Chapter 3 – Nodal and Loop Analysis
Prof. Jong-Woo Kim
건국대학교 전력시스템 집적화 연구실
Konkuk Integrated Power Systems (KIPS) Lab.
공학관 C동 223호
jongwookim@konkuk.ac.kr
MV
Wireless
CHAPTER 3 Nodal and Loop Analysis
• Learning goals
•
Calculate the branch currents and node voltages in circuits containing multiple nodes using KCL
and Ohm’s law in nodal analysis
•
Calculate the mesh currents and voltage drops and rises in circuits containing multiple loops
using KVL and Ohm’s law in loop analysis
•
Identify the most appropriate analysis technique that should be utilized to solve a particular
problem
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Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Loop Analysis
• Second systematic technique to determine every voltage and current in a circuit
• Dual to node analysis – it first determines all currents in a circuit and then it uses Ohm’s law to
compute necessary voltages
• There are situation where node analysis is not efficient and where the number of equations
required by this new method is significantly smaller
• It is important to determine which technique is appropriate before we start
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3.2 Loop Analysis
• Think about node analysis with this circuit
+ VR1 − + VR2 −
+
-
12V
R2
R1
+
-
R3
18V
•
•
•
•
4 nodes except for the reference
1 node connected to the ground through V source
1 supernode
We need 3 equation
+ VR3 −
• BUT, there is only ONE current flowing through all components
• If we find that current, all voltages can be computed with Ohm’s law
STRATEGY:
1. Apply KVL
(sum of voltage drops =0)
− 12[V ] + VR1 + VR 2 + 18[V ] − VR 3 = 0
2. Use Ohm’s Law to express
voltages in terms of the “loop current.”
− 12[V ] + R1 I + R2 I + 18[V ] + R3 I = 0
• We can write the second equation directly
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Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Loops, meshes and loop currents
a
1
2
b
3
• Each component is characterized by its voltage across and its
c
current flowing through
I1
I2
7
4
e
d
f
6
5
A BASIC CIRCUIT 𝑰𝟑
• A loop is a closed path not going twice over any node → 3 loops
• Mesh is a loop not enclosing any other loop → 2 meshes
• A loop current is a current assumed to flow around a loop
→ I1, I2, I3 ARE LOOP CURRENTS
• A mesh current is a loop current associated to a mesh. I1, I2 are mesh currents.
• CLAIM: IN A CIRCUIT, THE CURRENT THROUGH ANY COMPONENT CAN BE EXPRESSED IN
TERMS OF THE LOOP CURRENTS.
Ex) Iaf=–I1–I3, Ibe=I1–I2, Ibc=I2+I3, Ied=?
a
1
2
b
3
• FACT: NOT EVERY LOOP CURRENT IS REQUIRED TO COMPUTE
c
ALL THE CURRENTS THROUGH COMPONENTS.
I1
7
4
• For every circuit, there is a minimum number of loop currents
necessary to compute every current in the circuit. Such a
e
d
f
6
5
A BASIC CIRCUIT 𝑰𝟑
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collection is called a minimal set (of loop currents).
• #Loops in a minimal set = #Branches – ( #Nodes – 1 )
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3.2 Loop Analysis
• Example: find loop equations to solve the following circuit
• #B=7, #N=6
• L=7 – (6 – 1) = 2
• NOTE: Mesh currents are always independent
• KVL on left mesh
• Ohm’s law
v1 = i1 R1 , v 2 = i1 R2 , v 3 = ( i1 − i2 ) R3
• KVL on right mesh
• Replacing and rearranging
v4 = i2 R4 , v5 = i2 R5
• Matrix form
 R1 + R2 + R3

− R3

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− R3
  i1   v S 1 
=
R3 + R4 + R5   i2   − v S 2 
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3.2 Loop Analysis
• Developing a shortcut
• #B=8, #N=7
• L = 8 – (7 – 1) = 2
• Strategy
1. Draw the mesh currents. Orientation can be in any
direction, but CLOCKWISE is the convention
2. Now write KVL for each mesh and apply Ohm’s
law to every resistor
3. At each loop, follow the passive sign convention
using loop current’s reference direction.
• Again, when an element has more than one loop
current flowing through it (R2), we compute net
current in the direction of travel for NOW
• In KVL equation for i1 → (i1–i2)R2
In KVL equation for i2 → (i2–i1)R2
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3.2 Loop Analysis
• Learning example: find IO using loop analysis
• Using mesh equations
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• Alternative selection of loops → IO=I1
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3.2 Loop Analysis
• Mesh equations “BY INSPECTION”
• MUST HAVE ALL MESH EQUATIONS WITH THE SAME ORIENTATION → CLOCKWISE!!
• In loop k, the coefficient of Ik is the sum of resistances around the loop
• The coefficient of Ij is the sum of resistances common to both ik and ij, with a negative sign
• The right hand side is the sum of voltage sources around the loop
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LOOP 1
12kI1 − 6kI 2 = 12
LOOP 2
− 6kI1 + 9kI 2 = −3
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3.2 Loop Analysis
• A practice example: find mesh equations “BY INSPECTION”
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3.2 Loop Analysis
• Learning extension: use mesh equations to find VO
• Procedure
1. Draw the mesh currents
2. Write mesh equations
3. Solve them!
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3.2 Loop Analysis
• Circuits with independent current sources: find both VO and V1 in
the following circuit
• CURRENT SOURCES THAT ARE NOT SHARED
BY OTHER MESHES (OR LOOPS) DEFINES A
MESH (LOOP) CURRENT AND REDUCE THE
NUMBER OF REQUIRED EQUATIONS
→ I1=2mA directly!
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3.2 Loop Analysis
• Find VO using mesh analysis
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3.2 Loop Analysis
• Find VO using mesh analysis
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3.2 Loop Analysis
• THE SUPERMESH APPROACH – current sources shared by loops
• We need to define the voltage Vx in order to have KVL equations for the mesh I2 and I3
• Then we need to combine two equations to remove Vx
• Instead, we can define a SUPERMESH like we used supernode for two nodes connected by a
voltage source.
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3.2 Loop Analysis
• THE SUPERMESH APPROACH – solving strategy
SUPERMESH
• 1. Define a supermesh with two meshes sharing the current source, removing the shared current
sources
2. Write the equation INSIDE the supermesh I 2 − I3 = 4mA
3. Write equations for the other meshes
4. Write KVL for the supermesh
I1 = 2mA
− 6 + 1kI3 + 2kI2 + 2k ( I 2 − I1 ) + 1k ( I3 − I1 ) = 0
or by inspection:
• 3 eqs with 3 unknowns. Solve the equations!
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Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Current sources shared by meshes – the general loop approach
THE STRATEGY IS TO DEFINE LOOP CURRENTS THAT DO NOT SHARE CURRENT SOURCES
- EVEN IF IT IS NOT A MESH!
• Start using mesh currents until reaching a shared source : I1 and I2
• Define a new loop after reaching a shared source : 4 mA source → I3 loop
• The new loop needs to include components not counted previously in order to guarantee
that it is an effective equation
• A possible strategy is to create a loop by opening the current source
• The loop equations for the loops with current sources are: I1=2mA, I2=4mA
• Only one equation for I3 loop is required:
– 6 – (1k+2k) I1 + (2k+2k) I2 + (1k+2k+2k+1k) I3 = 0
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Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Circuits with dependent sources
• Same as before:
Treat the dependent source like it is independent, then add one equation for the controlling variable
MESH CURRENTS
DETERMINED BY SOURCES
I1 = 4mA
VX
2k
MESH 3 : − 1kI x + 2k ( I3 − I1 ) + 1k ( I3 − I 4 ) = 0
MESH 4 : 1k ( I 4 − I3 ) + 1k ( I 4 − I 2 ) + 12V = 0
CONTROLLIN G VARIABLES
I x = I4 − I2
V x = 2k ( I 3 − I1 )
I2 =
I1 = 4
I1 + I 2 − I 3 = 0
I 2 + 3I3 − 2 I 4 = 8
− I 2 − I 3 + 2 I 4 = −12
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Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Example: use mesh analysis to find VO
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Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Learning extension: circuits with dependent sources
• We want to find Vx and VO in the following circuit. What would be the expression for Vx and VO in
each selection of loops?
The selection of loop currents simplifies
expression for Vx and computation of Vo.
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Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Example: use mesh analysis to find VO
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3.2 Loop Analysis
• Use NODE analysis to find VO in the following circuit
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3.2 Loop Analysis
• Use LOOP analysis to find VO in the following circuit
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Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Summary – Loop Analysis
• Second systematic technique to determine every voltage and current in a circuit
• Dual to node analysis – it first determines all currents in a circuit and then it uses Ohm’s law to
compute necessary voltages
• There are situation where node analysis is not efficient and where the number of equations
required by this new method is significantly smaller
• It is important to determine which technique is appropriate before we start
Basic Circuit Theory
Prof. Jong-Woo Kim
24
3.2 Loop Analysis
• Summary – developing loop equations
• #B=8, #N=7
• L = #B –(#N–1) = 8 – (7 – 1) = 2
• Strategy
1. Draw the mesh currents. Orientation can be in any
direction, but CLOCKWISE is the convention
2. Now write KVL for each mesh and apply Ohm’s
law to every resistor
3. At each loop, follow the passive sign convention
using loop current’s reference direction.
• Again, when an element has more than one loop
current flowing through it (R2), we compute net
current in the direction of travel for NOW
• In KVL equation for i1 → (i1–i2)R2
In KVL equation for i2 → (i2–i1)R2
Basic Circuit Theory
Prof. Jong-Woo Kim
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3.2 Loop Analysis
• Summary – Supermesh technique
SUPERMESH
• 1. Define a supermesh with two meshes sharing the current source, removing the shared current
sources
2. Write the equation INSIDE the supermesh
3. Write equations for the other meshes
4. Write KVL for the supermesh or by inspection
• 3 eqs with 3 unknowns. Solve the equations!
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Prof. Jong-Woo Kim
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