EE484: Mathematical Circuit Theory + Analysis

Node and Mesh Equations

By: Jason Cho20076166

1

Overview

Review of Kirchhoff’s Circuit Laws

Node Equations

Mesh Equations

Why these methods?

Summary

Questions

2

DefinitionsNode: a point where two or more elements or branches

connect.

a point where all the connecting branches have the same voltage.

Branch: any path between two nodes.

Mesh: a set of branches that make up a closed loop path in a circuit where the removal of one branch will result in an open loop.

3

Kirchhoff’s Circuit Laws

Kirchhoff’s Current Law (KCL)

.. which states that the algebraic sum of all currents entering or leaving a node is zero for all time instances.

This law can be derived by using the Divergence Theorem, Gauss’ Law, and Ampere’s Law.

4

Kirchhoff’s Circuit Laws Enclose a node with a Gaussian surface, and apply Gauss’

Law, and the Divergence Theorem

0)()(

t

DJB

tJ

tJ

0

(cont’d)

Take the divergence of Ampere’s Law

dVJAdJVS )(

… (1) J = current density (vector)

B = magnetic field (vector)D = electric displacement (vector)

… (2) ρ = charge density (scalar)

5

Kirchhoff’s Circuit Laws (cont’d)

dVt

AdJVS

00

S AdJt

Substitute Eq. 2 into Eq. 1

Apply conservation of charge

So the final equation states that the sum of all current densities entering and leaving the enclosed surface is always zero.

6

Kirchhoff’s Circuit Laws (cont’d)

Intuitively, the divergence of a vector field measures the magnitude of the vector fields source or sink. Integrating all these sinks and sources inside this closed surface yields the net flow. Since our answer was zero, this means the sum of all sinks and the sum of all sources are equal.

7

Kirchhoff’s Circuit Laws

Kirchhoff’s Voltage Law (KVL)

.. which states that the algebraic sum of all the voltage drops or rises in any closed loop path is zero for all time instances.

(cont’d)

8

This law can be derived from Faraday’s Law of Induction.

Kirchhoff’s Circuit Laws Define a closed loop path in a circuit.

(cont’d)

SS

AdBdt

ddE

E = electric fieldB = magnetic field

9

Since there is no fluctuating magnetic field linked to the loop, the equation becomes

0S dE

The LHS of the above equation is also known as the electric potential equation.

So the above equation just states that the electric potential in the closed loop path is 0.

Faraday’s Law of Induction.

Node EquationsNode voltage analysis is one of many methods used in circuit analysis. This method involves a series of equations known as node equations. Each equation is expressed using Kirchhoff’s Current Law and Ohm’s Law. Therefore, this method can be thought of as a system of KCL equations, in terms of the node voltages. This method allows one to solve for the currents and voltages at any point in a circuit.

10

11

Node Equations (cont’d)

Step 1: Identify and label the nodes.Step 2: Determine a reference node.Step 3: Apply KCL at each non-reference node.

GND

V1

@ V1:

@ V2: 05.251

)4(

05.242

)10(

1222

2111

VVVVV

VVVVV

AVV

AVV

45

8

5

2

520

8

20

23

21

21

V2

12

Node Equations (cont’d)

A

A

V

V

4

5

5

8

5

220

8

20

23

2

1

A

A

V

V

5.6

5

5

210

20

8

20

23

2

1

VV

VV

5476.1

8095.3

2

1

Step 4: Solve the system of equations.

GND

V1 V2

Mesh EquationsMesh current analysis is another method used to solve for the voltages and currents at any point in a circuit. Mesh current analysis involves a series of equations known as mesh equations. Each equation is expressed using Kirchhoff’s Voltage Law, and Ohm’s Law. Therefore, this method can be thought of as a system of KVL equations, in terms of the mesh currents. The equations are similar to KVL in the way that it is also written as the algebraic sum of voltage rises or drops around a mesh.

13

14

Mesh Equations (cont’d)

0))(1())(2(7

0))(4())(2(28

221

121

IIIV

IIIVLoop 1:Loop 2: VII

VII

7))(3())(2(

28))(2())(6(

21

21

Step 1: Identify and label the mesh loops, and choose direction of current flow.

Step 2: Apply KVL to each mesh loop.

15

Mesh Equations (cont’d)

V

V

I

I

7

28

32

26

2

1

V

V

I

I

7

28

70

26

2

1

AI

AI

5

1

1

2

Step 4: Solve the system of equations.

Net current flow down the middle branch is (-1A) + 5A = 4A (upwards).

16

Why? Consider a larger network.

• Branch current method:

V1 V2

I2 I4

I3 I5I1

5 different branch currents2 non-reference nodes, 3 independent loops

GND

• Mesh current methodOr Node voltage method:

3 mesh loops, 2 non-reference nodes

V1 V2

GND

l3

NOO!! 3 KVL + 2 KCL = 5 equations with 5 variables!!

3 KVL or 2 KCL = 3 equations with 3 variablesOR 2 equations with 2 variables.

Summary

Revisted Kirchhoff’s Circuit LawsKirchhoff’s Current Law (KCL)Kirchhoff’s Voltage Law (KVL)

Node Equations

Mesh Equations

Why these methods?

17

Thank You!

18