Download Circuits Lecture 8: Node Analysis (2)

Circuits
Lecture 3: Mesh Analysis
李宏毅 Hung-yi Lee
Mesh 1
Introduction
Mesh 2
Mesh 3
• Node analysis:
Current +
Voltage
Voltage
Potential
Current
Mesh Current
• Mesh analysis:
Current +
Voltage
Mesh: closed current path that contains no closed paths
within it (Textbook, P149)
Meshes are “holes” in the circuits.
Mesh analysis
i1  i2
i2  i3
i1  i3
Target of Mesh analysis is to find the mesh currents.
Mesh analysis
Consider KCL:
ib  ic  id  0
Represent ib, ic and id by mesh current
ib  i1 ic  i2  i1 id  i2
KCL is already fulfilled
Mesh analysis
Consider KVL:
va  vb  vc  vs
Represent va, vb and vc by mesh current
va  Ra i1  is 
vb  Rbi1
vc  Rc i1  i2 
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 vs as
unknown variables
vs
vd  vb  vs  0
vc  ve  vs
……
is
Represent the voltages vd, vb, vc and
ve …. (except vs) by mesh current
Solve mesh currents and vs
One more equation: is
 i2  i1
Mesh analysis - current sources
• 2. Interior current sources
Method 2: when the current source is
parallel with a resistor
Textbook:
Chapter 2.5
Mesh analysis - current sources
• 2. Interior current sources
Method 2: when the current source is
parallel with a resistor
Another point of view:
The voltage of is is equal to Rs.
vs  Rs im  ix 
Mesh analysis - current sources
• 2. Interior current sources
Method 3:
supermesh
Rd i1
i3
Rb i1  i3 
vs
Rc i2  i3   Rc is  i1  i3 
is
Rei2
 Re is  i1 
is  i2  i1
i2  is  i1
Example 4.9
10i1
6i1  5
3i1  4 
6(i1-5) + 10i1 + 3(i1+4) = 20
→i1=2A
Exercise 4.11
Special case:
current sources are
parallel with resistors
Exercise 4.11
i1  3mA
6k
i1  2mA
i1
5k
i1  5k  i1  3m  4k  i1  2m  6k  0
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
A
C
B
Non-planar Circuit
• Some non-planar circuits can be transformed into
planar circuit
Non-planar Circuit
• Why mesh analysis cannot be applied on nonplanar 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
• 感謝 林楷恩 (b02)
• 糾正錯誤的作業答案