We will consider that our electrical circuits consist of two or more circuit
elements interconnected by perfect conductors. The circuit elements can
be resistors, transistors, capacitors, inductors, integrated circuits, or any
other device which has an effect on voltage or current. Our
“perfect” conductors can be wires that allow current to flow from one
element to another. This representation of electrical circuits contains an
underlying assumption: only the circuit elements have an effect on the energy
storage or transfer in the circuit^{1}. Since the energy is lumped at
discrete points in the circuit (in the components), a circuit which is
represented this way is called a lumped-parameter circuit^{2}.
Figure 1 shows an example of a lumped parameters circuit. The circuit elements
A, B, C, and D affect the energy in the circuit; the connecting lines between
them are perfect conductors —they can transfer any amount of current
without storing or dissipating energy.
Circuit Nodes
Circuit elements in lumped parameters circuits are connected at nodes.
Identification of circuit nodes will be extremely important to us when we are
creating and analyzing circuits. Every node has a single unique voltage, so there
can be no voltage drops (and thus no circuit elements) within a node. Perfect
conductors do not cause voltage drops, so a node can contain perfect conductors.
An example circuit is shown in Fig. 2(a). A common error would be to identify
seven nodes in the circuit, as indicated by points a, b, c, d, e, f, and
g. However, nodes b and c are connected only by a perfect conductor, so there is
no voltage difference between those two points; they are part of the same node.
Likewise, a perfect conductor connects points f and g. Therefore, there are only
five nodes in the circuit, as shown in Fig. 2(b).
Definitions:
A node is an electrical “point” where two or more circuit
elements are connected. Since perfect conductors don't necessarily count
as circuit elements, nodes can contain perfect conductors.
Circuit nodes can also be identified by the fact that they are portions
of a circuit which are all at the same voltage. Since there is no
voltage drop across a perfect conductor, any points in a circuit which
are connected by perfect conductors will be at the same voltage, and will
thus be part of the same node.
Physically, circuit elements can be interconnected in a number of ways. We will
be creating our circuits on solderless breadboards, however, so we only have a
couple of options:
On a solderless breadboard, all of the holes in a single row are
connected by a conductor. Thus, all five of these holes automatically
form a single node, as shown in Fig. 3. This allows us to interconnect
components by inserting their terminals in holes in the same row.
We can also make connections with jumper wires. The jumper wires are
good conductors, so we will assume that points connected with jumper
wires become a single node. Figure 4 shows how this approach can be used
to create nodes with more than five holes on a breadboard.
Obviously, identifying any physical connector (such as a wire) as a
“perfect” conductor will be incorrect; any wire will have some
non-zero resistance, and will cause an amount of voltage difference between its
ends. The trick is that the resistance of the conductor should be negligible
compared to the resistance of the other components in the circuit in order for it
to be considered “close enough” to zero to make no difference. For
example, a wire with a resistance of 0.1Ω can probably be neglected
if it connects two 10kΩ resistors.
Important Points
Lumped-parameters circuits consist of discrete circuit components
connected by, what are assumed to be, perfect conductors. An underlying
assumption of lumped parameters circuits is that the perfect conductors
instantaneously transfer current with no voltage drop.
The lumped-parameters circuit model is inappropriate if the distances
involved are large or the voltages and currents are changing very
rapidly. In these cases, distributed-parameters models must be used.
These models are considerably more mathematically complex than
lumped-parameters models.
A node is an electrical “point” where two or more circuit
elements are connected. Nodes can be spread out from a single geometric
point with perfect conductors.
Circuit nodes can also be identified by the fact that they are portions
of a circuit which are all at the same voltage. Since there is no
voltage drop across a perfect conductor, any points in a circuit which
are connected by perfect conductors will be at the same voltage, and will
thus be part of the same node.
Test Your Knowledge!
How many nodes does the circuit below have?
How many nodes does the circuit below have?
Identify the nodes in the circuit below.
Identify the nodes in the circuit below.
Selected Answers
Three, as shown.
Five, as shown.
Five nodes, as indicated on the figure below.
Four nodes, as indicated on the figure below. (Note that both ground
terminals are connected internally within the Analog Discovery™.
This makes them the same node.)
^{1}Remember that our goal in creating any electrical circuit is
to transfer energy to perform some useful task. Since only the circuit elements
have an effect on the energy, they are the only things that really matter to us.
The conductors (or wires) interconnecting the circuits only exist to transfer the
energy between elements.
^{2}There are cases when the effects of the conductors between
circuit elements have to be accounted for. This occurs when the time required to
get the electricity from one point to another in a circuit becomes significant,
as when we are transmitting electricity over long distances (as in cross-country
power transmission lines) or when voltages and currents are changing extremely
rapidly (such as voltage changes associated with bit switching in modern
computers). Circuits of this type are called distributed-parameters circuits.
The mathematics required to analyze these types of circuits is, as you may
expect, considerably more difficult than for lumped-parameter circuits.