Lecture 6

Nodal/Mesh Analysi4.2,4.3, 4.5 &4.6

Definitions

Nodes

Node: a point where two or more circuit elements join

Essential Node

Essential Node: a node where three or more circuit elements join

Step 1: Number of Nodal Voltage Equations

• ne is essential nodes

• Number of nodal voltage Equations is ne-1

ne = 3ne-1 =2 nodal voltage equations are required.

Step 2: Designate a node as the reference node.

• Suggestion: Select the node with the most branches.

(3 branches)(3 branches)

(4 branches)

Step 3:Define the Node Voltage on the Diagram

Step 4: Apply KCL

• Apply KCL to essential nodes.

R1 R2

R10R5VS IS

Simulation Results

Solve ia, ib and ic

Hints

ne=2Need ne-1=1 equation

Mesh Analysis

Definitions

Branch

Branch: a path that connects two nodes.

Essential Branch

Essential branch: a path which connects two essential nodes without passing through an essential node.

Mesh

Mesh: a loop that does not enclose any other loops

Step 1: Determine the number of essential nodes

ne=3

Step 2: Determine the number of essential branches

be=5

# of equations: be-(ne-1)=5-(3-1)=3

Step 3: Apply KVL Around Loop b

a. Focus initially on ia. 40-iaR2-(ia )8=0b. Account for ib. 40-iaR2-(ia-ib)8=0

+ -

+

-

Step 4: Apply KVL Around Loop b

a. Focus initially on ib. -(ib )8-ib6-(ib )6=0b. Account for ia. -(ib-ia)8-ib6-(ib )6=0c. Account for ic. -(ib-ia)8-ib6-(ib-ic )6=0

+ -

+

-+

-

Step 5: Apply KVL Around Loop c

a. Focus initially on ic. -(ic )6-ic4-20=0b. Account for ib -(ic- ib)6-ic4-20=0

+ -

+

-

Solve 3 EQ and 3 Unknowns Using Mathematica

3 Unknown equations 3 unknowns

Get Mathematica Through SSU

Step 1

# of essential branch: 6# of essential nodes: 4# of equations: 6-(4-1)=3

Step 2

See in the handout.

Use Mathematica to Solve Equations

Format: Solve[{equations separated by a comma},{list of unknowns}]To solve an equation: Evaluation→Evaluate Cells

Degenerated Common Emitter Amplifier

(Small signal model)

What if we drive the base with a small signal?

Vin, m=1 mV Vout, m=46 mV

Mesh Analysis

ne=2 essential nodesbe=3 essential branches

3-(2-1)=2 equations

Mesh Analysis

1 2

Clockwise around loop 1:+Vin-i1rπ-(i1+i2)RE=0i2=gmi1rπ

Vout=-i2RC

Loop 1: clockwiseLoop 2: counter-clockwise