>> 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.