Transcript The Kirchoff Laws

Topic 4
Engineering 1333: Electrical
Circuits
The Kirchhoff Laws
(Sections 2.3 & 2.4)
9/16/20059/12/2007
Kirchoff Laws
1
A Circuit
R1
Here’s a circuit with one
voltage source and
several resistors
i4
+
+
vs
R3
-
We’d like to “solve” the circuit.
That is we’d like to find out:
(a) the voltage drop
across each resistor
or
R4 v4
-
Either will do since we
can use Ohm’s Law to
find the other
For example
v4  R4i4
(b) the current that
flows through it.
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R2
Kirchoff Laws
To do that we need to know
how voltages and currents
combine
2
The Kirchhoff Current Law
+
vs
R1
R2
i1
i2
i3
Tee
Junction
R4
R3
-
i2
i1
What goes in
must come out
i1  i2  i3
i3
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Kirchoff Laws
3
Effect of Reference Directions
+
vs
R1
R2
i1
i2
i3
Arbitrarily changed
current reference
directions
R4
R3
-
What goes in
must come out
i2
i1
i3
i1  i2  i3  0
or
i1  i2  i3
How many ways could we choose
reference directions for this junction?
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Kirchoff Laws
8
4
In General
i2
i1
At any node
(a node is a point
where two or more
elements meet)
i3
i4
The sum of the
incoming currents
=
The sum of the
outgoing currents
i1  i2  i3  i4
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Kirchoff Laws
5
Alternatively
i1
i4
i2
i3
The sum of all currents (treating
incoming currents as positive and
outgoing currents as negative)
is
zero
i1  i2  i3  i4  0
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Kirchoff Laws
6
A Loop
Start at any node trace a closed path through the circuit
coming back to the original node without passing through
any intermediate node more than once.
There are 3
loops
There are 4 nodes
in the circuit below
R2
R1
2
1
3
c
+
a
vs
b
R3
R4
0
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Kirchoff Laws
7
Kirchhoff Voltage Law
The algebraic sum of all voltages around
any closed path (loop) in a circuit
We need to assign
some voltages
v1
+ R1
a
-
v3
v2
+ R2
Take your best shot
-
…and let the algebra sort it out.
+ c
R3
b
Loop a
R4 v4
-
Loop c
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zero
…but which way is uphill
and which is downhill?
+
+
vs
-
is
Kirchoff Laws
Loop b
v1  v3  vs  0
v2  v4  v3  0
v1  v2  v4  vs  0
8
Improved Model for a Battery
Voltage drops as
current increases
vo
vB
Ri
io
+
vi
io
-
Internal
Resistance
+
+
Ri
vo
vB
-
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By KVL:
Solving for vo:
Applying
Ohm’s Law:
Kirchoff Laws
vi  vo  vB  0
vo  vB  vi
vo  vB  Riio
9