#import "../../core/core.typ"

=== 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:

#image("static/00110.png")

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:

$ E_(2 - 1) = +45 V $

When a voltage is specified with a double subscript (the characters
\"2-1\" in the notation \"E#sub[2-1]\"), it means the voltage at the
first point (2) as measured in reference to the second point (1). A
voltage specified as \"E#sub[cg]\" 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\".

#image("static/00435.png")

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:

$ 
  E_(3-2) = -10 V \
  E_(4-3) = -20 V \
  E_(1-4) = -15 V
$

#image("static/00436.png")

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:

#image("static/10108.png")

This principle is known as #emph[Kirchhoff\'s Voltage Law] (discovered
in 1847 by <NAME>, a German physicist), and it can be
stated as such:

#quote[
 The algebraic sum of all voltages in a loop must equal zero
]

By #emph[algebraic], I mean accounting for signs (polarities) as well as
magnitudes. By #emph[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:

#image("static/10109.png")

This may make more sense if we re-draw our example series circuit so
that all components are represented in a straight line:

#image("static/00111.png")

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 #emph[resisting] the flow of
electrons being pushed by the battery. In other words, the \"push\"
exerted by the resistors #emph[against] the flow of electrons
#emph[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:

#image("static/00112.png")

If we were to take that same voltmeter and read voltage across
combinations of components, starting with only R#sub[1] on the left and
progressing across the whole string of components, we will see how the
voltages add algebraically (to zero):

#image("static/00113.png")

The fact that series voltages add up should be no mystery, but we notice
that the #emph[polarity] of these voltages makes a lot of difference in
how the figures add. While reading voltage across R#sub[1],
R#sub[1]--R#sub[2], and R#sub[1]--R#sub[2]--R#sub[3] (I\'m using a
\"double-dash\" symbol \"--\" to represent the #emph[series] connection
between resistors R#sub[1], R#sub[2], and R#sub[3]), 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 R#sub[1], R#sub[2], and R#sub[3] 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 R#sub[1]: 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 #emph[electrically
common] to each other. And, as such, the voltage between those two
electrically common points #emph[must] be zero.

Kirchhoff\'s Voltage Law (sometimes denoted as #emph[KVL] for short)
will work for #emph[any] circuit configuration at all, not just simple
series. Note how it works for this parallel circuit:

#image("static/00114.png")

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:

#image("static/10110.png")

Note how I label the final (sum) voltage as E#sub[2-2]. 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 (E#sub[2-2]), 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:

#image("static/00115.png")

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 #emph[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:

#image("static/00114.png")

#image("static/10111.png")

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:

#image("static/00116.png")

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
#emph[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:

$
  &E_(4-3) + E_(9-4) + E_(8-9) + E_(3-8) = 0 \
  &E_(4-3) + 12 + 0 + 20 = 0 \
  &E_(4-3) + 32 = 0 \
  &E_(4-3) = -32 V 
$

#image("static/00358.png")

#image("static/00359.png")

#image("static/00360.png")

#image("static/00361.png")

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:

#image("static/00117.png")

In other words, the initial placement of our \"meter leads\" in this KVL
problem was \"backwards.\" Had we generated our KVL equation starting
with E#sub[3-4] instead of E#sub[4-3], stepping around the same loop
with the opposite meter lead orientation, the final answer would have
been E#sub[3-4] \= +32 volts:

#image("static/00437.png")

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.

#core.review[
  #quote(block: true, attribution: [Kirchhoff\'s Voltage Law (KVL)])[
    The algebraic sum of all voltages in a loop must equal zero
  ]
]

