engineering 1333: electrical circuits 9/16/20059/12/2007kirchoff laws1 topic 4 the kirchhoff laws...
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9/16/20059/12/2007 Kirchoff Laws 1
Engineering 1333: Electrical Circuits
Topic 4
The Kirchhoff Laws
(Sections 2.3 & 2.4)
9/16/2005 Kirchoff Laws 2
A Circuit
+
-
+
-
R1 R2
R3R4
vs
Here’s a circuit with one voltage source and
several resistors
We’d like to “solve” the circuit. That is we’d like to find out:
(a) the voltage drop across each resistor
(b) the current that flows through it.
or
Either will do since we can use Ohm’s Law to
find the other
For example
4 4 4v R i
v4
+
-
i4
To do that we need to know how voltages and
currents combine
9/16/2005 Kirchoff Laws 3
The Kirchhoff Current Law
+
-
+
-
R1 R2
R3R4
vs
i1i3
i2
Tee Junction
i1 i2
i3
What goes in must come out
1 2 3i i i
9/16/2005 Kirchoff Laws 4
Effect of Reference Directions
+
-
+
-
R1 R2
R3R4
vs
i1i3
i2
i1 i2
i3
What goes in must come out
1 2 3 0i i i
1 2 3i i i
Arbitrarily changed current reference
directions
How many ways could we choose reference directions for this junction?
8
or
9/16/2005 Kirchoff Laws 5
In General
At any node
The sum of the incoming currents
The sum of the outgoing currents
=
1 2 3 4i i i i
(a node is a point where two or more
elements meet)
i2i1
i3i4
9/16/2005 Kirchoff Laws 6
Alternativelyi2i1
i3i4
The sum of all currents (treating incoming currents as positive and
outgoing currents as negative)
is zero
1 2 3 4 0i i i i
9/16/2005 Kirchoff Laws 7
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.
+
-
+
-
R1 R2
R3R4
vs
0
1 2 3
a b
There are 4 nodes in the circuit below
There are 3 loops
c
9/16/2005 Kirchoff Laws 8
Kirchhoff Voltage Law
The algebraic sum of all voltages around any closed path (loop) in a circuit is zero
+
-
+
-
R1 R2
R3R4
vs
…but which way is uphill and which is downhill?
We need to assign some voltages
v4
v1 v2
v3
+
-
--
-
++
+
Take your best shot
…and let the algebra sort it out.
a
2 4 3 0v v v
b1 3 0sv v v
c
1 2 4 0sv v v v Loop c
Loop b
Loop a
9/16/2005 Kirchoff Laws 9
Improved Model for a Battery
vo
io
vB
Voltage drops as current increases
+
-
+
-
vBvo
io
Ri
+
-
Ri
Internal Resistance
vi+ -
By KVL: 0i o Bv v v Solving for vo: o B iv v v
o B i ov v R i Applying Ohm’s Law: