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# RC Passive Low-Pass Filter

A **low-pass** filter passes frequencies below a certain cutoff frequency
and attenuates those beyond that frequency.

The first circuit we shall analyze is that of an RC low-pass filter, as shown in the figure above. Before launching into a mathematical analysis, we can deduce some of the electrical properties by visual inspection of the circuit.

If the applied voltage is of very low frequency, the reactance of *C*
will be very high compared with *R*, and *C* may be considered
an open circuit. Therefore, at low frequencies the input voltage
**V**_{in} will appear virtually unattenuated at the output.
Hence, we have the name low-pass filter. As the input frequency increases,
the reactance *X*_{C} becomes smaller, causing the input to
be increasingly attenuated. At an infinitely high frequency,
*X*_{C} = 0 and therefore the output voltage *V*_{out} = 0.

To analyze the circuit mathematically, we would use the voltage-divider relationship and write

We shall, however, solve the ratio of **V**_{out} to
**V**_{in}, since we generally wish to express the
filter gain or loss. This ratio is called the **transfer function**.
As a transfer function, we then have

where *ω*_{C} = 1/*RC* is the **characteristic frequency**.

The transfer function can be expressed in polar form

## Frequency response

A most useful means of displaying the frequency characteristics of a filter
is to plot the magnitude of the transfer function (amplitude characteristic)
versus frequency on one curve and the phase characteristic as a separate
curve but with the same frequency axis. The amplitude characteristic, which
may vary over a wide range, can be conveniently plotted in terms of decibels.
Curves with this type of display are known as **Bode plots**
and find wide application in the analysis of AC circuits.

## Amplitude Characteristic

Consider first the amplitude characteristic (spectrum) corresponding to the equation above. This is the absolute value (magnitude) of the transfer function, or

On a decibel basis, this becomes

Let us examine the equation above for very low and for very high frequencies. For low frequencies, we have

Thus, the low-frequency behavior is essentially independent of frequency and
can be represented by a horizontal straight line at 0 dB as in the figure below.
The actual amplitude characteristic given by the transfer function is
asymptotic to this straight line for small *ω*.

For the other extreme, we have

This is of the form *G*_{dB} = -20*x*,
where *x* = log_{10} (*ω*/*ω*_{C}).
The straight line so defined is the high-frequency asymptote of the actual
characteristic. The slope of the asymptote is
*dG*_{dB}/*dx* = -20; that is, when *x* increases one unit,
*G*_{dB} decreases by 20 dB. But

and so *ω*/*ω*_{C} must increase by a factor of
10, or one *decade*, to make *x* increase one unit. Therefore,
the slope of the high-frequency asymptote is -20 dB per decade.
Some people prefer to use the octave (frequency ratio of 2:1). The
corresponding slope is -6 dB per octave. The two straight-line asymptotes
intersect at *ω*/*ω*_{C} = 1, for then the
amplitude characteristic has the value zero. The two asymptotes are shown
dashed in the figure above. Their point of intersection,
*ω* = *ω*_{C}, in addition to being termed the
characteristic frequency of the circuit, is also called the **break point**,
or **cutoff frequency**. Together, the two asymptotes form a broken-line
approximation to the actual characteristic. Depending upon the accuracy desired,
neither line may be a sufficiently good approximation to the actual
characteristic in the neighborhood of *ω* = *ω*_{C}.
It can be shown that the maximum error occurs at
*ω*/*ω*_{C} = 1 and is approximately 3 dB.
Further more, an octave away from this point
(at *ω*/*ω*_{C} = 0.5 and
*ω*/*ω*_{C} = 2)
the error is approximately 1 dB. From this, it is easy to sketch the actual
amplitude characteristic with reasonable accuracy. The actual characteristic
is shown by the solid line in the figure above. The low-pass
characteristic of the circuit is easily seen in this figure.

## Phase Characteristic

Let us now consider the angle of the transfer function of the filter (sometimes called the phase spectrum), which is

The phase angle starts at zero for *ω* = 0 and approaches -π/2
radians at large *ω*. The phase characteristic can be approximated
reasonably well by three straight-line segments, as shown in the figure
below: a low-frequency approximation at 0 radians, a high-frequency
approximation at -π/2 radians, and an intermediate-frequency approximation
which is tangent to the curve at -π/4 radians. It can be shown that
the middle segment intersects the low-frequency approximation at
*ω*/*ω*_{C} = 1/4.81 and intersects the
high-frequency approximation at *ω*/*ω*_{C} = 4.81.
The actual characteristic is shown by the solid line in the figure.