DC circuit analysis technique
Any
black box containing resistances only and voltage and current sources can be replaced by an equivalent circuit consisting of an equivalent current source in parallel connection with an equivalent resistance.
Edward Lawry Norton
In
direct-current
circuit theory , Norton's theorem , also called the Mayer–Norton theorem , is a simplification that can be applied to
networks made of
linear time-invariant
resistances ,
voltage sources , and
current sources . At a pair of terminals of the network, it can be replaced by a current source and a single resistor in parallel.
For
alternating current (AC) systems the theorem can be applied to
reactive
impedances as well as resistances.
The Norton equivalent circuit is used to represent any network of linear sources and impedances at a given
frequency .
Norton's theorem and its dual,
Thévenin's theorem , are widely used for circuit analysis simplification and to study circuit's
initial-condition and
steady-state response.
Norton's theorem was independently derived in 1926 by
Siemens & Halske researcher
Hans Ferdinand Mayer (1895–1980) and
Bell Labs engineer
Edward Lawry Norton (1898–1983).
[1]
[2]
[3]
[4]
[5]
[6]
To find the equivalent, the Norton current I no is calculated as the current flowing at the terminals into a
short circuit (zero resistance between A and B ). This is I no . The Norton resistance R no is found by calculating the output voltage produced with no resistance connected at the terminals; equivalently, this is the resistance between the terminals with all (independent) voltage sources short-circuited and independent current sources
open-circuited . This is equivalent to calculating the Thevenin resistance.
When there are dependent sources, the more general method must be used. The voltage at the terminals is calculated for an injection of a 1 amp test current at the terminals. This voltage divided by the 1 A current is the Norton impedance R no (in ohms). This method must be used if the circuit contains dependent sources, but it can be used in all cases even when there are no dependent sources.
Example of a Norton equivalent circuit
The original circuit Calculating the equivalent output current Calculating the equivalent resistance Design the Norton equivalent circuit
In the example, the total current I total is given by:
I
t
o
t
a
l
=
15
V
2
k
Ω
+
1
k
Ω
∥
(
1
k
Ω
+
1
k
Ω
)
=
5.625
m
A
.
{\displaystyle I_{\mathrm {total} }={15\,\mathrm {V} \over 2\,\mathrm {k} \Omega +1\,\mathrm {k} \Omega \parallel (1\,\mathrm {k} \Omega +1\,\mathrm {k} \Omega )}=5.625\,\mathrm {mA} .}
The current through the load is then, using the
current divider rule :
I
n
o
=
1
k
Ω
+
1
k
Ω
1
k
Ω
+
1
k
Ω
+
1
k
Ω
⋅
I
t
o
t
a
l
=
2
/
3
⋅
5.625
m
A
=
3.75
m
A
.
{\displaystyle {\begin{aligned}I_{\mathrm {no} }&={1\,\mathrm {k} \Omega +1\,\mathrm {k} \Omega \over 1\,\mathrm {k} \Omega +1\,\mathrm {k} \Omega +1\,\mathrm {k} \Omega }\cdot I_{\mathrm {total} }\\[5pt]&=2/3\cdot 5.625\,\mathrm {mA} =3.75\,\mathrm {mA} .\end{aligned}}}
And the equivalent resistance looking back into the circuit is:
R
n
o
=
1
k
Ω
+
(
2
k
Ω
∥
(
1
k
Ω
+
1
k
Ω
)
)
=
2
k
Ω
.
{\displaystyle R_{\mathrm {no} }=1\,\mathrm {k} \Omega +(2\,\mathrm {k} \Omega \parallel (1\,\mathrm {k} \Omega +1\,\mathrm {k} \Omega ))=2\,\mathrm {k} \Omega .}
So the equivalent circuit is a 3.75 mA current source in parallel with a 2 kΩ resistor.
Conversion to a Thévenin equivalent
To a Thévenin equivalent
A Norton equivalent circuit is related to the
Thévenin equivalent by the equations:
R
t
h
=
R
n
o
V
t
h
=
I
n
o
R
n
o
V
t
h
R
t
h
=
I
n
o
{\displaystyle {\begin{aligned}&R_{\rm {th}}=R_{\rm {no}}\\[8pt]&V_{\rm {th}}=I_{\rm {no}}R_{\rm {no}}\\[8pt]&{\frac {V_{\rm {th}}}{R_{\rm {th}}}}=I_{\rm {no}}\end{aligned}}}
Queueing theory
The passive circuit equivalent of "Norton's theorem" in
queuing theory is called the
Chandy Herzog Woo theorem .
[3]
[4]
[7] In a
reversible queueing system , it is often possible to replace an uninteresting subset of queues by a single (
FCFS or
PS ) queue with an appropriately chosen service rate.
[8]
See also
References
^
Mayer, Hans Ferdinand (1926). "Ueber das Ersatzschema der Verstärkerröhre" [On equivalent circuits for electronic amplifiers]. Telegraphen- und Fernsprech-Technik (in German). 15 : 335–337.
^
Norton, Edward Lawry (1926). "Design of finite networks for uniform frequency characteristic".
Bell Laboratories . Technical Report TM26–0–1860.
^
a
b Johnson, Don H. (2003).
"Origins of the equivalent circuit concept: the voltage-source equivalent" (PDF) .
Proceedings of the IEEE . 91 (4): 636–640.
doi :
10.1109/JPROC.2003.811716 .
hdl :
1911/19968 .
^
a
b Johnson, Don H. (2003).
"Origins of the equivalent circuit concept: the current-source equivalent" (PDF) .
Proceedings of the IEEE . 91 (5): 817–821.
doi :
10.1109/JPROC.2003.811795 .
^ Brittain, James E. (March 1990).
"Thevenin's theorem" .
IEEE Spectrum . 27 (3): 42.
doi :
10.1109/6.48845 .
S2CID
2279777 . Retrieved 2013-02-01 .
^
Dorf, Richard C. ; Svoboda, James A. (2010).
"Chapter 5: Circuit Theorems" . Introduction to Electric Circuits (8th ed.). Hoboken, NJ, USA:
John Wiley & Sons . pp. 162–207.
ISBN
978-0-470-52157-1 . Archived from
the original on 2012-04-30. Retrieved 2018-12-08 .
^
Gunther, Neil J. (2004).
Analyzing Computer System Performance with Perl::PDQ (Online ed.). Berlin:
Springer Science+Business Media . p. 281.
ISBN
978-3-540-20865-5 .
^
Chandy, Kanianthra Mani ; Herzog, Ulrich; Woo, Lin S. (January 1975).
"Parametric Analysis of Queuing Networks" .
IBM Journal of Research and Development . 19 (1): 36–42.
doi :
10.1147/rd.191.0036 .
External links