Mesh Analysis Techniques in Electrical Circuits

Circuits Lecture 3: Mesh Analysis Hung-yi Lee

Mesh 1 Introduction Mesh 2 Mesh 3 Node analysis: Current + Voltage Voltage Potential Mesh analysis: 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 analysis i i i i 2 3 1 2 i i 1 3 Target of Mesh analysis is to find the mesh currents.

Mesh analysis + + = 0 i i i Consider KCL: b c d Represent ib, icand idby mesh current 1i ib= 2 i ic = = id 2i i 1 KCL is already fulfilled

Mesh analysis + + = v v v v Consider KVL: a b c s Represent va, vband vcby mesh current ( ) s a i i R = 1 v ( ) = b= v R i i R 1i v 1 2 c c b a

Mesh analysis For mesh 1: Ra(i1-is) + Rbi1 + Rc(i1-i2) =vs For mesh 2: Rc(i2-i1) + Rdi2 + Re(i2-i3) = 0 For mesh 3: Re(i3-i2) + Rfi3 + vs = 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 vsas unknown variables si sv + + + = = 0 v v v v v v Represent the voltages vd, vb, vcand ve . (except vs) by mesh current Solve mesh currents and vs One more equation: d b s c e s = is i i 2 1

Mesh analysis - current sources 2. Interior current sources when the current source is parallel with a resistor Method 2: Textbook: Chapter 2.5

Mesh analysis - current sources 2. Interior current sources when the current source is parallel with a resistor Method 2: Another point of view: The voltage of is isequal to Rs. ( ) = v R i i s s m x

Mesh analysis - current sources 2. Interior current sources Method 3: supermesh 3i ( ) ( ) ( ) = + R i i R i i i Rb i i 2 3 1 3 c c s 1 3 si sv Rd 1i R = i 2 e ( ) + R i i 1 e s = = s+ is i i i i i 2 1 2 1

Example 4.9 10i 1 ( ) 5 1 i 6 ( ) 4 31+ i 6(i1-5) + 10i1 + 3(i1+4) = 20 i1=2A

Exercise 4.11 Special case: current sources are parallel with resistors

Exercise 4.11 1 mA 3 i 6 k 1+ 2 i mA 1i 5 k ( ) ( ) + + + = 5 3 4 2 6 0 i k i m k i m k 1 1 1

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 B A C C A B

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 v - + If node analysis is used, it is tedious to find i If mesh analysis is used, it is tedious to find 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 (b02)