L7 5 1 Source Transformations

Transforming voltage source and series impedance with a current source and a parallel impedance

>> In the next few videos, we're
going to extend the concepts of

equivalent circuits into the phasor domain,

in terms of impedances and
phasor voltages and phasor currents.

So to do that,

we're going to start by looking
at the source transformations,

transforming a voltage source with

a series impedance into a current source
with a parallel impedance and back.

Then we'll also extend the concept of

Thevenin equivalent circuits to include
phasors and complex impedances.

So by review, what we mean when we
say two circuits are equivalent,

we mean that they have in this case
the same terminal characteristics.

By that we mean that
an external circuit connected to

a voltage source with a series
impedance will experience

the same voltage and current
as that same external circuit

would experience if it were connected
to a parallel current source,

connected in parallel with an impedance.

Both instances, we're going to have
the source impedance be the same value,

and what we wanted to do is
determine the relationship between

V sub s and I sub s.

So that loads or

external circuits connected to either

of these would not be able
to tell the difference.

So they have the same
terminal characteristics.

That means same voltage and same current.

To accomplish that, this load here
is going to experience the same V,

what reference V12, the voltage from

node one to node two in both the circuits.

In other words, this V12 and
this V12 will be the same.

So to do that,

we need to have the open-circuit voltage.

The voltage that you would
experience if there was

no load connected here to be the same.

In this case, since it's
open circuit there'll be

no current flowing through here,

and the voltage V12 will simply equal V,

the open circuit will equal

V sub s. Down here when
the terminals one and two are open,

no current is coming this way.

So in this case, all of
the current from the source is

going through this parallel impedance,

and the voltage that you
would then measure here,

this V open circuit,

would equal just I sub s times
Z sub s. From this then,

we can write directly what
the relationship needs to be.

In order for these two open-circuit
voltages to be the same,

this V_OC which is V sub s,

must equal this open- circuit voltage here,

I sub s times Z sub s. So in transforming

a current source with

a parallel impedance into
a voltage source with a series impedance,

the voltage source here
would be equal to I sub

s times Z sub s. Simply rearranging it,

we can come up with
an expression for I sub s in

terms of V sub s. That would
be I sub s equals V sub s

over Z sub s. So if we had
a series voltage source and impedance,

we could replace those with a parallel
current source and impedance.

If I sub s here was equal to
the quantity V sub s divided by Z sub s,

we get a little bit better
feel for that by looking

at what is referred to as
the short circuit current.

If you short this out here
and call it I short circuit,

we should expect to experience
the same I short circuit,

the same current through this short here.

Well in this circuit here,

I short circuit, in other words
zero resistance there,

the current there is just
going to be V sub s divided by

Z sub s. On the other hand down
here with this being shorted,

it shorts out the impedance and so
none of the current goes through here.

The short-circuit current then would
be simply I sub s or I short circuit

equals I sub s. Here we then see that in
order for these two to be equivalent,

I sub s equals V sub s over
Z sub s as we saw there.