L(2-3) Equivalent circuits

Equivalent circuits; series & parallel circuits (or resistance R, conductance G, voltage V and current I); voltage and current dividers (and adders); See www.ece.utah.edu/~ece1250 for more information

Hello, this is Dr.
Cynthia Furse from the University of Utah.

Today we're going to be talking
about Equivalent Circuits.

First we're going to talk about
what an equivalent circuit is.

It's basically a circuit that
gives you the same voltage and

current that another circuit
would have given you.

We'll talk about what it means to be
series and parallel for resistance and

conductance, and also for
voltage and current sources.

We'll talk about voltage and current
dividers and then equivalent sources.

Equivalent circuits are two circuits
that are the same if the voltage and

the current characteristics
at the nodes are identical.

Then the circuits are considered
equivalent, what does that mean?

So right here are two nodes,
here's V1 and here's is V2.

If the currents i1, i2 and v1,

v2 are the same, then it means that
we have an equivalent circuit.

Basically we're going to be changing
the part that's right here, and

the rest of the circuit.

So that we have two circuits that give us
the same voltage and current situation.

So let's say here's a very simple example.

If I had a circuit right here,
notice this is the part on the left and

it has two nodes, v1 and v2.

And there are five resistors,
R1, R2, R3, R4, and R5.

If we combine all of these resistors and
series right there,

we'd have an equivalent resistance.

The equivalent resistance is basically
just the sum of all five of the resistors.

And if we did that, this circuit and

this circuit would have
exactly the same front-end.

The voltage and the current at
the front would be exactly the same.

These two circuits would be equivalent.

So if we have circuits that are in series,

it means they have the same
current as shown here at the top.

If we have circuits that are unparallel,
it means they have the same voltage,

as shown here at the bottom.

Resistors that are in series add,
that's what we just did.

So if we had four resistors in series, and

we want to find their equivalent
resistance, we just add them up.

The equivalent resistance is just
the sum of all the four resistors.

A voltage divider uses resistors in series
to be able to divide up the voltage.

We take the original voltage, VS, and
we run it across a couple of resistors.

And you can see right here that one of the
resistors is used to divide the voltage so

that we can get v2.

V2 is going to be equal to R2
over R1 + R2 as shown here.

If we had many resistors, the voltage
would be the resistor that the voltage

has taken across divided by all
of the other resistor in series.

Here's an example of the voltage divider
cards that you have in your package and

you can see that we can take one voltage
such as the voltage across this battery

and divide it into two parts by
running it across two resistors.

Now, the other thing that we can do
with the voltage divider is we can take

two voltages and
put them together in series, so

basically stack up two batteries and
we'll get the sum of the two voltages.

A voltage divider works as
both a voltage divider and

in the other direction
as a voltage summer.

So voltages in series add.

Here we have three voltages in a circuit.

And the way they're going to add is
we'll just go in this direction.

Here is -v1 + v2- v3 and

so right here the voltage

equivalent is v1- v2 + v3.

The equivalent resistance is found
by summing up the two resistors that

are in series, so the bottom circuit
is an equivalent of a top circuit.

Currents in series have to be the same or
else something blows up.

So this circuit, this right here, you
we have a 6 amp in series to the 4 amp.

That's unrealizable impossible circuit.

This can't happen or
else the circuit will blow up.

Resistors that are in parallel
add in a different way.

The equivalent resistance is taken by,
take the inverse of each resistor,

1 divided by R1 + 1 divided
by R2 + 1 divided by R3, and

invert all of that and that's going
to be the equivalent resistance.

So no matter how many resistors we have,
we can find the equivalent resistance for

this circuit right here.

There's another way to think of that,
and that's in terms of conductance.

Conductance is 1 divided by the
resistance, so here is the conductance,

1 divided by R1,
that you can see right here.

Adding up the resistors in parallel is the
same thing as adding the conductance in

parallel except it' very simple.

The conductances in parallel add,
G equivalent = G1 + G2 + G3.

Remember the R equivalent
1 divided by G equivalent.

So this is often used in electromagnetics
as well as other aspects of electrical

engineering.

The current divider is what happens
when we have resistors in parallel.

We bring in a single current, is, and
it divides into two currents, i1 and i2.

According to this equation here,

you can see that i1 is going
to be dependent upon R2,

and i2 is dependent upon R1.

Here's your current divider card.

Now a current divider can also
be used as a current adder.

If we had one current,
we could divide it into two or

if we had two currents coming in,
they could be combined into one current.

Now currents can be added up in
parallel but voltages can't.

So voltages and parallel have to be
the same or else the socket blows up.

Here, for example, are three different
batteries, this is a 1.5 volt battery,

this is a 9 volt battery, and
this is a 12 volt battery.

We can't put those in parallel
unless they were equal, or

else we blow up our circuit.

So this is also an unrealizable circuit.

In order to transform a source, we often
do this in order to simplify our circuit

or better understand what's
going on in the circuit.

So if, for example, we had had a voltage
source, remember how we had the voltage

source card for a realistic voltage
that had a resistor in series with it?

If we want to convert that instead
to a realistic current source,

not an ideal current source,
a realistic current source,

here's the transformation
that we would do.

Is would be vs divided by R1, and R1 and
R2 in these pictures would be equal.

We can go back and forth between
these two equivalent circuits and

have different source transformations.

Now, how might we use this.

I want you to take a look in your
text book of example 2-10 and

go through this example
in some level of detail.

Basically, transforming back and
forth between current and

voltage sources allows us to more
easily analyze this particular circuit.

So we started out with
a current source and

several resistors in series and
in parallel.

Then, if we convert the current
source to a voltage source,

this allows us to combine
these two resistors in series.

Then, if we convert this new voltage
source back to a current source,

that's going to allow us to more easily
include these resistors in parallel and

so on until we're finished.

This analysis is often done when we're
doing filter design and development.

So in short,
we've talked about equivalent circuits.

Basically, an equivalent circuit
is if we have the same voltage and

current at the front end we know
that two circuits are equivalent.

We talked about series and parallel.

Remember that resistances in series
add and conductances in parallel add.

Voltages can be added in series, and

current sources can be added in parallel,
but not the other way around.

We also talked about voltage and
current dividers.

Remember that those can
also be used as summers and

then we talked about equivalent sources.

The picture from the front is from the rim
of Snow Canyon in Saint George, Utah.