Simple parallel circuits
Let's start with a parallel circuit consisting of three resistors
and a single battery:
The first principle to understand about parallel circuits is that
the voltage is equal across all components in the circuit. This is
because there are only two sets of electrically common points in a
parallel circuit, and voltage measured between sets of common points
must always be the same at any given time. Therefore, in the above
circuit, the voltage across R1 is equal to the voltage
across R2 which is equal to the voltage across R3
which is equal to the voltage across the battery. This equality of
voltages can be represented in another table for our starting
values:
Just as in the case of series circuits, the same caveat for Ohm's
Law applies: values for voltage, current, and resistance must be in
the same context in order for the calculations to work correctly.
However, in the above example circuit, we can immediately apply
Ohm's Law to each resistor to find its current because we know the
voltage across each resistor (9 volts) and the resistance of each
resistor:
At this point we still don't know what the total current or total
resistance for this parallel circuit is, so we can't apply Ohm's Law
to the rightmost ("Total") column. However, if we think carefully
about what is happening it should become apparent that the total
current must equal the sum of all individual resistor ("branch")
currents:
As the total current exits the negative (-) battery terminal at
point 8 and travels through the circuit, some of the flow splits off
at point 7 to go up through R1, some more splits off at
point 6 to go up through R2, and the remainder goes up
through R3. Like a river branching into several smaller
streams, the combined flow rates of all streams must equal the flow
rate of the whole river. The same thing is encountered where the
currents through R1, R2, and R3
join to flow back to the positive terminal of the battery (+) toward
point 1: the flow of electrons from point 2 to point 1 must equal
the sum of the (branch) currents through R1, R2,
and R3.
This is the second principle of parallel circuits: the total
circuit current is equal to the sum of the individual branch
currents. Using this principle, we can fill in the IT
spot on our table with the sum of IR1, IR2,
and IR3:
Finally, applying Ohm's Law to the rightmost ("Total") column, we
can calculate the total circuit resistance:
Please note something very important here. The total circuit
resistance is only 625 Ω: less than any one of the individual
resistors. In the series circuit, where the total resistance was the
sum of the individual resistances, the total was bound to be
greater than any one of the resistors individually. Here in the
parallel circuit, however, the opposite is true: we say that the
individual resistances diminish rather than add to
make the total. This principle completes our triad of "rules" for
parallel circuits, just as series circuits were found to have three
rules for voltage, current, and resistance. Mathematically, the
relationship between total resistance and individual resistances in
a parallel circuit looks like this:
The same basic form of equation works for any number of
resistors connected together in parallel, just add as many 1/R terms
on the denominator of the fraction as needed to accommodate all
parallel resistors in the circuit.
Just as with the series circuit, we can use computer analysis to
double-check our calculations. First, of course, we have to describe
our example circuit to the computer in terms it can understand. I'll
start by re-drawing the circuit:
Once again we find that the original numbering scheme used to
identify points in the circuit will have to be altered for the
benefit of SPICE. In SPICE, all electrically common points must
share identical node numbers. This is how SPICE knows what's
connected to what, and how. In a simple parallel circuit, all points
are electrically common in one of two sets of points. For our
example circuit, the wire connecting the tops of all the components
will have one node number and the wire connecting the bottoms of the
components will have the other. Staying true to the convention of
including zero as a node number, I choose the numbers 0 and 1:
An example like this makes the rationale of node numbers in SPICE
fairly clear to understand. By having all components share common
sets of numbers, the computer "knows" they're all connected in
parallel with each other.
In order to display branch currents in SPICE, we need to insert
zero-voltage sources in line (in series) with each resistor, and
then reference our current measurements to those sources. For
whatever reason, the creators of the SPICE program made it so that
current could only be calculated through a voltage source.
This is a somewhat annoying demand of the SPICE simulation program.
With each of these "dummy" voltage sources added, some new node
numbers must be created to connect them to their respective branch
resistors:
The dummy voltage sources are all set at 0 volts so as to have no
impact on the operation of the circuit. The circuit description
file, or netlist, looks like this:
Parallel circuit
v1 1 0
r1 2 0 10k
r2 3 0 2k
r3 4 0 1k
vr1 1 2 dc 0
vr2 1 3 dc 0
vr3 1 4 dc 0
.dc v1 9 9 1
.print dc v(2,0) v(3,0) v(4,0)
.print dc i(vr1) i(vr2) i(vr3)
.end
Running the computer analysis, we get these results (I've
annotated the printout with descriptive labels):
v1 v(2) v(3) v(4)
9.000E+00 9.000E+00 9.000E+00 9.000E+00
battery R1 voltage R2 voltage R3 voltage
voltage
v1 i(vr1) i(vr2) i(vr3)
9.000E+00 9.000E-04 4.500E-03 9.000E-03
battery R1 current R2 current R3 current
voltage
These values do indeed match those calculated through Ohm's Law
earlier: 0.9 mA for IR1, 4.5 mA for IR2, and 9
mA for IR3. Being connected in parallel, of course, all
resistors have the same voltage dropped across them (9 volts, same
as the battery).
In summary, a parallel circuit is defined as one where all
components are connected between the same set of electrically common
points. Another way of saying this is that all components are
connected across each other's terminals. From this definition, three
rules of parallel circuits follow: all components share the same
voltage; resistances diminish to equal a smaller, total resistance;
and branch currents add to equal a larger, total current. Just as in
the case of series circuits, all of these rules find root in the
definition of a parallel circuit. If you understand that definition
fully, then the rules are nothing more than footnotes to the
definition.
- REVIEW:
- Components in a parallel circuit share the same voltage: ETotal
= E1 = E2 = . . . En
- Total resistance in a parallel circuit is less than any
of the individual resistances: RTotal = 1 / (1/R1
+ 1/R2 + . . . 1/Rn)
- Total current in a parallel circuit is equal to the sum of the
individual branch currents: ITotal = I1 + I2
+ . . . In.
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