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At last, we come to circuits, or at least some laws that apply to them.
Let's start by introducing Gustav Kirchhoff. Kirchhoff was a physicist that made some rather
large contributions to the fields of thermodynamics (like the term black-body radiation), spectroscopy,
and circuit theory in the mid 1800's. For our purposes, he is famous for setting things
to zero.
Let's start our discussion of Kirchhoff's Laws with Kirchhoff's current law (or KCL
and electrical engineers lazily and affectionately refer to it).
The law, simply stated, says that the current entering something must equal the current
leaving it.
So, if I were to draw an arbitrary boundary with two currents entering it, say Iin1 and
Iin2, then the currents leaving it, say Iout1, Iout2, and Iout3 must be equal to the input
currents. Or mathematically, Iin1+Iin2=Iout1+Iout2+Iout3.
More formally stated, Kirchhoff's Current Law says: the sum of currents entering any
node or closed boundary must add up to zero.
This law should make sense to us, if we have a little bit of background in physics. The
principle here would be that of the conservation of charge, that is, charge cannot be created
or destroyed (at least under normal circumstances). This makes sense in light of the law of conservation
of energy, but that's a discussion for another day.
We usually apply this law at a node (this was defined in the last video), though there
are occasions where we do extend it to a closed boundary.
How this might look for a piece of a circuit is this: here we have five resistors connected
at a single node. If we assume that the other ends of the resistors are connected somewhere
to provide each of the currents, then using Kirchhoff's current law, all five currents
must add to zero. In this case it will require that at least one of the five currents is
negative, but that is not a really big deal. All a negative current means is that it goes
in the opposite direction the arrow is pointing.
Let's apply this to a simple circuit. Here we have an unknown current through a resistor.
If we use the top node to apply KCL, I might define currents that enter the node as positive,
and those that leave the node as negative. So I can say that 3A-4A+7A-Ix=0. Then Ix must
equal 6A.
Kirchhoff also came up with another circuit law called Kirchhoff's Voltage Law, or KVL.
This law states that the sum of voltages around a closed path must add to zero.
A common way of restating this law is that the voltage rises around a closed loop must
equal the voltage drops.
This might look like this: if we take an arbitrary loop with arbitrary elements and indicate
the voltages across each element, if we add them all up, we must get zero volts. The question
then remains, how do we define a voltage increase? That is a matter of choice. There are two
possible ways to do this. One is to go around the loop, and mark the voltage in the equation
with the sign we first encounter. So if I start here, I encounter the positive sign
on V1, so I would call that positive. Continuing on, V2 would be positive, V3 negative, V4
negative, and V5 negative.
Another would be to define the voltages by the change in voltage across the element.
So if we go from a negative sign to a positive sign that would indicate a voltage increase.
But, if we go from a positive sign to a negative sign, that is a voltage drop. Either method
works, as long as you are consistent. I will choose the latter method because it reflects
my perception of reality more.
Using the latter method, I will go around the circuit clockwise first. Starting at this
point, V1 goes from positive to a negative, so I will call that a voltage drop. V2 does
also. V3 goes from negative to a positive, so I will call that a voltage rise and put
it on the other side of the equation. The same is true for V4 and V5. So, we have the
voltage drops, V1+V2, must equal the voltage rises V3+V4+V5.
If we go around the loop counter clockwise, we get a similar result, however, the voltage
rises and drops switch. Starting at V3, we go from a positive to a negative, so that
is a drop. V2 goes from negative to positive. It is a rise, as is V1. Continuing on V5 and
V4 are drops. The equation, in the end, is the same, so
the results are equivalent.
Let's apply this to an actual circuit. Given this circuit, if we go around the loop clockwise
and apply KVL, here we go from a negative to a positive so that is an increase of a
12 volts. Here we go from a positive to a negative. That is a decrease of 3 volts. Again,
here we go from positive to negative that is a decrease of two volts. Here we go from
negative to a positive for an increase of four volts. And here we go from a negative
to positive, for an increase of the Vx. Now we have completed the loop and, by KVL, this
must be equal to zero. We can solve for Vx and get a -11 volts.
So there we have it Kirchhoff laws. Kirchhoff's current law says that the currents that enter
and leave a closed boundary must sum to zero. While, Kirchhoff's voltage law says that the
sum of voltages around any closed path must add to zero. I also told you that out of laziness
and affection these laws are commonly referred to as KCL and KVL.
Next time, we'll go over how Kirchhoff's laws in combination with Ohm's law can be used
to solve for all the voltages and currents in any circuit.
Until next time, go out and make it a great one!