Introduction of transfer function(frequency response).
The ratio of the output and input amplitudes for [link], known as the transfer
function or the frequency response, is given
by
Implicit in using the transfer function is that the input is a
complex exponential, and the output is also a complex
exponential having the same frequency. The transfer function
reveals how the circuit modifies the input amplitude in creating
the output amplitude. Thus, the transfer function
completely describes how the circuit
processes the input complex exponential to produce the output
complex exponential. The circuit's function is thus summarized
by the transfer function. In fact, circuits are often designed
to meet transfer function specifications. Because transfer
functions are complex-valued, frequency-dependent quantities, we
can better appreciate a circuit's function by examining the
magnitude and phase of its transfer function
([link]).
Simple Circuit
A simple
circuit.
Magnitude and phase of the transfer function
Magnitude and phase of the transfer function of the RC circuit
shown in [link] when
.
This transfer function has many important properties and provides all the insights needed to determine how the circuit functions.
First of all, note that we can compute the frequency response for both positive and
negative frequencies. Recall that sinusoids consist of the sum
of two complex exponentials, one having the negative frequency
of the other. We will consider how the circuit acts on a
sinusoid soon. Do note that the magnitude has even
symmetry: The negative frequency portion is a mirror
image of the positive frequency portion:
.
The phase has odd symmetry:
. These properties of this specific example apply for
all transfer functions associated with
circuits. Consequently, we don't need to plot the negative
frequency component; we know what it is from the positive
frequency part.
The magnitude equals
of its maximum gain (1 at
)
when
(the two terms in the denominator of the magnitude are
equal). The frequency
defines the boundary between two operating ranges.
For frequencies below this frequency, the circuit does
not much alter the amplitude of the complex exponential
source.
For frequencies greater than
, the circuit strongly attenuates the amplitude.
Thus, when the source frequency is in this range, the
circuit's output has a much smaller amplitude than that of
the source.
For these reasons, this frequency is known as the cutoff
frequency. In this circuit the cutoff frequency depends
only on the product of the resistance and
the capacitance. Thus, a cutoff frequency of 1 kHz occurs when
or
. Thus resistance-capacitance combinations of 1.59
kΩ and 100 nF or 10 Ω and 1.59 μF result in the
same cutoff frequency.
The phase shift caused by the circuit at the cutoff frequency
precisely equals
.
Thus, below the cutoff frequency, phase is little affected, but at
higher frequencies, the phase shift caused by the circuit becomes
. This phase shift corresponds to the difference
between a cosine and a sine.
We can use the transfer function to find the output when the
input voltage is a sinusoid for two reasons. First of all, a
sinusoid is the sum of two complex exponentials, each having a
frequency equal to the negative of the other. Secondly, because
the circuit is linear, superposition applies. If the source is
a sine wave, we know that
Since the input is the sum of two complex exponentials, we know
that the output is also a sum of two similar complex
exponentials, the only difference being that the complex
amplitude of each is multiplied by the transfer function
evaluated at each exponential's frequency.
As noted earlier, the transfer function is most conveniently
expressed in polar form:
.
Furthermore,
(even symmetry of the magnitude) and
(odd symmetry of the phase). The output voltage expression
simplifies to
The circuit's output to a sinusoidal input is also a
sinusoid, having a gain equal to the magnitude of the
circuit's transfer function evaluated at the source frequency
and a phase equal to the phase of the transfer function at the
source frequency. It will turn out that this
input-output relation description applies to any linear
circuit having a sinusoidal source.
This input-output property is a special case of a more
general result. Show that if the source can be written as
the imaginary part of a complex exponential—
— the output is given by
.
Show that a similar result also holds for the real part.
The key notion is writing the imaginary part as the
difference between a complex exponential and its complex
conjugate:
The response to
is
,
which means the response to
is
.
As
,
the Superposition Principle says that the output to the
imaginary part is
.
The same argument holds for the real part:
.
The notion of impedance arises when we assume the sources are
complex exponentials. This assumption may seem restrictive;
what would we do if the source were a unit step? When we use
impedances to find the transfer function between the source
and the output variable, we can derive from it the differential
equation that relates input and output. The differential equation
applies no matter what the source may be. As we have argued, it is
far simpler to use impedances to find the differential equation
(because we can use series and parallel combination rules) than
any other method. In this sense, we have not lost anything by
temporarily pretending the source is a complex exponential.
In fact we can also solve the differential equation using
impedances! Thus, despite the apparent restrictiveness of
impedances, assuming complex exponential sources is actually
quite general.