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EE484: Mathematical Circuit Theory + Analysis
Node and Mesh Equations
By: Jason Cho
20076166
1
Overview
Review of Kirchhoff’s Circuit Laws
Node Equations
Mesh Equations
Why these methods?
Summary
Questions
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Definitions
Node: 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.
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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.
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Kirchhoff’s Circuit Laws (cont’d)
 Enclose a node with a Gaussian surface, and apply Gauss’ Law,
and the Divergence Theorem
 

SJ  dA  V (  J ) dV … (1)
J = current density (vector)
 Take the divergence of Ampere’s Law


 D
  (  B)    ( J 
)0
t
 
  J 
0
t


… (2)
  J  
t
B = magnetic field (vector)
D = electric displacement (vector)
ρ = charge density (scalar)
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Kirchhoff’s Circuit Laws (cont’d)
 Substitute Eq. 2 into Eq. 1
 

J

d
A


S
V t dV
 Apply conservation of charge
 

 0   J  dA  0
S
t
So the final equation states that the sum of all current densities
entering and leaving the enclosed surface is always zero.
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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.
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Kirchhoff’s Circuit Laws (cont’d)
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.
This law can be derived from Faraday’s Law of Induction.
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Kirchhoff’s Circuit Laws (cont’d)
 Define a closed loop path in a circuit.
 Faraday’s Law of Induction.

d  
SE  d   dt SB  dA
E = electric field
B = magnetic field
Since there is no fluctuating magnetic field linked to the loop, the
equation becomes

E
  d  0
S
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.
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Node Equations
Node 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.
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Node Equations
V1
(cont’d)
V2
GND
Step 1: Identify and label the nodes.
Step 2: Determine a reference node.
Step 3: Apply KCL at each non-reference node.
V1  (10V ) V1 V1  V2


0
2
4 2.5
V  (4V ) V2 V2  V1
@ V2: 2


0
1
5 2.5
@ V1:
23V1 8V2

 5A
20 20
 2V1 8V2


 4 A
5 5

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Node Equations
(cont’d)
Step 4: Solve the system of equations.
 23

  20
2

 5

8
8
 23

20  V1    5 A 
20  V1    5 A    20

21  V2   6.5 A
8  V2   4 A
0


5 
5 

V1
V1  3.8095V
V 2 1.5476V
V2
GND
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Mesh Equations
Mesh 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.
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Mesh Equations
(cont’d)
Step 1: Identify and label the mesh loops, and choose direction
of current flow.
Step 2: Apply KVL to each mesh loop.
Loop 1:  28V  (2)( I1  I 2 )  (4)( I1 )  0
Loop 2:  7V  (2)( I1  I 2 )  (1)( I 2 )  0
 (6)( I1 )  (2)( I 2 )  28V
 (2)( I1 )  (3)( I 2 )  7V
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Mesh Equations
(cont’d)
Step 4: Solve the system of equations.
6 2  I1  28V 






I
2
3
7
V

 2  

6 2   I1  28V 






I
0

7
7
V

 2  

 I 2  1A
I1  5 A
Net current flow down the middle branch is (-1A) + 5A = 4A (upwards).
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Why?
Consider a larger network.
V1
I1
V2
I3
I5
l3
I2
I4
GND
• Branch current method: 5 different branch currents
2 non-reference nodes, 3 independent loops
 3 KVL + 2 KCL = 5 equations with 5 variables!!
• Mesh current method
3 mesh loops, 2 non-reference nodes NOO!!
Or Node voltage method:  3 KVL or 2 KCL = 3 equations with 3 variables
OR 2 equations with 2 variables.
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Summary
Revisted Kirchhoff’s Circuit Laws
Kirchhoff’s Current Law (KCL)
 Kirchhoff’s Voltage Law (KVL)

Node Equations
Mesh Equations
Why these methods?
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Thank You!
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