Download AC Circuits and RMS Values

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Basic Information About
D.C. CIRCUITS
Basic Information About
A.C. CIRCUITS
A DC power supply is designed to produce an electric field in one
direction through the wires connected to the supply. That field motivates
charge carriers to move in one direction and one direction only in the
circuit.
An AC power supply is designed to produce an electric field that alternates
back and forth in direction through the wires connected to the supply. That
field motivates charge carriers to jiggle back and forth in the circuit.
The symbol for a DC power supply is:
The symbol for a AC power supply is:
DC power is produced by setting up a fixed potential difference between
two terminals using a battery or other such source.
AC power is produced by rotating a coil in a magnetic field.
The concept of current in a AC setting is less obvious. As charge carriers
move back and forth, a count of the number of charge carriers that pass by
a point per unit time yields zero net (“average”) current in the branch. As
we do want to assign current-like information to AC circuits, how we can do
this is something we have to discuss in more detail.
The idea of current in a DC setting makes perfect sense. As charge
carriers move in one direction and one direction only, a count of the
number of charge carriers that pass by a point per unit time gives us the
current in the branch.
1.
2.
The relationship between angular frequency
Remembering that a battery’s voltage in a DC circuit really tells us the
potential difference between the battery’s terminals (in other words, it’s
really a !V term), the function that identifies the “voltage” across an AC
source must tell us what the potential difference is between the AC
source’s terminals. What makes this a bit exciting is the fact that this
voltage changes with time.
! and frequency ! is:
! = 2"#
So the voltage function is often written as:
V ( t ) = Vo sin ( 2!" t )
The time dependent function that identifies the voltage across an AC
source’s terminals is:
That means a resistor circuit in which an AC source exists looks like:
V ( t ) = Vo sin ( ! t )
R
where Vo is the amplitude of the voltage (i.e., the largest voltage the
source can provide), ! is the angular frequency of the source and
identifies how fast the function is changing, and the product !t is an
angular measure in radians (sine functions act on angles).
V ( t ) = Vo sin ( 2!" t )
3.
4.
NOTE that the frequency used in a typical household wall socket is 60
Hz, so the voltage function for that kind of source becomes:
Now consider the AC circuit shown to
the right, and let’s additionally assume
the resistor is really a light bulb. The
AC source is providing a certain amount
of power to the light bulb, lighting the
bulb.
V ( t ) = Vo sin ( 2!(60) t )
= Vo sin ( 377 t ) .
We still don’t know that amplitude for this function (no, it’s not 110 volts),
but we will make short shrift of that quantity in short order.
R
V ( t ) = Vo sin ( 2!" t )
The question is: How large would a DC battery have to be, and how
much DC current would that battery have to supply to the circuit so
that the power being provided to the circuit was the same as the power
being provided by the AC source?
That is the question we need to answer.
5.
The key is to determine the DC-equivalent
voltage and current--the DC voltage and
current that mimics the AC voltage and
current, and to do that we need to look at
the power relationship.
In a DC circuit, the power being provided by the
battery is equal to:
6.
R
V1
Reiterating, iDC-equivalent is the amount of current you would need in a DC
circuit to duplicate the power input being provided by the AC source. But
how to calculate that quantity? We can’t take the average of the current
in the AC circuit as that will be zero.
R
The point to notice is that the quantity we really want is not the average
current, it’s the average of the current squared. It is the current squared
that determines the power.
i1
P = ( i1 ) R,
2
Following suit for our AC circuit, the AC
source’s power must equal:
P = ( iDC effective ) R,
2
where the iDC effective quantity is related to
the AC current but is additionally equal to
the amount of DC current needed to
parallel the AC case.
V ( t ) = Vo sin ( 2!" t )
7.
8.
The square of a sine wave function is always positive, and although it
doesn’t look so in my sketch the walls of a sine function squared are
fairly vertical. As such, we can approximate the average of the AC
current-squared as the amplitude of the current-squared divided by 2.
From this, we can write:
(i
i( t)
io
!
!
!t
i( t)
DC-equivalent
!
)
2
= io 2 / 2
(i
DC-equivalent
)
2
= io 2 / 2
io
/2
iDC-equivalent = .707 io
iDC-equivalent =
2
io 2
As the process used to get this relationship was to take the square Root
of the Mean value of the amplitude Squared, it’s called the RMS value of
the AC current. In other words:
io 2 / 2
iRMS = .707 io
!t
9.
10.
For a more graphic presentation, the meters in the two circuits below
will read the same (assuming the resistance is the same in both
circuits). (See if you can’t see why.)
Likewise, there is a RMS value for voltage, also. It is written as:
VRMS = .707 Vo
VAC
It is important to understand what RMS values give you. In the case of
voltage generated by an AC power supply, it gives you the single, DC
voltage that would provide to the circuit the same amount of power the
AC power supply does. In other words, if you wanted to take the AC
source out of the circuit and replace it with a comparable DC source,
that’s how big the DC source would have to be to accommodate the job.
A AC
Additionally, as a DC meter in an AC circuit would just quiver (the
alternating electric field would make charge go one way, then the other
way, etc., and the DC meter wouldn’t be able to keep up with the
changes), AC meters are designed differently than DC meters. In any
case, AC meters read RMS values.
A DC
V ( t ) = Vo sin ( 2!" t )
11.
VDC
VRMS = .707Vo
12.
An even more graphic example is a 20 ohm resistor hooked into a wall
socket:
VAC
VDC
AC voltmeter reads
110 volts
DC voltmeter reads
110 volts
R = 20 !
R = 20 !
AC ammeter reads 5.5 amps
DC ammeter reads 5.5 amps
A DC [ as i=V/R=(110)/(20) ]
A AC (as RMS meters read DC-
equivalent values and
iRMS =VRMS /R=(110)/(20)
OR, iRMS =.707io =.707(7.78))
i
VRMS = 110 volts
as VRMS = .707 (155.6) ) = 110 volts
i(t)
i ( t ) = 7.78 sin ( 377 t )
(
V ( t ) = 155.6 sin ( 2! ( 60 Hz )) t
)
In short, you can start with the DC-equivalent values and determine
their AC counterparts, or go the other way.
13.
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