Kirchhoff's Voltage Law (KVL)
Let's take another look at our example series circuit, this time
numbering the points in the circuit for voltage reference:
If we were to connect a voltmeter between points 2 and 1, red
test lead to point 2 and black test lead to point 1, the meter would
register +45 volts. Typically the "+" sign is not shown, but rather
implied, for positive readings in digital meter displays. However,
for this lesson the polarity of the voltage reading is very
important and so I will show positive numbers explicitly:
When a voltage is specified with a double subscript (the
characters "2-1" in the notation "E2-1"), it means the
voltage at the first point (2) as measured in reference to the
second point (1). A voltage specified as "Ecg" would mean
the voltage as indicated by a digital meter with the red test lead
on point "c" and the black test lead on point "g": the voltage at
"c" in reference to "g".
If we were to take that same voltmeter and measure the voltage
drop across each resistor, stepping around the circuit in a
clockwise direction with the red test lead of our meter on the point
ahead and the black test lead on the point behind, we would obtain
the following readings:
We should already be familiar with the general principle for
series circuits stating that individual voltage drops add up to the
total applied voltage, but measuring voltage drops in this manner
and paying attention to the polarity (mathematical sign) of the
readings reveals another facet of this principle: that the voltages
measured as such all add up to zero:
This principle is known as Kirchhoff's Voltage Law
(discovered in 1847 by Gustav R. Kirchhoff, a German physicist), and
it can be stated as such:
"The algebraic sum of all voltages in a loop must equal
zero"
By algebraic, I mean accounting for signs (polarities) as
well as magnitudes. By loop, I mean any path traced from one
point in a circuit around to other points in that circuit, and
finally back to the initial point. In the above example the loop was
formed by following points in this order: 1-2-3-4-1. It doesn't
matter which point we start at or which direction we proceed in
tracing the loop; the voltage sum will still equal zero. To
demonstrate, we can tally up the voltages in loop 3-2-1-4-3 of the
same circuit:
This may make more sense if we re-draw our example series circuit
so that all components are represented in a straight line:
It's still the same series circuit, just with the components
arranged in a different form. Notice the polarities of the resistor
voltage drops with respect to the battery: the battery's voltage is
negative on the left and positive on the right, whereas all the
resistor voltage drops are oriented the other way: positive on the
left and negative on the right. This is because the resistors are
resisting the flow of electrons being pushed by the battery. In
other words, the "push" exerted by the resistors against the
flow of electrons must be in a direction opposite the source
of electromotive force.
Here we see what a digital voltmeter would indicate across each
component in this circuit, black lead on the left and red lead on
the right, as laid out in horizontal fashion:
If we were to take that same voltmeter and read voltage across
combinations of components, starting with only R1 on the
left and progressing across the whole string of components, we will
see how the voltages add algebraically (to zero):
The fact that series voltages add up should be no mystery, but we
notice that the polarity of these voltages makes a lot of
difference in how the figures add. While reading voltage across R1,
R1--R2, and R1--R2--R3
(I'm using a "double-dash" symbol "--" to represent the series
connection between resistors R1, R2, and R3),
we see how the voltages measure successively larger (albeit
negative) magnitudes, because the polarities of the individual
voltage drops are in the same orientation (positive left, negative
right). The sum of the voltage drops across R1, R2,
and R3 equals 45 volts, which is the same as the
battery's output, except that the battery's polarity is opposite
that of the resistor voltage drops (negative left, positive right),
so we end up with 0 volts measured across the whole string of
components.
That we should end up with exactly 0 volts across the whole
string should be no mystery, either. Looking at the circuit, we can
see that the far left of the string (left side of R1:
point number 2) is directly connected to the far right of the string
(right side of battery: point number 2), as necessary to complete
the circuit. Since these two points are directly connected, they are
electrically common to each other. And, as such, the voltage
between those two electrically common points must be zero.
Kirchhoff's Voltage Law (sometimes denoted as KVL for
short) will work for any circuit configuration at all, not
just simple series. Note how it works for this parallel circuit:
Being a parallel circuit, the voltage across every resistor is
the same as the supply voltage: 6 volts. Tallying up voltages around
loop 2-3-4-5-6-7-2, we get:
Note how I label the final (sum) voltage as E2-2.
Since we began our loop-stepping sequence at point 2 and ended at
point 2, the algebraic sum of those voltages will be the same as the
voltage measured between the same point (E2-2), which of
course must be zero.
The fact that this circuit is parallel instead of series has
nothing to do with the validity of Kirchhoff's Voltage Law. For that
matter, the circuit could be a "black box" -- its component
configuration completely hidden from our view, with only a set of
exposed terminals for us to measure voltage between -- and KVL would
still hold true:
Try any order of steps from any terminal in the above diagram,
stepping around back to the original terminal, and you'll find that
the algebraic sum of the voltages always equals zero.
Furthermore, the "loop" we trace for KVL doesn't even have to be
a real current path in the closed-circuit sense of the word. All we
have to do to comply with KVL is to begin and end at the same point
in the circuit, tallying voltage drops and polarities as we go
between the next and the last point. Consider this absurd example,
tracing "loop" 2-3-6-3-2 in the same parallel resistor circuit:
KVL can be used to determine an unknown voltage in a complex
circuit, where all other voltages around a particular "loop" are
known. Take the following complex circuit (actually two series
circuits joined by a single wire at the bottom) as an example:
To make the problem simpler, I've omitted resistance values and
simply given voltage drops across each resistor. The two series
circuits share a common wire between them (wire 7-8-9-10), making
voltage measurements between the two circuits possible. If we
wanted to determine the voltage between points 4 and 3, we could set
up a KVL equation with the voltage between those points as the
unknown:
Stepping around the loop 3-4-9-8-3, we write the voltage drop
figures as a digital voltmeter would register them, measuring with
the red test lead on the point ahead and black test lead on the
point behind as we progress around the loop. Therefore, the voltage
from point 9 to point 4 is a positive (+) 12 volts because the "red
lead" is on point 9 and the "black lead" is on point 4. The voltage
from point 3 to point 8 is a positive (+) 20 volts because the "red
lead" is on point 3 and the "black lead" is on point 8. The voltage
from point 8 to point 9 is zero, of course, because those two points
are electrically common.
Our final answer for the voltage from point 4 to point 3 is a
negative (-) 32 volts, telling us that point 3 is actually positive
with respect to point 4, precisely what a digital voltmeter would
indicate with the red lead on point 4 and the black lead on point 3:
In other words, the initial placement of our "meter leads" in
this KVL problem was "backwards." Had we generated our KVL equation
starting with E3-4 instead of E4-3, stepping
around the same loop with the opposite meter lead orientation, the
final answer would have been E3-4 = +32 volts:
It is important to realize that neither approach is "wrong." In
both cases, we arrive at the correct assessment of voltage between
the two points, 3 and 4: point 3 is positive with respect to point
4, and the voltage between them is 32 volts.
- REVIEW:
- Kirchhoff's Voltage Law (KVL): "The algebraic sum of all
voltages in a loop must equal zero"
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