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In the last video, we went over the definitions of series and parallel. Today we will see
how those definitions might help us to simplify circuits.
Let's start today by introducing the idea of equivalent circuits. If we have two circuits,
or portions of circuits, that give us the same relationship between current and voltage,
for all values of current and voltage, those segments or circuits are equivalent.
For example, we might look at an arrangement of resistors with a source attached. If we
were to analyze this circuit, we would find that the current being drawn from the source
is 2A.
If we were to look at this much simpler circuit, we would find that the current from the source
is 2A.
I am not going to take the time right now to prove that this would be the case for any
value of source voltage. It is. If you doubt, pause the video and try a few different values.
Confirm for yourself that the current drawn from the source will be the same for both
circuits at various source voltages.
Since these two configurations of resistors produce the same result, as far as the current
voltage relationship from the source, we can say that they are equivalent. Namely, this...is
equivalent to this.
How can we determine the equivalent circuit, or resistance?
Let's start by drawing a simple circuit with two resistors. We have a voltage Vin and resistors
R1 and R2. If we call the current from the source Iin and current through the first resistor
IR1 and the current through the second resistor IR2, we can then by, using Kirchhoff's current
law, or KCL, write Iin equals IR1 equals IR2. Since all three currents are the same, let’s
simply call it ‘I’.
We can then indicate the voltages across R1 as V1 and the voltage across R2 as V2. Then
by Kirchhoff's voltage law, or KVL, Vin equals V1 plus V2.
We can substitute Ohms Law, V=RI. Then we have the Vin equals R1 times I plus R2 times
‘I’. Since ‘I’ is common to both terms we can factor it out. Then Vin equals I times
the quantity R1+R2. Alternatively, we can say that Vin=I times an equivalent resistance,
Req.
We can do the same thing with three resistors. If we have a circuit with R1, R2, and R3,
with a voltage source Vin, we can, again, call the current leaving the source Iin. We
could define the voltages across the resistors as V1, V2, and V3.
Since all resistors are in series, by KCL, Iin goes through all of them. So, there is
only one current in the circuit. Then using KVL, we can write Vin equals V1 plus V2 plus
V3. Substituting in Ohm’s Law we have R1 times Iin plus R2 times Iin, plus R3 times
Iin. Iin is common to all terms, so we can factor it. Then we have Iin times the quantity
R1 plus R2 plus R3. Which we can rewrite as Iin times R equivalent.
Let’s generalize this. If we take an arbitrary loop of series resistors, R1, R2, etc. up
to RN, and define the current through all of them as Iin, we can write that input voltage
is equal to the sum of the voltage drops across the resistors. By Ohm's Law, each of those
terms is the resistance times Iin. Iin is common to all the terms, so we can factor
it out. If we substitute Req in for all the resistors, we have defined the equivalent
resistance of all the resistors. So, the equivalent resistance of resistors in series is simply
the sum of all the individual resistors that are in series.
We can summarize this by saying that series resistances add.
This might make qualitative sense to us. Resistance is a measure of how much something opposes
the flow of electricity, that is, current. Connecting resistors in series is like stacking
those resistances, so the total resistance increases when we add resistors.
Before we go on to parallel combinations, I want to talk about another concept, that
of conductance. The idea is a straightforward one.
Sometimes it is convenient to talk about how well a material opposes the flow of electricity.
That is resistance. Alternatively, we can talk about how well a material allows the
flow of electricity. That is conductance.
Mathematically, conductance, G, is the reciprocal of resistance, R. If we take this expression
and substitute it into Ohm's Law, we get a new expression voltage times conductance equals
current.
I bring this up because with resistors, or conductors, in parallel it is sometimes easier
to work with conductances.
Maybe this will be apparent if we just try it.
Let's start with two resistors in parallel with a current source. We'll label them R1
and R2. Then we can then define the current through each of them IR1 and IR2. By KCL the
input current must equal the sum of the current through each of the resistors.
Since there are only two nodes in the circuit, there is only one voltage, let's call it Vin.
Remembering that Ohm’s Law can be written as current is equal to voltage divided by
resistance. We can write the input current is equal to voltage divided by R1 plus the
voltage divided by R2.
We can factor out Vin. Then, we have the input current is equal to the voltage times the
quantity one over R1 plus one over R2.
Or we can say that the input current is equal to the input voltage times one over an equivalent
resistance.
Then one over the equivalent resistance is equal to one over R1 plus one over R2.
Which we can solve for the equivalent resistance as one over one over R1 plus one over R2.
Although there's no mathematical difference between this and what I'm about show you,
conceptually the following maybe a little bit easier. Or, at least the equations are
cleaner.
Let's take a look at the same circuit, however, this time let’s treat the resistors as conductors
instead. Current through the first conductance we will call I1. The current through the second
conductance will be I2. By KCL, the input current will be equal to
the sum of those two currents. There are only two nodes in the circuit so there's only one
voltage Vin.
So, if we substitute the Ohm’s Law relationship, current is equal to conductance times voltage,
into the equation, we have the input current is equal to G1 times Vin plus G2 times Vin.
Vin factors out and we can write this as the equivalent conductance times Vin; where the
equivalent conductance is equal to the sum of the conductances that are in parallel.
We can take the same idea and extend it to an arbitrary number of conductances in parallel.
Given a current Iin connected to parallel conductances G1 and G2, with an arbitrary
number of conductances in parallel out to GN, the same analysis applies. The supply
current Iin will be equal to the sum of all currents through the parallel conductances.
The circuit still has only two nodes and therefore only one voltage, Vin.
We can write the KCL equation in terms of voltages by writing each conductance times
the voltage Vin, all the way out to conductance n times Vin. But, the voltage that is common
to every term, so it factors out.
We end up with the input current equals the voltage times the sum of the conductances.
So, the equivalent conductance of parallel conductances is simply the sum of the parallel
conductances.
Since the parallel conductances add, the conductance of the combination will always go up. This
might make sense to us by remembering that by connecting conductors in parallel we are
providing more paths for the current to travel through, so the current will be higher for
the same applied voltage.
In this video, we covered series and parallel combinations of resistors, though for the
parallel resistors we turned them into conductances. We discovered that series resistors can be
combined into an equivalent resistance by adding them. Since we are adding resistors,
the equivalent resistance will always be higher than any of the individual resistances in
the combination.
We also discovered that parallel resistors can easily be combined by converting the resistances
to conductances. Then the equivalent conductances can be found by adding all of the conductances
that are in parallel. That means that parallel conductances always result in a higher conductance
than any of the individual components. If we wanted to think about the parallel combinations
in terms of resistances, then the total resistance of the parallel combination will be smaller
than the smallest resistance that is being combined.
That's all for today, go out and make it a great one!