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- Kirchoff's Laws - KCL
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Kirchhoff's
Current Law - Introduction
Kirchhoff's Current Law - KCL - is one of two fundamental laws in electrical
engineering, the other being Kirchhoff's Voltage Law (KVL).
KCL is a fundamental law,
as fundamental as Conservation of Mass in mechanics, for example, because
KCL is really conservation of charge.
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.
People and computer
programs both use KVL and KCL for circuit analysis. Spice (in all its incarnations)
starts with KCL.
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 KCL at every node in the circuit.
Be able to solve the KCL equations, especially for simple circuits.
These goals are very important. If you can't write KCL 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
Current Law
At this point,
you have learned the fundamentals of charge and current. There is
one important law, Kirchhoff's Current Law that you will need to learn.
It is not as complex as it might seem. All you really need to know
is that charge is conserved, so KCL is really based on one simple fact.
Charge can neither be
created nor destroyed.
From that basic fact we
can get to Kirchhoff's Current Law. Despite that simplicity, it is
a fundamental, widely used law, that you need to know to go very far in
electrical engineering.
Let's examine a circuit simulation. It's shown below. Charge
(current) is flowing through the circuit. The simulation shows some
charge - the large red blob - flowing through a battery (where it picks
up energy, but that's another story. Click
here for that lesson.) That charge flows through Element #1 in
the simulation. After the charge flows through Element #1 it splits.
Some of the charge goes through Element #2, and some goes through Element
#3. (Notice that it does not split equally! Sometimes
it does. Sometimes it doesn't.) When, in the course of its
flow through the circuit, there is no possibility of splitting, all of
the charge entering a node will flow through the next element. (That
element is said to be in series. Element #3 and Element #4 are in
series because all of the current going through #3 goes through #4.
Elements #1 and #2 are not in series.)
There is one node in the simulation where charge flowing through two elements
comes together and "reunites" and flows back into the battery.
Note that this simulation emphasizes the
conservation of charge. When charge flows through Element #1 when
it gets to the end of Element #1 it splits into two. However, what
arrives at that node is what leaves that node, so the amount of charge
that enters the node - the big red blob - equals the amount of charge that
leave that node - the sum of the charge on the medium sized red blob and
the charge on the small red blob.
Problem
1. In
this circuit, charge flows from the battery, through Element #1 to the
node. Willy Nilly observes that 35 coulombs flows through Element
#1 in 20 seconds, and that, in that same time, 17 coulombs flows through
Element #2. How much charge flows through Element #3 in that time?
2. How
much charge flows through Element #4 in that time?
KCL
Charge usually flows through some sort of metallic wire, flowing through
the atomic lattice. Although it is physically unlike water flowing
in a pipe, that analogy is sometimes drawn. Like water confined to the
interior of a pipe, charge is confined to flow within a wire, and it doesn't
leave the surface of the wire. You may want to think in those terms as
you interpret current flow in the sketches and diagrams that follow.
We will be developing rules for current flow in circuits in this section.
You will need to know about that in order to be able to analyze larger
circuits with lots of elements.
In practice current flows in wires and often splits between two or more
devices. We need to consider what happens in networks of conductors
in which current can split. Single wires carrying current aren't
the most important case we can look at, and you need to learn about Kirchhoff's
Current Law which describes those situations where we have large networks
of interconnected elements carrying current. Those kinds of circuits
will have many connection points (called nodes) where current can split
into smaller currents. Shown below is part of a circuit. Current
(I) comes in from the left and splits into two parts, I_{1} and
I_{2}. There is one simple relationship between these two
currents and the current, I, flowing in from the left below.
Here, a red dot has
been placed over both of the nodes in the picture.
Focus attention on a very short time, DT. Assume all currents constant
during DT.
A current, I, flows into
the top node, and I_{1} and I_{2} are flowing out of the
node. No charge accumulates!
During time DT,
the total amount of charge that flows into the node is zero so:
IDT
- I_{1}DT
- I_{2}DT
= 0
And during DT,
IDT
is the charge flowing in.
I_{1}DT
is the charge flowing through the left resistor.
I_{2}DT
is the charge flowing through the right resistor.
So, we have - for the
period of time DT,
the total amount of charge that flows into the node is zero so:
IDT
- I_{1}DT
- I_{2}DT
= 0
Cancelling the DT's
everywhere, we get:
I - I_{1}
- I_{2} = 0
Which can be rephrased
as:
The sum of the currents
flowing into the node is zero.
or
I = I_{1}
+ I_{2}
which says that "The
current entering the node equals the current leaving the node."
We need to be more precise in this.
When we have the expression:
I - I_{1}
- I_{2} = 0
Or when we think "The
sum of the currents flowing into
the node is zero."
We interpret I as a current
entering and - I_{1}
and - I_{2}
also as currents entering.
Note the negative signs!
Since I_{1}
is leaving the node, then we can think of - I_{1}
as the value of the current entering.
We do the same for I_{2}
and - I_{2}.
Again, we need to be more
precise
when we express things the other way.
When we have the expression:
I = I_{1}
+ I_{2} = 0
Or when we think "The
sum of the currents flowing into
the node equals the sum of the currents leaving
the node."
We interpret
I
as a current
entering
and
I_{1}
and I_{2}
as currents leaving.
Either of the above formulations is Kirchhoff's Current Law, otherwise
known as KCL.
If you understand that "The sum of the currents entering a node is zero.",
then you know KCL. With that, it's time for you to answer a few questions.
Problems
3. In
this circuit - which you saw above - determine the current I_{2},
in terms of trhe other two currents. You will need to write KCL at
the node marked with a red dot. Notice that we have defined current
symbols and polarities for all the currents involved.
4. Now,
determine a value for the current, I_{2}, when you have
numerical values for the other currents.
I_{1} = .75 A
I_{3}= - .45 A
(nobody said the value had to be positive!)
5. Here
is another circuit.
You need to determine
a value for the current, I_{4}, given the following numerical values
for some other currents. First, you'll need to get an algebraic expression
for I_{4}. Click on the corrrect expression.
Now, determine the numerical
value for I_{4} when I_{3} is 0.45A.
6. Here's
a problem for you. In 10 seconds, an observer - Willy Nilly - notices
that 35 coulombs of charge leaves node "n" in this circuit, heading for
node "x". (Vn is the voltage at node "n", etc.) In the same
ten seconds, 22 coulombs of charge leaves node "n" heading for node "z".
Determine the current, I_{y}.
KCL
There are just a few
points about Kirchhoff's Current Law that need to be made.
The complete expression
of KCL is "The sum of all the currents entering
a node is equal to the sum of all the currents leaving the node."
Kirchhoff's Current Law
holds
at every node in a network.
Kirchhoff's Current Law
holds
at every instant of time.
If you remember each of these items, you'll be able to figure out a great
many things about circuits that you encounter. It's a very wide ranging
and fundamental law in electrical engineering.
We introduced this lesson with a simulation. That simulation seems
to say a lot, and it really shows what KCL means. We'll let you look
at it again.