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Electric Circuits and Interconnection Laws
Summary: A circuit is the electrical/physical analogue of a system. Kirchoff's laws provde connection rules about "nodes" in a circuit to determine relations between various circuit elements.
A circuit connects circuit elements together in a
specific configuration designed to transform the source signal
(originating from a voltage or current source) into another
signal—the output—that corresponds to the current
or voltage defined for a particular circuit element. A simple
resistive circuit is shown in Figure 1. This circuit is the electrical embodiment of a
system having its input provided by a source system producing
v
in
t
v
in
t
 Subfigure 1.1  Subfigure 1.2  Subfigure 1.3 Figure 1:
The circuit shown in the top two figures is perhaps the
simplest circuit that performs a signal processing
function. On the bottom is the block diagram that corresponds
to the circuit. The input is provided by the voltage source
|
To understand what this circuit accomplishes, we want to
determine the voltage across the resistor labeled by its value
R
2
R
2
v
out
v
out
What we need to solve every circuit problem are
mathematical statements that express how the circuit elements are interconnected. Said another way, we need the laws that govern the
electrical connection of circuit elements. First of all, the
places where circuit elements attach to each other are called
nodes. Two nodes are explicitly indicated in Figure 1; a third is at the bottom
where the voltage source and resistor
R
2
R
2
Kirchoff's Current Law
At every node, the sum of all currents entering a node must
equal zero. What this law means physically is that charge
cannot accumulate in a node; what goes in must come out. In
the example, Figure 1,
below we have a three-node circuit and thus have three KCL
equations.
- i -
i
1
= 0
i
i
1
0
i
1
-
i
2
= 0
i
1
i
2
0
i +
i
2
= 0
i
i
2
0
Given any two of these KCL equations, we can find the other by
adding or subtracting them. Thus, one of them is redundant
and, in mathematical terms, we can discard any one of
them. The convention is to discard the equation for the
(unlabeled) node at the bottom of the circuit.
Figure 2:
The circuit shown is perhaps the simplest circuit that
performs a signal processing function. The input is provided
by the voltage source
|
Problem 1
In writing KCL equations, you will find that in an
n n
Solution 1
KCL says that the sum of currents entering or leaving a
node must be zero. If we consider two nodes together as a
"supernode", KCL applies as well to currents entering the
combination. Since no currents enter an entire circuit,
the sum of currents must be zero. If we had a two-node
circuit, the KCL equation of one must
be the negative of the other, We can combine all but one
node in a circuit into a supernode; KCL for the supernode
must be the negative of the remaining node's KCL equation.
Consequently, specifying n - 1 n 1
Kirchoff's Voltage Law (KVL)
The voltage law says that the sum of voltages around every
closed loop in the circuit must equal zero. A closed loop has
the obvious definition: Starting at a node, trace a path
through the circuit that returns you to the origin node. KVL
expresses the fact that electric fields are conservative: The
total work performed in moving a test charge around a closed
path is zero. The KVL equation for our circuit is
v
1
+
v
2
- v = 0
v
1
v
2
v
0
Figure 3:
The circuit shown is perhaps the simplest circuit that
performs a signal processing function. The input is provided
by the voltage source
|
For the example
circuit, we have three
v-i relations, two KCL equations, and
one KVL equation for solving for the circuit's six voltages and
currents.
v-i:
v =
v
in
v
v
in
v
1
=
R
1
i
1
v
1
R
1
i
1
v
out
=
R
2
i
out
v
out
R
2
i
out
- i -
i
1
= 0
i
i
1
0
i
1
-
i
out
= 0
i
1
i
out
0
- v +
v
1
+
v
out
= 0
v
v
1
v
out
0
We have exactly the right number of equations! Eventually, we
will discover shortcuts for solving circuit problems; for now,
we want to eliminate all the variables but
v
out
v
out
v
in
v
in
v
in
=
v
1
+
v
out
v
in
v
1
v
out
v
in
=
R
1
i
1
+
R
2
i
out
v
in
R
1
i
1
R
2
i
out
i
1
=
i
out
i
1
i
out
v
in
=
R
1
i
out
+
R
2
i
out
=
R
1
+
R
2
i
out
v
in
R
1
i
out
R
2
i
out
R
1
R
2
i
out
i
out
=
v
in
R
1
+
R
2
i
out
v
in
R
1
R
2
v
out
=
R
2
R
1
+
R
2
v
in
v
out
R
2
R
1
R
2
v
in
Problem 2
Referring back to
Figure 1,
a circuit should serve some useful purpose. What kind of
system does our circuit realize and, in terms of element
values, what are the system's parameter(s)?
Solution 2
The circuit serves as an amplifier having a gain of
R
2
R
1
+
R
2
R
2
R
1
R
2
More about this content: Metadata | Version History | Cite This Content
This work is licensed by Don Johnson. See the Creative Commons License about permission to reuse this material.
Last edited by Don Johnson on May 10, 2007 8:49 am GMT-5.
"Electrical Engineering Digital Processing Systems in Braille."