305221, 226231
Computer Electrical Circuit
Analysis
การวิเคราะห ์วงจรไฟฟ้าทาง
คอมพิวเตอร ์
3(2-3-6)
ณรงค์ชยั
มุ่งแฝงกลาง
คมกริ ช มาเที่ยง
สัปดาห์ที่ 3
Nodal and Mesh Analysis
Outline
1
Objectives
2
Reviews
3
Nodal Analysis
4
Mesh Analysis
5
Source Transform
Objectives
 เพื่อให้ นิสิตมีความรู้ ความเข้ าใจเกีย่ วกับ Nodal Analysis
 เพื่ อให้นิ สิ ต มี ค วามรู ค
้ วามเข้า ใจเกี่ยวกับ
Mesh Analysis
 เพื่อให้ นิสิตมีความรู้ ความเข้ าใจเกีย่ วกับ Supernodal Analysis
เพื่อให้ นิสิตมีความรู้ความเข้ าใจเกีย่ วกับ Supermesh Analysis
เพื่อให้ นิสิตมีความรู้ความเข้ าใจเกีย่ วกับ Source Transform
Outline
1
Objectives
2
Reviews
3
Nodal Analysis
4
Mesh Analysis
5
Source Transform
Ohm’s Law (cont.)
 Ohm’s law states that “the voltage v across a resistor is
directly proportional to the current i flowing through the
resistor”.
v  i
v  iR
R 
Note : 1  = 1 V/A
v
i
Power
P  vi
P 
2
v
v
; i
R
R
P  i R; v  iR
2
Example of Power Consumption
 From the figure, determine the current i and the power.
Branch
 Matthew Sadiku, define A
branch represents a single
element such as a voltage source or a resistor.
 William Hayt, define a branch as a single path in a network,
composed of one simple element and the node at each end of
that element.
Branch
Node
Matthew Sadiku, a node is the point of connection
between two or more branches.
William Hayt, a point at which two or more elements
have a common connection is called “a node”.
Node
Loop
 Matthew Sadiku, A loop is any closed path in a
circuit.
William Hayt, If the node at which we started is the
same as the node on which we ended, then the path
is, by definition, a closed path or “a loop”.
Kirchhoff’s Current Law
 Kirchhoff’s current law (KCL) states that “the
algebraic sum of currents entering a node (or
leaving the node) is zero”.
Kirchhoff’s Voltage Law
Kirchhoff’s voltage law (KVL) states “that the
algebraic sum of all volt-ages around a closed path
(or loop) is zero”.
Equivalent of Series Resistors
Req
 R1  R2
 The equivalent resistance of any number of resistors
connected in series is the sum of the individual resistances.
Req
 R1  R2 
N
 RN   Rn
n 1
Voltage Division
v1

R1
v ; v2
R1  R2

R2
v
R1  R2
Voltage Division Example
Parallel Resistors
1
Req
1
Req
Req

1 1

R1 R2

R1  R2
R1 * R2

R1 * R2
R1  R2
The equivalent resistance of two parallel resistors is
equal to the product of their resistances divided by their
sum.
Equivalent Conductances
 The equivalent conductance of resistors connected
in parallel is the sum of their individual conductances.
Current Division
i1

R2
i ; i2
R1  R2

R1
i
R1  R2
Current Division Example
Outline
1
Objectives
2
Reviews
3
Nodal Analysis
4
Mesh Analysis
5
Source Transform
Nodal Analysis

Step to determine voltages:
1. Select a node as the reference node. Assign voltages v1,v2,…vn-1 to
the remaining n-1 nodes. The voltages are referenced with respect
to the reference node.
2. Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law
to express the branch current in terms of node voltage.
3. Solve the resulting simultaneous equations to obtain the unknown
node voltages.
Nodal Analysis : Step 1
Symbol for reference node
Assign voltages
Reference Node
Node
Nodal Analysis : Step 2 and 3

Apply ohm’s law
 Apply KCL:

Then solve equation
Nodal Analysis Example
2. Assign Voltages
1. Reference Node
Nodal Analysis Example (cont.)
At node 1 :
1
At node 2 :
2
At node 3 :
3
Nodal Analysis Example (cont.)
Adding Eq.1 and 3 obtain :
4
Adding Eq.2 and 3 obtain :
5
Nodal Analysis Example (cont.)
Substituting Eq. 5 into Eq. 4 :
From Eq. 3:
Answer :
Nodal Analysis with Voltage Source
A supernode is formed by enclosing a (dependent or
independent) voltage source connected between two
non-reference nodes and any elements connected in
parallel with it.
Nodal Analysis with Voltage Source (cont.)
Apply KCL:
Or :
Nodal Analysis with Voltage Source (cont.)
Redraw :
Apply KVL :
Nodal Analysis with Voltage Source Example
Supernode
Nodal Analysis with Voltage Source Example
i
Apply KCL :
1
Apply KVL :
2
Substituting Eq. 2 into Eq. 1 :
Summary of Supernode Analysis Procedure
1.
Count the number of nodes (N).
2. Designate a reference node. The number of terms in your nodal equations
can be minimized by selecting the node with the greatest number of
branches connected to it.
3. Label the nodal voltages(there are N − 1 of them).
4. If the circuit contains voltage sources, form a supernode about each one.
This is done by enclosing the source, its two terminals, and any other
elements connected between the two terminals within a broken-line
enclosure.
5. Write a KCL equation for each nonreference node and for each supernode
that does not contain the reference node. Sum the currents flowing into a
node/supernode from current sources on one side of the equation. On the
other side, sum the currents flowing out of the node/supernode through
resistors. Pay close attention to “−” signs.
Summary of Supernode Analysis Procedure
6. Relate the voltage across each voltage source to nodal voltages. This is
accomplished by simple application of KVL; one such equation is needed
for each supernode defined.
7. Express any additional unknowns (i.e., currents or voltages other than
nodal voltages) in terms of appropriate nodal voltages. This situation can
occur if dependent sources appear in our circuit.
8. Organize the equations. Group terms according to nodal voltages.
9. Solve the system of equations for the nodal voltages (there will be N − 1 of
them).
Outline
1
Objectives
2
Reviews
3
Nodal Analysis
4
Mesh Analysis
5
Source Transform
Mesh Definition
 A Mesh is a loop which does not contain any other
loop within it.
Mesh Analysis

Steps to determine Mesh current:
1. Assign mesh currents i1, i2, …, in to the n meshes.
2. Apply KVL to each of the n meshes. Use Ohm’s law to express the
voltages in terms of the mesh current.
3. Solve the resulting n simultaneous equations to get the mesh
currents.
Mesh Analysis : Step 1
Assign mesh current
Mesh Analysis : Step 2
Apply KVL to each mesh
Mesh i1 :
Mesh i2 :
Mesh Analysis : Step 3
Solve the resulting :
Mesh Analysis Example
Assign mesh current
Mesh Analysis Example (cont.)
Apply KVL for mesh 1 :
Or
1
Apply KVL for mesh 2 :
Or
2
Mesh Analysis Example (cont.)
Substituting Eq.2 into Eq.1 :
From Eq.2 :
Answers:
Mesh Analysis with Current Source
CASE 1 : When set i2=-5A and write a mesh equation for other mesh:
Mesh Analysis with Current Source
CASE 2:
Apply KVL :
Or
Mesh Analysis with Current Source
At node 0, Apply KCL :
Solving equation, obtain :
Mesh Analysis with Current Source Example
Supermesh
Mesh Analysis with Current Source Example
Apply KVL to supermesh :
Or
1
Apply KCL to node P :
2
Mesh Analysis with Current Source Example
Apply KCL to node Q :
But :
3
Apply KVL to mesh 4 :
Or
4
Mesh Analysis with Current Source Example
From Eq.1 to Eq.4 :
Summary of Supermesh Analysis Procedure
1. Determine if the circuit is a planar circuit. If not, perform nodal
analysis instead.
2. Count the number of meshes(M). Redraw the circuit if necessary.
3. Label each of the M mesh currents. Generally, defining all mesh
currents to flow clockwise results in a simpler analysis.
4. If the circuit contains current sources shared by two meshes, form a
supermesh to enclose both meshes. A highlighted enclo-sure helps when
writing KVL equations.
5. Write a KVL equation around each mesh/supermesh. Begin with a
convenient node and proceed in the direction of the mesh current. Pay close
attention to “ −” signs. If a current source lies on the periphery of a mesh, no
KVL equation is needed and the mesh current is determined by inspection.
Summary of Supermesh Analysis Procedure
6. Relate the current flowing from each current source to mesh currents.
This is accomplished by simple application of KCL; one such equation is
needed for each supermesh defined.
7. Express any additional unknowns such as voltages or currents other than
mesh currents in terms of appropriate mesh currents. This situation can
occur if dependent sources appear in our circuit.
8. Organize the equations. Group terms according to nodal voltages.
9. Solve the system of equations for the mesh currents (there will be M of
them).
Outline
1
Objectives
2
Reviews
3
Nodal Analysis
4
Mesh Analysis
5
Source Transform
Source Transform
 Source Transform is the process of replacing a
voltage source Vs in series with a resistor R by a
current source is in parallel with a resistor R, or vice
versa.
Source Transform Example
Source Transform Example
Source Transform Example (cont.)
Source Transform Example (cont.)
Assignment s
 Matthew Sadiku : 3.4, 3.15, 3.18, 3.19 and 3.46
William Hayt : 10, 12, 13, 15, 22, 23, 40, 45 and 46