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 8 (Norton’s
c 2010 Theorem)
Oct 5th, 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
1
1
1
Signals and Systems
Boise State University
Overview
Circuits
• In this Electric
chapter, the concept
of
superposition will be introduced.
• Source transformation will also be
Chalmers
covered. Communications
Systems Group,
Signals and Systems,
• Thevenin and
Norton’s
theorems will be
Chalmers University of Technology,
covered.
Sweden
c 2010
• Examples of applications for these
August
20, 2015
concepts will be
presented.
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
22 1
Signals and Systems
Boise State University
Norton’s Theorem
Electric Circuits
• Similar to Thevenin’s
theorem, Norton’s theorem
states that a linear two
terminal circuit mayChalmers
be
Communications Systems Group,
replaced with an Signals
equivalent
and Systems,
Chalmers
of Technology,
circuit containing
aUniversity
resistor
and a current sourceSweden
c 2010
• The Norton resistance will
August
2015
be exactly the same
as20,the
Thevenin
Hani Mehrpouyan (hani.mehr@ieee.org)
17
3 1
Boise State c 2015
Signals and Systems
Boise State University
Norton’s Theorem II
Electric Circuits
• The Norton current IN is found by short
circuiting the circuit’s terminals and
Chalmers current
measuring the resulting
I =i
Communications Systems Group,
N andscSystems,
Signals
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
18
4 1
Signals and Systems
Boise State University
Norton vs. Thevenin
Electric Circuits
• These two equivalent circuits can be related to
each other
• One need only lookChalmers
at source transformation
to understandCommunications
this
Systems Group,
Signals and Systems,
• The Norton current
and Thevein voltage are
Chalmers University of Technology,
related to each otherSweden
as follows:
VThc 2010
RTh 20, 2015
August
IN =
Hani Mehrpouyan (hani.mehr@ieee.org)
19
5 1
Boise State c 2015
Signals and Systems
Boise State University
Norton vs. Thevenin II
Electric Circuits
• With VTH, IN, and (RTH=RN) related, finding the
Thevenin or Norton equivalent circuit requires
that we find:
Chalmers
• The open-circuit
voltageSystems
across
Communications
Group,terminals a
Signals and Systems,
and b.
Chalmers University of Technology,
• The short-circuit current
at terminals a and b.
Sweden
c 2010
• The equivalent or input
resistance at terminals
a and b when allAugust
independent
20, 2015 sources are
turned off.
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
20
6 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
7
1
8
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
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
9
1
Signals and Systems
Boise State University
Maximum Power Transfer
Electric Circuits
• In many applications, a circuit is
designed to power a load
Chalmers
• Among those applications
there are
Systems Group,
many casesCommunications
where
we
wish
to maximize
Signals and Systems,
Chalmers
University
of
Technology,
the power transferred to the load
Sweden
• Unlike an ideal source,
c 2010 internal
resistance will restrict the conditions
August 20, 2015
where maximum power is transferred.
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
21
10 1
Signals and Systems
Boise State University
Maximum Power Transfer II
Electric Circuits
• We can use the Thevenin
equivalent circuit for finding the
maximum power in a linear circuit
Chalmers
• We will assume that the
load
Communications
resistance can be
varied Systems Group,
Signals and Systems,
• Looking at the Chalmers
equivalent
circuit
University
of Technology,
with load included, the power
Sweden
transferred is:
c 2010
2
⎛ VTh ⎞August 20, 2015
p=⎜
⎟ RL
⎝ RTh + RL ⎠
Hani Mehrpouyan (hani.mehr@ieee.org)
22
11 1
Boise State c 2015
Signals and Systems
Boise State University
Maximum Power Transfer III
Electric Circuits
• For a given circuit, VTH and
RTH are fixed. By varying
the load resistance RL, the
power delivered to the
load
Chalmers
varies as shown
Communications Systems Group,
Signals
as
RLand Systems,
• You can see that
Chalmers University of Technology,
approaches 0 and ∞ the
Sweden
power transferred goesc 2010
to
zero.
August 20, 2015
• In fact the maximum power
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
23
12 1
Signals and Systems
Boise State University
Pspice?
Electric
Circuits
• The Thevenin
and Norton
equivalent
circuits are useful in understanding the
behavior of realistic sources
Chalmers
• Ideal voltage
sources
have no internal
Communications Systems Group,
Signals and Systems,
resistance
Chalmers University of Technology,
• Ideal current sources
Swedenhave infinite
internal resistance c 2010
20, 2015 circuits
• The Thevenin August
and Norton
introduce deviations from these ideals
Hani Mehrpouyan (hani.mehr@ieee.org)
24
13 1
Boise State c 2015
Signals and Systems
Boise State University
Source Modeling
Electric Circuits
• Take the Thevenin circuit with load resistor:
• The internal resistor and the load act a voltage
divider.
Chalmers
• The lower theCommunications
load resistance,
the more
Systems Group,
and Systems,
voltage drop thatSignals
occurs
in the source
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
25
14 1
Signals and Systems
Boise State University
Source Modeling II
Electric Circuits
• This means that as the load resistance
increases, the voltage source comes
closer to operating
like the ideal source.
Chalmers
Communications Systems Group,
• Similarly, with aSignals
realistic
current source,
and Systems,
University
Technology, with the
the internalChalmers
resistor
inofparallel
Sweden
ideal source acts toc 2010
siphon away current
that would otherwise go to the load.
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
26
15 1
Boise State c 2015
Signals and Systems
Boise State University
Source Modeling III
Electric Circuits
• Here, the load and the
internal resistor act as a
current divider.
Chalmers
• From that perspective,
the
Communications Systems Group,
Signals and Systems,
lower the load resistance,
the
University of Technology,
more currentChalmers
passes
through
Sweden
it.
c 2010
• Thus lower load resistance
20, 2015
leads to behaviorAugust
closer
to
the ideal source.
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
27
16 1
Signals and Systems
Boise State University
Resistance Measurement
Electric Circuits
• Although the ohmmeter is the most common
method for measuring resistance, there is a
more accurate method
• It is called the Wheatstone
Chalmers bridge
Communications
Systems
• It is based on theSignals
principle
ofGroup,
the voltage
and Systems,
divider
Chalmers University of Technology,
Sweden
c 2010
August 20, 2015
Hani Mehrpouyan (hani.mehr@ieee.org)
28
17 1
Boise State c 2015
Signals and Systems
Boise State University
Resistance Measurement
Electric Circuits
• Using three known resistors and a
galvanometer, an unknown resistor can be
tested
Chalmers
• The unknownCommunications
resistor isSystems
placed
at the position
Group,
Signals and Systems,
R4
Chalmers University of Technology,
R2 is adjusted until the
• The variable resistorSweden
2010
galvanometer shows czero
current
• At this point, the August
bridge20,is2015
“balanced” and the
voltages from the two dividers are equal
Hani Mehrpouyan (hani.mehr@ieee.org)
Boise State c 2015
29
18 1
Signals and Systems
Boise State University
Balanced Bridge
Electric Circuits
• When balanced, the unknown resistor’s value
R3
is
Rx =
R1
R2
• The key to the highChalmers
accuracy lies in the fact
that any slightCommunications
difference
in the voltage
Systems Group,
Systems,flow
dividers will lead Signals
to a and
current
Chalmers University of Technology,
Sweden
V
R + Rm
AugustTh20, 2015
I=
Hani Mehrpouyan (hani.mehr@ieee.org)
c 2010
Th
30
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