Download AC RLC Circuits (M - O – U – S – Eeeee)

AC RLC Circuits (M - O – U – S – Eeeee)
The following is discussion of modeling a series circuit incorporating an inductance, a resistance
and a capacitance. It is assumed that the behavior of these attributes in the presence of a static voltage (a
DC circuit) is understood.
The dominant contribution to any one of these quantities can often be specifically identified and
isolated (a solenoid will often constitute the dominant inductance, for example, and a resistor the dominant
source of resistance). In general, however, these attributes will be difficult or impossible to isolate from
one another (the wire of which the solenoid is composed will constitute another source of resistance and
cannot be isolated from the inductance). Thus, any analysis which incorporates only one or two of these
attributes will necessarily be an approximation, albeit perhaps a good one. To begin, I will address each
element by itself (that is, I will do what I just counseled against – model a circuit as consisting of only one
element). In all cases, the assumption is that there is a sinusoidal voltage source (or generator) in series
with each element (and finally in series with all elements).
AC Voltage with Resistance
In this case, the voltage will drive a current and at any instant of time, the voltage drop across the
resistor (given by IR) will be equal to that provided by the voltage source. And so the current delivered to
the circuit will be given by:
I (t ) =
V (t ) VMAX
=
cos(ωt )
R
R
(1)
Note that the peak current is independent of frequency and that the voltage across the resistor is in
phase with the voltage. Also, keep in mind for later that the voltage across the resistor is the product of the
current and the resistance. Simple enough so far.
AC Voltage with Capacitance
Again, the voltage will deliver a current that serves to deposit a charge on the capacitor. At any
instant, the loop rule implies that the voltage drop across the capacitor will equal that across the generator:
V(t) = VMAX cos(ωt) = VCAP = Q/C, where Q is the charge on the capacitor at that instant. The rate at which
charge is deposited, dQ/dt, will equal the current delivered by the generator, I(t). So we have that:
I (t ) =
dQ
= −ωCVMAX sin (ωt )
dt
(2)
First, notice that the maximum current is linear in the frequency: the higher the frequency, the
higher the current. If you were to plot the voltage and the current delivered in time, you would see that the
current lags the voltage by ¼ of a period, as distinct from the resistive circuit, for which the two quantities
were in phase. Also, see that the maximum current is given by ωCVMAX, as compared to the resistive case,
for which the maximum current was given by VMAX / R. We can make this one look like that one if we
define something called the capacitive reactance as XC = 1/ωC. Then the current in the circuit becomes:
I (t ) = −
VMAX
V
sin (ωt ) = − MAX sin (ωt )
1 ωC
XC
(3)
The analogy with Eq. (1) is clear, as are the units for reactance (figure out what they have to be
and then prove it explicitly). Again, for later, note that the voltage across a capacitor is the product of the
current and the capacitive reactance.
AC Voltage with Inductance
As before, the generator will deliver a time-varying current to the circuit that will traverse the
inductance and induce a voltage that will appear across the inductor. From Faraday’s law, we know that
the voltage drop is given by LdI/dt, and so the generator voltage satisfies V(t) = VMAX cos(ωt) = VIND =
LdI/dt. We can easily get the current as:
I (t ) = ∫
V
dI
dt = MAX sin (ωt )
ωL
dt
(4)
Now, notice that the maximum current varies inversely with the frequency: the higher the
frequency, the lower the current. If you were to plot the voltage and the current delivered, you would see
that the current leads the voltage by ¼ of a period, as distinct from the resistive circuit, for which the two
quantities were in phase, or the capacitive circuit, in which the current lagged. Also, see that the maximum
current is given by VMAX / ωL. Compare this to the resistive and capacitive circuits, for which the
maximum current was given by VMAX / R and VMAX / (1/ωC), respectively. Again, we can make this one
look like that one if we define something called the inductive reactance as XL = ωL. Then the current in
the circuit becomes:
I (t ) =
VMAX
V
sin (ωt ) = MAX sin (ωt )
ωL
XL
(5)
The analogy with Eq. (1) is again clear, as are the units for reactance (and again you should figure
out what they have to be and then prove it). And note that the voltage across an inductor is the product of
the current and the inductive reactance.
AC Voltage with Inductance, Capacitance and Resistance
The total voltage in a series RLC circuit can be written as:
VTOT = VR + VL + VC = I MAX R cos(ωt ) + I MAX X L sin (ωt ) − I MAX X C sin (ωt )
(6)
where I have used the ideas developed above and assumed that the current in the circuit varies
sinusoidally (i.e., as I(t) = IMAX sin(ωt) ). We will discover in the fullness of time how the voltage must
behave for such an outcome. If I introduce a quantity Z (called the impedance, whose meaning we will
presently discuss) defined as:
Z = R 2 + (X L − X C )
2
(7)
then I can rewrite Eq. (6) as:
X − XC
R

VTOT = I MAX Z  cos(ωt ) + L
sin (ωt ) .
Z
Z

(8)
Note that the coefficients of the (two trig functions inside the parentheses) satisfy
(R/Z)2 + ((XL-XC)/Z)2 = 1 (because of the way I defined Z), so that they can be called the sin and cosine of
some made up angle (which we will interpret directly) Θ; that is, I can define cosΘ = R/Z and sinΘ =
(XL-XC)/Z. Then I can rewrite Eq. (8) as:
VTOT = I MAX Z (cos Θ cos(ωt ) + sin Θ sin (ωt )) = I MAX Z cos(ωt − Θ )
(9)
That is, the total voltage across the circuit varies sinusoidally with a maximum value of VMAX =
IMAXZ, a frequency the same as that of the current, and a phase shifted from the current by Θ. If the phase
shift Θ is positive (meaning that XL > XC or that the circuit is more inductive than capacitive), then we
know that the current leads the voltage, consistent with our earlier analysis of a purely inductive circuit.
Conversely, if the phase shift is negative (meaning that XL < XC or that the circuit is more capacitive than
inductive), then we know that the current lags the voltage, consistent with our earlier conclusions about a
purely capacitive circuit. Finally, there’s a magical place, when XL = XC, for which Θ = 0 and thus the
current and voltage are in phase (like a resistive circuit). What exactly is this magical place? Putting in the
definitions of inductive and capacitive reactance, the condition of XL = XC corresponds to the condition:
ω=
1
= ω0
LC
(10)
where ω0 is the natural frequency of oscillation of the inductor and capacitor when in circuit alone.
That is, the voltage and current are in phase if the driving frequency, ω, is equal to the natural frequency of
the system, ω0. How big is that current, IMAX, for a given input voltage, VMAX? In general, the current
amplitude is given by:
I MAX =
VMAX
R + (X L − X C )
2
2
=
ωVMAX
L
1
(2ωγ )2 + (ω 2 − ω 02 )2
(11)
where I’ve defined γ = 2R/L, as in the past. If we impose the condition of Eq. (11), this is:
I MAX , RES =
ωVMAX
L
1
(2ωγ )
=
VMAX
R
(12)
That is, the maximum current for this particular condition is that expected if the circuit were
purely resistive. So if you choose to drive the circuit at its natural frequency, the circuit behaves as though
it has no inductance or capacitance, only a resistance. This condition is called resonance, and a little
thought (or a graph or mathematical proof) shows that this is the largest current amplitude possible. For
any other driving frequency, the current amplitude will be smaller than this value. With this in mind, you
can (and should) rewrite Eq. 11 as:
I MAX =
VMAX
R
1
ω −ω
1 + 
 2ωγ
2
2
0



2
(13)
which shows more clearly the behavior described. (You should also prove that it shows that at
zero frequency [DC], the current is zero [right? Does this make sense? Also, what is the dependence near
DC] and for frequencies far above the resonant frequency, the current amplitude goes as VMAX/ωL, like an
inductive circuit). There are so many ways to go here (high and low pass filters, band pass filters,
impedance matching, power coupling, etc.), but we will content ourselves with a laboratory investigating
this resonance condition. The basic question? Is Eq. 13 right – does it represent the world accurately? So,
how do we measure that? How do we prove or disprove Lucky Eq. 13?
Begin.