermesh Applying KVL to the supermesh Applying KCL to node P Applying KCL to node Q Applying KVL to mesh 4 4 variables solved by 4 equations

How to choose? Nodal Analysis –More parallel-connected elements, current sources, or supernodes –N node < N mesh –If node voltages are required Mesh Analysis –More series-connected elements, voltage sources, or supermeshes –N mesh < N node –If branch currents are required

Applications: Transistors Bipolar Junction Transistors (BJTs) Field-Effect Transistors (FETs)

Bipolar Junction Transistors (BJTs) Current-controlled devices

DC Equivalent Model of BJT

Example of Amplifier Circuit