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The world is a complex place. Even a very general look around reveals this. In every
discipline, everywhere we look, we see enormous complexity.
This is true in biology, this is true in geology, this is
true in technology, this is true in astronomy, and even in rocket science.
In fact, the world is such a complex place that sometimes when we think about trying
to understand it the mere thought of it is overwhelming.
Even something that we think we understand like Ohm's law can become very complicated
if we begin considering that resistance can be affected
by things like light, electron density, or temperature.
So how do we even begin to analyze any of these things?
The answer is we can't, at least not absolutely and comprehensively. We can, however, begin
by looking at small pieces of systems, or by putting
reasonable restrictions on the conditions under which we do an
analysis.
This function is clearly quadratic. Quadratic functions are not lines. But if we take a
small portion of a quadratic function, it is very close to a line. Even if we zoom in
quite a bit, the quadratic function and the line are very close to each other. So, as
long as the variation from the quadratic function is not too large, we can use the line to approximate
the quadratic function over a small range of values.
Looking at a small range of values of a non-linear function to obtain an approximately linear
function is called linearization. Linearization is a technique we use to simplify the analysis
of a complex system over a small range of values.
We could also look at Ohm's Law. Ohm's law will hold true for a resistor, as long as
we used that resistor in a way it is designed to be used.
If the resistor is designed to dissipate up to a quarter watt of power, as long as we
don't try to dissipate more power than 1/4 watt, it
will behave as we expect. And we expect voltage, resistance, and
current to obey Ohm's law.
Ohm's law is a linear equation, so we expect to see a linear relationship between current
and voltage when applied to a resistor. This does hold true.
So our basic electrical systems have a property of
linearity.
All linearity means is that our system, in our case a circuit, is governed by linear
relationships. There are limitations to this linearity, but we are
not going to concern ourselves with that for a while (if ever).
When we can describe a system as linear, there are properties the system has that make
it easier to analyze.
One of those properties is scalability. Scalability can be illustrated in the following way, let's
say we have a circuit with a 12 volt source and that source caused the output voltage
to be four volts. If we take the same circuit and apply a 36 volt
source, since the input has increased by a factor of three, the
output will be increased by a factor of three. The output of the circuit would then be 12
volts.
Sometimes we can actually use this idea to analyze a circuit.
If we have this circuit, and we want to determine the voltage marked VO; we could
start by pretending we don't know what the value of the source is.
Then we can assume a voltage across the 20 ohm resistor. We could, for example, assume
20 volts and then determine what the consequences would be.
Starting with 20 volts across 20 ohms, this would give us a current of one
amp. The current through the 20 ohm resistor must come
from the 10 ohm resistor. One amp through 10 ohms gives 10 volts. Then we can go around
the loop, 20 volts plus 10 volts equals 30 volts. 30
volts across five ohms gives us six amps. Those two currents
must come through the 20 ohm. So, one amp plus six amps equals seven amps.
Seven amps through 20 ohms is the 140 volts. 30 volts plus 140 volts equals 170 volts.
So if the source had been 170 volts, the output would be 20 volts. By linearity the ratio
of the actual VO to the source of 16 volts must be the same.
So a 16 volt source results in an output voltage of 1.88 volts.
Now, this is not a very practical way to analyze circuits but it does illustrate the point.
In a linear system scalability is true.
Another property of linear systems is that if a linear system has multiple stimuli, the
response of this system will be the sum of its response to
each of the individual stimuli. This probably would not
surprise us if we took the last circuit and split the 16 volt source into two 8 volt sources.
We can clearly see that two 8 volt sources in series result
in a 16 volt source. So using two eight volt sources will give us the same result.
However this may be more difficult to see if the sources are not touching or are of
different types. Let's look at the following example just two illustrate this concept.
I do want to emphasize that right now I'm just trying to illustrate a concept. I'm not
trying to give you an efficient way to solve circuits. With that in mind, lets first and
analyze this circuit using node voltage analysis so that we know the answer we should get.
This circuit can be analyzed with a single node equation.
As always, we start by choosing a reference node, and then identify the other nodes. The
node on the left is connected to a voltage source that is connected to the reference
node, so the voltage at the non-reference node must simply be the value of the source.
The voltage at the other node is clearly VX. If then I choose a direction for the current
through the six kiloohm resistor, and then draw the current through the 14 kiloohm resistor
so that it agrees with the passive sign convention, we can write a node equation.
Solving this node equation results in a value of VX that is 35 volts.
So now let's analyze this circuit looking at the contributions of each source individually
to VX. To look at an individual sources contribution, I need to turn off all but one source in the
circuit. And I am using the phrase turn off all but one source very intentionally. If
we keep in mind that we are turning off the source, we will avoid some very common errors.
If I turn off a voltage source that means where that source was will now have zero volts.
Saying that I have zero volts between two nodes is the same as saying those nodes are
connected. So turning off the voltage source results in a wire being placed where the voltage
source was.
Also, since I am not analyzing the whole circuit right now, I'm going to put a prime on all
the variables just to keep them straight.
If I want to know the voltage VX prime, it will be very easy to determine if we know
the current through the 14 kiloohm resistor. We can then use Ohm's Law.
Just so we are sure we are looking at the circuit correctly, let's identify the nodes.
Since there are only two nodes in the circuit, everything is in parallel.
The current through the 14 kiloohm resistor is easily determined using current division.
This results in a current of ix prime equals 1.5 milliamps. Which results VX prime being
equal to 21 volts.
Going back to the original circuit, let's now look at the contribution to VX from the
20 volt source. To do that, we need to turn off the five milliamp source. Turning off
a current source results in no current through the branch that contained the source. This
is the same as replacing that source with an open circuit.
Let's call the voltage from this portion of the circuit VX double prime so that we don't
confuse it with the other variables.
We can solve for VX double prime by doing voltage division. This results in vx double
prime equals 14 volts.
Going back to the original circuit, the total value of VX is equal to the individual contributions
from the sources. So VX will be equal to VX prime plus VX double prime, or 21 volts plus
14 volts which is equal to 35 volts.
This principle, that the total response of a linear network is equal to the sum of responses
to each of the individual sources in a circuit is called Superposition.
About superposition as a method of solution.
In general, this is not a method of analysis that I would consider a shortcut or a quick
trick to get a solution. In fact, when we are first in learning this principle analyzing
circuits with Superposition generally increases the number of steps and the time it takes
to get a solution.
Why do we bother learning this if it does not simplify our analysis of circuits?
Well, because there are situations where it is necessary for analysis.
A simple example of this would be a circuit that contains both AC and DC sources. In this
case, it becomes extremely difficult to analyze the effect of both sources at the same time.
In more complicated circuits it may be nearly impossible to analyze the circuit with all
the sources in the circuit.
There are also many components that respond differently to DC voltages and currents vs.
AC voltages and currents. An example of this would be a capacitor, which we will learn
about later. To DC currents and voltages a capacitor looks like an open circuit. To time-varying
signals, a capacitor can look like anything from a short circuit to nearly an open circuit
depending upon how quickly the signal varies in time.
One of the most common situations for us to apply the principle of Superposition is in
semi-conductor circuits. Semi-conductor circuits are those that contain things like diodes
and transistors.
When we analyze circuits that have semiconductor components, we begin by finding a DC solution.
Then the DC solution is used to determine the parameters for our AC analysis.
The topic of semiconductors brings up something important we have to discuss before we leave
the topic of Superposition. If you remember several videos back I was talking about some
things called dependent sources. Dependent sources were not the invention of the evil
maniacal teachers, but instead, they are used to model the response of and to simplify the
analysis of more complex components like transistors.
If dependent sources are used to model a response, that means they must always be present in
a circuit to determine if they have a response. This means we can never turn off the dependent
source or remove it from the circuit when doing an analysis.
Something that can help us remember this comes from looking at the different types of dependent
sources.
Looking at the four different types of dependent sources, the current-controlled current source,
the current-controlled voltage source, the voltage controlled current source, and the
voltage-controlled voltage source, we see they all have one thing in common and that
is the symbol representing them is a diamond. And we all know that 'Diamonds are forever.'
In case I have not made this clear, you can never turn off the dependent source.
I'm guessing an example problem would be helpful right now.
So here is a circuit with a dependent source. Once again I would like to point out that
Superposition is not an efficient means of solving this problem, but this is a simple
example to demonstrate how Superposition would be done with a dependent source in the circuit.
To use Superposition to solve for VQ, we need to analyze the contribution to VQ from each
of the independent sources.
Let's start by looking at the response due to the three milliamp source. To do that we
turn of the 30 volt source. We turn off the source by making the voltage of the source
zero volts and that is same as replacing it with a wire.
Then let's start by highlighting the nodes. This makes it easy to see that there are only
two nodes in the circuit, so there is only one voltage VQ prime. We can then write a
KCL equation at the node highlighted with blue. If we set the currents entering the
node equal to the currents leaving the node, we get IY prime plus four IY prime plus three
milliamps equals VQ prime over 10 kiloohms plus VQ prime over 20 kiloohms. A second equation
can be found by relating the controlling parameter to VQ prime. That results in IY prime equals
negative VQ prime over five kiloohms. A little bit of algebra will get us to twenty three
VQ prime equals 60 volts. So, VQ prime equals 2.61 volts.
Now let's look of the circuit again, this time analyzing the contribution to VQ from
the 30 volt source. To do this we will turn off the independent current source. Turning
off the current source results in no current, so that is the same as making the branch of
the current source an open circuit.
Let's begin this analysis also by highlighting the nodes.
If I designate the green node as the reference node, the purple ish node would be 30 volts,
and the yellow node would be VQ double prime.
We could then do an informal node voltage analysis. At the yellow node, we would have
the current through the five kiloohm resistor, plus the current through the 20 kiloohm resistor,
plus four times IY double prime equal to the current through the ten kiloohm resistor.
We also have that the controlling parameter, IY double prime, is equal to 30 volts minus
VQ double prime over five kiloohms. Taking these two equations and performing
a little bit of algebra results in 23 times VQ double prime equal 630 volts.
Or VQ double prime equals 27.39 volts.
So if we take the results from the first analysis that VQ prime equals 2.61 volts and the results
from the second analysis, VQ double prime equals 27.39 volts
VQ must be the sum of them resulting in VQ equals 30 volts.
I encourage you to go back and verify this by analyzing the original circuit using a
different method.
Let me be emphasize that this was a horrible approach to this problem. We could have applied
node voltage analysis to the first circuit and been done in less than half the time.
That is not why I did this. This analysis is only meant to demonstrate how to do Superposition
when there is a dependent source in the circuit. In fact I can safely say that most of them
problems you're going to be assigned to learn about Superposition could be better approached
using another method of analysis. That is NOT the point of the problems. They are there
to help you learn how superposition works.
It's kind of like learning how to parallel park. We don't start on a busy street with
an expensive car trying to park between two expensive cars. We learn how to parallel park
in a parking lot between two shopping carts. In real life you would never parallel park
your car between two shopping carts. That would just be silly. But it is a relatively
safe way to learn a difficult skill.
Anyway, using the principle of Superposition we can analyze the response of the network
by looking at the response to each of the individual independent sources in the circuit.
This will be a necessary method of analysis for many circuits we will encounter in the
future.
That's all for today. Thanks for watching. Go out and make it a great one.