Kirchhoff's Voltage Law - KVL - is one of two fundamental laws in electrical
engineering, the other being Kirchhoff's Current Law (KCL).
KVL is a fundamental law, as
fundamental as Conservation of Energy in mechanics, for example, because KVL is
really conservation of electrical energy.
KVL and KCL are the starting
point for analysis of any circuit.
KCL and KVL always hold and are
usually the most useful piece of information you will have about a circuit after
the circuit itself.
Goals For
This Lesson
What should you be able to do after this lesson? Here's the basic
objective.
Given an electrical circuit:
Be
able to write KVL for every loop in the circuit.
Be
able to solve the KVL equations, especially for simple circuits.
These goals are very important. If you can't write KVL equations and solve
them, you may well be lost when you take a course in electronics in a few years.
It will be much harder to learn that later, so be sure to learn it well now.
Kirchhoff's Voltage Law
Here's
a simple circuit. It has three components - a battery and two other
components. Each of the three components will have a current going through
it and a voltage across it. Here we want to focus on the voltage across
each element, and how those three voltages are related.
We could measure voltage:
Anywhere along the wire shown
in purple
Anywhere along the wire shown
in green
Anywhere along the wire shown
in blue.
Note: Any point along the green wire is at the
same voltage, and the same situation pertains for the blue wire and the purple
wire.
Here's
the same circuit. Here, with the button, you can move the dot representing
charge around the circuit.
Answer these questions about what happens as that
charge moves.
Problems
Q1.
As the charge moves from the top of the battery to the top of Element #1 (along
the wire shown in purple), how much energy does the charge lose?
Q2.
As the charge moves from the top of Element #1 through Element #1 to the bottom
of element #1, how much energy does the charge lose?
Q3.
As the charge moves from the bottom of Element #1 to the top of Element #2, how
much energy does the charge lose?
Q4.
As the charge moves from the top of Element #2 through Element #2 to the bottom
of element #2, how much energy does the charge lose?
Q5.
As the charge moves from the bottom of Element #2 to the bottom of the battery,
how much energy does the charge lose?
Q6.
As the charge moves from the bottom of the battery through the battery to the
top of the battery, how much energy does the charge lose?
The last question is tricky because the charge actually gains energy as it goes
through the battery. Now, we can track the energy acquired and given up by
the charge as it traverses the circuit. And, as the charge completes one
round trip around the circuit - returning to its starting point - there can be
no net gain of energy or no net loss. That's really a statement of
conservation of energy. What you put in is what you get out.
TANSTAAFL! (There Ain't No Such Thing As A Free Lunch. It's not good
English, but it says something that can't be said easily otherwise!) Let's
formalize that.
The energy put into the charge
as it goes through the battery is Vb
* Q.
The charge loses
V1
* Q. as it goes through Element #1.
The charge loses
V2
* Q. as it goes through Element #2.
The net energy put into the
charge (What's put in minus what it loses!) is:
Vb * Q - V1
* Q - V2 * Q = (Vb
- V1 - V2)Q = 0
We can note that the amount of
charge is irrelevant in what we have learned here. What we have learned
is:
Vb - V1
- V2 = 0, or
Vb = V1
+ V2 = 0
This can be paraphrased several ways:
Voltage across the battery =
Voltage across Element #1 + Voltage across Element #2.
The algebraic sum of the
voltages around a closed loop is zero.
There are
some things to note about this conclusion - either way it is phrased. Note
the following:
The conclusion does not depend
on what the elements in the loop are. They can be anything at all but
still, the algebraic sum of the voltages around a closed loop will be zero.
If you have a circuit with many
loops, the algebraic sum of the voltages around any
loop in the circuit is zero.
That last note will need a little explanation and
work to be sure you understand it. Consider this circuit.
Q7.
How many loops are there in the circuit above?
Given an electrical circuit:
Be
able to write KVL for every loop in the circuit.
