Signals and Systems
Boise State University
Electric Circuits
Electric Circuits
Chalmers
Chalmers
Communications
Systems Group,
Hani
Mehrpouyan,
Signals and Systems,
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Department of
Electrical and Computer Engineering,
Chalmers University
of Technology,
Sweden
Boise State
University
Sweden
c 2010
Lecture 5 (Mesh
c 2010 Analysis)
Sep 18th, 2015
August 20, 2015
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
Boise State c 2015
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Signals and Systems
Boise State University
Overview
Electric Circuits
• With Ohm’s and Kirchhoff’s law established, they may now be
applied to circuit analysis.
• Two techniques will be presented in this chapter:
– Nodal analysis, which is based on Kichhoff current law
(KCL)
Chalmers
– Mesh analysis, which is based on Kichhoff voltage law
Communications Systems Group,
(KVL)
Signals and Systems,
University of Technology,
• Any linear circuit Chalmers
can be analyzed
using these two techniques.
• The analysis will result in a Sweden
set of simultaneous equations
c 2010
which may be solved by Cramer’s
rule or computationally
(using MATLAB for example)
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
Signals and Systems
Boise State University
Mesh Analysis
Electric Circuits
• Another general procedure for analyzing circuits is to
use the mesh currents as the circuit variables.
• Recall:
– A loop is a closed path Chalmers
with no node passed more than once
Communications
Systems
Group,
– A mesh is a loop
that does not
contain
any other loop within it
Signals and Systems,
• Mesh analysis Chalmers
uses KVL
to find
unknown currents
University
of Technology,
• Mesh analysis is limited Sweden
in one aspect: It can only
2010
apply to circuits that can cbe
rendered planar.
• A planar circuit can be drawn such that there are no
August 20, 2015
crossing branches.
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
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Signals and Systems
Boise State University
Planar vs Nonpalanar
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
The figure on the left is a nonplanar
circuit: The branch with the 13Ω
August
resistor prevents the circuit from being
drawn without crossing branches
Hani Mehrpouyan (hani.mehr@ieee.org)
The figure on the right is a planar
circuit:
20,
2015It can be redrawn to avoid
crossing branches
Boise State c 2015
Signals and Systems
Boise State University
Mesh Analysis Steps
Electric Circuits
• Mesh analysis follows these steps:
1.Assign mesh currents i1,i2,…in to the n
Chalmers
meshes
Communications Systems Group,
Signals and
2.Apply KVL to each
ofSystems,
the n mesh
Chalmers University of Technology,
currents.
Sweden
2010
3.Solve the resultingc n
simultaneous
equations to get
the
mesh
August 20, 2015 currents
Boise State c 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
5
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Signals and Systems
Boise State University
Mesh Analysis Example
Electric Circuits
• The above circuit has two
paths that are meshes
(abefa and bcdeb)
• The outer loop (abcdefa) is a
loop, but not a mesh
Chalmers
and i2
• First, mesh currents i1 Communications
Systems Group,
Signals and Systems,
are assigned to the two
Chalmers University of Technology,
meshes.
Sweden
• Applying KVL to the meshes:
c 2010
− V1 + R1i1 + R3 (i1 − i2 ) = 0
⇓
(R1 + R3 )i1 − R3i2 = V1
Hani Mehrpouyan (hani.mehr@ieee.org)
R i + V + R3 (i2 − i1 ) = 0
2
2
August 20, 22015
⇓
− R3i1 + (R2 + R3 )i2 = −V2
Boise State c 2015
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
7
1
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Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
Signals and Systems
Boise State University
Mesh Analysis with Current
Sources
Electric Circuits
• The presence of a current source makes the mesh
analysis simpler in that it reduces the number of
equations.
• If the current source is located on only one mesh, the
Chalmersby the source.
current for that mesh is defined
Communications Systems Group,
• For example:
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
• Here, the current i2 is equal to -5A
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
9
1
Signals and Systems
Boise State University
Supermesh
Electric Circuits
• Similar to the case of nodal analysis where a voltage source
shared two non-reference nodes, current sources (dependent or
independent) that are shared by more than one mesh need
special treatment
• The two meshes must be joined together, resulting in a
Chalmers
supermesh.
Systems
Group,
• The supermesh isCommunications
constructed by
merging
the two meshes and
Signals and Systems,
excluding the shared source and any elements in series with
Chalmers University of Technology,
it
Sweden
• A supermesh is required because mesh analysis uses KVL
c 2010
• But the voltage across a current source cannot be known in
advance.
August 20, 2015
• Intersecting supermeshes in a circuit must be combined to for a
larger supermesh.
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
10 1
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
11 1
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
12 1
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
13 1
Signals and Systems
Boise State University
Selecting an Appropriate Approach
Electric Circuits
• In principle both the nodal analysis and mesh
analysis are useful for any given circuit.
• What then determines
if one is going to be
Chalmers
more efficientCommunications
for solving
a circuit problem?
Systems Group,
Signals and Systems,
• There are two factors
that dictate the best
Chalmers University of Technology,
choice:
Sweden
c 2010 network is the first
– The nature of the particular
factor
August 20, 2015
– The second factor
is the information required
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
14 1
Signals and Systems
Boise State University
Mesh analysis when…
Electric Circuits
• If the network contains:
–
–
–
–
Many series connected elements
Voltage sources
Supermeshes
A circuit with fewer meshes
than nodes
Chalmers
• If branch or mesh
currents are
what
is being solved
Communications
Systems
Group,
Signals and Systems,
for.
Chalmers University of Technology,
• Mesh analysis is the only
suitable analysis for
Sweden
transistor circuits
c 2010
• It is not appropriate for operational amplifiers because
August
20, 2015
there is no direct way
to obtain
the voltage across an
op-amp.
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
15 1
Signals and Systems
Boise State University
Nodal analysis if…
Electric Circuits
• If the network contains:
–
–
–
–
Many parallel connected elements
Current sources
Supernodes
Chalmers
Circuits with fewer
nodes than
meshes
Communications
Systems
Group,
Systems,
• If node voltages areSignals
whatand
are
being solved for
Chalmers University of Technology,
• Non-planar circuits can only
be solved using nodal
Sweden
analysis
c 2010
• This format is easier to solve by computer
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
24
16 1
Signals and Systems
Boise State University
Circuit Analysis with PSpice
Electric Circuits
• PSpice is a common program used for circuit analysis.
• It is capable of determining all of the branch voltages
and currents if the numerical values for all circuit
Chalmers
components are known.
Communications Systems Group,
• Analysis using PSpice
begins with drawing a
Signals and Systems,
schematic viewChalmers
of the University
circuit. of Technology,
Sweden can be indicated during
• The node voltages of interest
c 2010
the schematic setup using
‘VIEWPOINTS’
• The values are obtained by running ‘Analysis/
August 20, 2015
Simulate’
Hani Mehrpouyan (hani.mehr@ieee.org)
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Boise State c 2015
Signals and Systems
Boise State University
Rendering a circuit in PSpice
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August
2015 schematic (bottom) is fairly
Converting a standard schematic
(top) 20,
to a PSpice
straightforward. Note the explicit definition of the reference node by using the
ground symbol labeled with a ‘0’ and the nodal voltages of interest displayed
with the ‘viewpoints’
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
26
18 1
Signals and Systems
Boise State University
Application: DC transistor circuit
•
•
•
Electric Circuits
In general, there are two types of transistors
commonly used: Field Effect (FET) and
Bipolar Junction (BJT).
A BJT is a three terminal device, where the
input current into one terminalChalmers
(the base)
affects the current flowing out of a second
Communications Systems Group,
terminal (the collector).
Signals and Systems,
The third terminal (the
emitter) is the
Chalmers University of Technology,
common terminal for both currents
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
27
19 1
Boise State c 2015
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
20 1
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
21 1
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
22 1
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
23 1
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
24 1
Signals and Systems
Boise State University
Electric Circuits
Chalmers
Communications Systems Group,
Signals and Systems,
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
25 1