Circuits Lecture 3: Mesh

Analysis 李宏毅 Hung-yi Lee

Introduction

• Node analysis:

• Mesh analysis:

Current + Voltage Voltage Potential

Current + Voltage Current Mesh Current

Mesh: closed current path that contains no closed paths within it (Textbook, P149)

Meshes are “holes” in the circuits.

Mesh 1

Mesh 2 Mesh 3

Mesh analysis

21 ii 32 ii

31 ii

Target of Mesh analysis is to find the mesh currents.

Mesh analysis

Consider KCL: 0 dcb iiiRepresent ib, ic and id by mesh current

KCL is already fulfilled

1iib 12 iiic 2iid

Mesh analysis

Consider KVL: scba vvvv

Represent va, vb and vc by mesh current

saa iiRv 1 1iRv bb 21 iiRv cc

For mesh 1: Ra(i1-is) + Rbi1 + Rc(i1-i2) =vs

Mesh analysis

For mesh 3: Re(i3-i2) + Rfi3 + vs = 0

For mesh 2: Rc(i2-i1) + Rdi2 + Re(i2-i3) = 0

Can we always represent voltage by mesh currents?

Mesh analysis – 3 cases

• Represent the voltage of (1) voltage sources, (2) resistor, and (3) current sources by mesh current

Mesh analysis - current sources • 1. Current sources on the perimeter of the circuit

Mesh analysis - current sources • 2. Interior current sources

Method 1:

Consider vs as unknown variables

0 sbd vvv

sec vvv ……

sisv

Represent the voltages vd, vb, vc and ve …. (except vs) by mesh currentSolve mesh currents and vs

12 iiis One more equation:

Mesh analysis - current sources • 2. Interior current sources

Method 2:

Textbook: Chapter 2.5

when the current source is parallel with a resistor

Mesh analysis - current sources • 2. Interior current sources

Method 2:

xmss iiRv

Another point of view:

when the current source is parallel with a resistor

The voltage of is is equal to Rs.

• 2. Interior current sources

Mesh analysis - current sources

Method 3:

sisv

12 iiis

supermesh

12 iii s

3i

1iRd

31 iiRb 3132 iiiRiiR scc

1

2

iiR

iR

se

e

Example 4.9

6(i1-5) + 10i1 + 3(i1+4) = 20→i1=2A

56 1 i

110i

43 1 i

Exercise 4.11

Special case: current sources are parallel with resistors

Exercise 4.11

k6

k51i

mAi 31

mAi 21

062435 111 kmikmiki

Non-planar Circuit

• Mesh analysis cannot be applied on non-planar circuit

Cannot transform this circuit into planar circuit

Non-planar Circuit

• Some non-planar circuits can be transformed into planar circuit

A

C

B

A BC

Non-planar Circuit

• Some non-planar circuits can be transformed into planar circuit

Non-planar Circuit

• Why mesh analysis cannot be applied on non-planar circuit?

6 meshes? 5 meshes

Node v.s. Mesh

Node v.s. Mesh

• What is the final target?• Node analysis is not suitable for current• Mesh analysis is not suitable for voltage

i

If node analysis is used, it is tedious to find i

If mesh analysis is used, it is tedious to find v

v+ -

Node v.s. Mesh

• Number of Equation: Number of Nodes v.s. Number of Meshes

• Number of Voltage sources v.s. Number of Current sources• Use both to check your results

Homework

• 4.38• 4.42

Thank you!

Answer

• 4.38• V1=24, v2=-16, v3=20

• 4.42• I1=3, i2=3.5, i3=3, v1=0, v2=15

Acknowledgement

• 感謝 林楷恩 (b02)• 糾正錯誤的作業答案