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Since the square of the voltage
appears in the energy formula, the energy stored is always positive. You can't
have a negative amount of energy in the capacitor. That means you can put
energy into the capacitor, and you can take it out, but you can't take out more
than you put in.
-
Power in to the capacitor can be
negative. Voltage can be positive while current is negative. Imagine a capacitor
that is charged. You could charge a capacitor by putting a battery across the
capacitor, for example. Then, if you placed a resistor across the capacitor,
charge would leave the capacitor - current would flow out of the capacitor - and
the energy in the capacitor would leave the capacitor only to become heat energy
in the resistor. When energy leaves the capacitor, power is negative.
Frequency Dependent Behavior For A Capacitor
We start with a
capacitor with a sinusoidal voltage across it.
where:
-
vC(t) = Voltage
across the capacitor
-
iC(t) = Current
through the capacitor
-
C = Capacitance (in farads)
We will assume
that the voltage across the capacitor is sinusoidal:
vC(t) =
Vmax sin(wt)
Knowing the voltage across the capacitor allows us to calculate the current:
iC(t) =
C dvC(t)/dt = w C
Vmax cos(wt)
= Imax cos(wt)
where
Imax
= wC
Vmax
Comparing the expressions for
the voltage and current we note the following.
-
The voltage and the current are both
sinusoidal signals (a sine function or a cosine function)
at the same frequency.
-
The
current leads the voltage.
In other words, the peak of the current occurs earlier in time than the peak of
the voltage signal.
-
The current leads the voltage by
exactly 90o.
It will always be exactly 90o in a capacitor.
-
The magnitude of the current and the
magnitude of the voltage are related:
Vmax/Imax
= 1/ wC
Now, with these observations in hand, it is possible to see that there may be an
algebraic way to express all of these facts and relationships. The method
reduces to the following.
-
If we have a circuit with
sinusoidally varying voltages and currents (as we would have in a circuit with
resistors, capacitors and inductors and sinusoidal voltage and current sources)
we associate a complex variable with every voltage and current in the circuit.
-
The complex variable for a voltage or
current encodes the amplitude and phase
for that voltage or current.
-
The voltage and current variables can
be used (using complex algebra) to predict circuit behavior just as though the
circuit were a resistive circuit.
We need to do two things here. First, we can illustrate what we mean with
an example. Secondly, we need to justify the claim above. We will
look at an example first, and we will do two examples. The first example
is jsut the capacitor - all by itself. The second example will be one that
you have considered earlier, a simple RC low-pass filter.
Example 1 - The Capacitor
In a capacitor with sinusoidal voltage and currents, we have:
where:
-
iC(t) =
wC Vmax cos(wt)
= Imax cos(wt)
We represent the voltage with a complex variable, V.
Considering this as a complex variable, it has a magnitude of
Vmaxand and angle of
0o. We would write:
V = Vmax/0o
Similarly, we can get a representation for the current. However, first
note:
iC(t) =
wC Vmax cos(wt)
= Imax cos(wt)
= Imax sin(wt
+ 90o)
(Here you must excuse the mixing of
radians and degrees in the argument of the sine. The only excuse is that
everyone does it!) Anyhow, we have:
I = Imax/90o
= j Imax = jwC
Vmax
Where j is the square root of -1.
Then we would write:
V/I = Vmax/jwC
Vmax = 1/jwC
and the quantity
1/jwC
is called the impedance
of the capacitor. In the next section we will apply that concept to a
small circuit - one you should have seen before.
Before moving
to the next section, a little reflection is in order. Here are some points
to think about.
-
A phasor summarizes information about
a sinusoidal signal. Magnitude and phase information are encoded into the
phasor. Frequency information is not encoded, and there is a tacit
assumption that all signals are of the same
frequency, which would be the case in a linear
circuit with sinusoidal voltage and current sources.
-
We looked at a case where we encoded
a signal Vmax
sin(wt)
into a phasor of Vmax/0o.
That was completely arbitrary, and many others would have encoded
Vmax
cos(wt)
into a phasor of Vmax/0o.
-
Phasors are intended only to show
relative phase information, and it doesn't matter which way you go.
Using Impedance
In the last
section we began to talk about the concept of impedance. Let us do that a
little more formally. We begin by defining terms.
A
sinusoidally varying signal (vC(t) = Vmax
sin(wt)
for example) will be represented by a phasor,
V, that incorporates the magnitude and phase angle
of the signal as a magnitude and angle in a complex number. Examples
include these taken from the last section. (Note that these phasors have
nothing to do with any TV program about outer space.)
vC(t) = Vmax
sin(wt)
is represented by a phasor
V = Vmax/0o
iC(t) = Imax
sin(wt + 90o)
is represented by a phasor
I = Imax/90o
va(t) = VA
sin(wt +
f)
is represented by a phasor
Va
= VA/f
Next, we can use the relationships for voltage and current phasors to analyze a
circuit. Here is the circuit.
Now, this circuit is really a frequency dependent voltage divider, and it is
analyzed differently in
another lesson. However, here we will use phasors. At the end of
this analysis, you should compare how difficult it is using phasors to the
method in the other lesson.
We start by noting that the current in the circuit - and there is only one
current - has a phasor representation:
I = Imax/0o
We will use the current phase as a
reference, and measure all other phases from the current's phase. That's
an arbitrary decision, but that's the way we will start.
Next we note that we can compute the voltage across the capacitor.
VC
= I/jwC
This expression relates the
current phasor to the phasor that represents the voltage across the capacitor.
The quantity 1/jwC
is the impedance of the capacitor. In the last section we justified
this relationship. We can also compute the phasor for the voltage across the resistor. VR
= IR
This looks amazingly like Ohm's law,
and it is, in fact, Ohm's law, but it is in phasor form. For that matter,
the relationship between voltage and current phasors in a capacitor - just above
- may be considered a generalized form of Ohm's law!
Now, we can also apply Kirchhoff's Voltage Law (KVL) to compute the phasor for
the input voltage.
VIN
= VR + VC = IR + I/jwC
= I(R + 1/jwC)
You should note the similarities in what happens here and what happens when you
have two resistors in series.
Example
Consider a
series circuit of a resistor and capacitor. The series impedance is:
Z = R + 1/jwC
That's the same as we showed just above. The
impedance can be used to predict relationships between voltage and current.
Assume that the voltage across the series connection is given by:
vSeries(t) = Vmax
cos(wt)
That corresponds to having a voltage phasor of:
V = Vmax/0o
We also know that the impedance establishes a relationship
between the voltage and current phasors in the series circuit. In
particular, the voltage phasor is the product of the current phasor and the
impedance.
V = I Z
For our particular impedance, we have:
V = I*(R + 1/jwC)
So, we can solve for the current
phasor:
I = V / (R + 1/jwC)
Now, we know the voltage phasor
and we know the impedance so we can compute the current phasor. Let us
look at some particular values.
Assume:
-
R = 1.0 kW
-
C = .1mf
= 10-7 f
-
f = 1 kHz, so
w= 2p103
-
Vmax
= 20 v
Then:
-
ZR = 1.0 kW
-
ZC = 1/(jwC)
= 1/(j2p103 10-7 ) = j 1.59 kW
And, the total impedance is:
-
Z = ZR
+ ZC = (1.0 + j 1.59) kW
This impedance value can also be
expressed in polar notation:
Now, compute the current phasor:
Substituting values, we find:
And, we need to examine exactly what
this means for the current as a function of time. But that isn't very
difficult. We can write out the expression for the current from what we
have above.
-
iC(t) = 10.65 cos(wt
- 62o) amps
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