Transcript Special Singularity Integrals Encountered In Electric Circuits

Calculation of the general impedance
between adjacent nodes of infinite
uniform N-dimensional resistive,
inductive, or capacitive lattices
Peter M. Osterberg & Aziz S. Inan
School of Engineering
University of Portland
ASEE 2009 (Austin, Texas)
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Outline
• Introduction and previous work
• Case 1: Infinite resistive lattice
• Case 2: Infinite inductive lattice
• Case 3: Infinite capacitive lattice
• Conclusion
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Introduction and previous work
• Special “adjacent node” circuit configurations of uniform, infinite extent are
excellent pedagogical vehicles for teaching and motivating EE students
(e.g., superposition, symmetry, etc).
• Infinite 2D square resistive lattice: Reff=R/2 (Aitchison)
• Infinite 2D honeycomb resistive lattice: Reff=2R/3 (Bartis)
• General N-dimensional resistive lattices: Reff=2R/M (Osterberg & Inan)
• This Presentation: Extend previous work to the most general problem of
finding the total effective impedance (Reff, Leff, or Ceff) between two adjacent
nodes of any infinite uniform N-dimensional resistive, inductive, or capacitive
lattice, where N=1, 2, or 3.
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Infinite 2D square lattice
a
b
(M=4)
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Infinite 2D honeycomb lattice
a
b
(M=3)
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Case 1: Infinite N-D resistive lattice
Ia = I
a
+
Va-b
Infinite resistive
lattice
b
Ib = I
Test circuit
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Infinite 2D honeycomb lattice
a
b
(M=3)
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Case 1: Infinite N-D resistive lattice
Va-b
 Ib 
 Ia 
  R   R
M 
M 
 I 
 2  R  Reff I
M 
Therefore:
Reff
Va-b 2 R


I
M
For an infinite 2D honeycomb resistive lattice where M=3, the
effective resistance between any two adjacent nodes is
Reff=2R/3 which agrees with Bartis.
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Case 2: Infinite N-D inductive lattice
Ia = I
a
+
la-b
Infinite inductive
lattice
b
Ib = I
Test circuit
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Infinite 2D honeycomb lattice
a
b
(M=3)
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Case 2: Infinite N-D inductive lattice
la-b
 Ia   Ib 
  L   L
M  M 
 I 
 2  L  Leff I
M 
Therefore:
Leff
la -b
2L


I
M
For an infinite 2D honeycomb inductive lattice where M=3, the
effective inductance between any two adjacent nodes is
Leff=2L/3.
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Case 3: Infinite N-D capacitive lattice
Ia = Qd(t)
a
+
Va-b
-
Infinite capacitive
lattice
b
Ib = Qd(t)
Test circuit
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Infinite 2D honeycomb lattice
a
b
(M=3)
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Case 3: Infinite N-D capacitive lattice

Qa M  Qb M 
V 

a -b
C
C

Q M
Q
2

C
Ceff
Therefore:
Ceff
Q
MC


Va-b
2
For an infinite 2D honeycomb capacitance lattice where M=3, the
effective capacitance between any two adjacent nodes is
Ceff=3C/2.
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Conclusions
•
Extended previous work to the most general problem of finding the total effective impedance
(Reff, Leff, or Ceff) between any two adjacent nodes of any infinite uniform N-dimensional
resistive, inductive, or capacitive lattice, where N=1, 2, or 3.
•
Using Kirchhoff’s laws, superposition and symmetry, along with Ohm’s law for a resistive lattice,
or the magnetic flux/current relationship for an inductive lattice, or the electrical charge/voltage
relationship for a capacitive lattice, a general and easy-to-remember solution is determined for
each of these cases, as follows, where M is the total number of elements connected to any
node of the lattice:
Reff=2R/M
Leff=2L/M
Ceff=MC/2
•
These general solutions are simple and of significant pedagogical interest to undergraduate EE
education.
•
They can serve as illustrations of the simplicity and elegance of superposition and symmetry
used with appropriate fundamental physical principles to facilitate more intuitive understanding
and simpler solution of such general problems.
•
The authors have recently added electrical circuit problems of this type to their first-semester
undergraduate electrical circuits course and received good feedback from the students.
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Any
questions?
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