Passive Low Pass Filter |
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Low Pass Filters
Basically, an electrical filter is a circuit that can be designed to modify, reshape or reject
all unwanted frequencies of an electrical signal and accept or pass only those signals wanted by the circuits designer.
In low frequency applications (up to 100kHz), passive filters are usually made from simple RC
(Resistor-Capacitor) networks while higher frequency filters (above 100kHz) are usually made from RLC
(Resistor-Inductor-Capacitor) components.
Simple First-order passive filters (1st order) can be made by connecting together a single resistor
and a single capacitor in series across an input signal, (Vin) with the output signal,
(Vout) taken from the junction of these two components. Depending on which way around we connect
the resistor and the capacitor with regards to the output signal determines the type of filter construction resulting in
either a Low Pass Filter or a High Pass Filter.
The function of any filter is to allow signals of a given band of frequencies to pass unaltered while
attenuating or weakening the others that are not wanted. As there are two passive components within this type of filter
design the output signal has a smaller amplitude than its corresponding input signal, therefore passive
RC filters attenuate the signal and have a gain of less than one, (unity).
The Low Pass RC Filter
A simple passive Low Pass Filter or LPF, can be easily made by connecting
together in series a single Resistor with a single Capacitor as shown below. In this type of filter arrangement the input
signal (Vin) is applied to the series combination (both the Resistor and Capacitor together) but
the output signal (Vout) is taken across the capacitor only. This type of filter is known generally
as a 1st order Filter, why 1st order?, because it has only "one" reactive component in the circuit, the Capacitor.
Low Pass Filter Circuit
As mentioned previously in the
Capacitive Reactance tutorial, the reactance
of a capacitor varies inversely with frequency, while the value of the resistor remains constant as the frequency changes.
At low frequencies the capacitive reactance, (Xc) of the capacitor will be very large compared
to the resistive value of the resistor, R and as a result the voltage across the capacitor,
Vc will also be large while the voltage drop across the resistor, Vr
will be much lower. At high frequencies the reverse is true with Vc being small and
Vr being large.
While the circuit above is that of an RC Low Pass Filter circuit, it can also
be classed as a frequency variable potential divider circuit similar to the one we looked at in the
Resistors tutorial. In that tutorial we used
the following equation to calculate the output voltage for two single resistors connected in series.
We also know that the capacitive reactance of a capacitor in an AC circuit is given as:
Opposition to current flow in an AC circuit is called impedance, symbol Z
and for a series circuit consisting of a single resistor in series with a single capacitor, the circuit impedance is calculated as:
Then by substituting our equation for impedance above into the resistive potential divider equation gives
us:
So, by using the potential divider equation of two resistors in series and substituting for impedance
we can calculate the output voltage of an RC Filter for any given frequency.
Example No1
A Low Pass Filter circuit consisting of a Resistor of 4k7Ω in
series with a Capacitor of C = 47nF is connected across a 10v DC supply.
Calculate the output voltage (Vout) at a frequency of 100Hz and again at frequency of 10,000Hz or 10kHz.
At a frequency of 100Hz.
Frequency Response
We can see above, that as the frequency increases from 100Hz to 10kHz, the output voltage (Vout)
decreases from 9.9v to 0.718v. By plotting the output voltage against the input frequency, the Frequency Response Curve or
Bode Plot function of the low pass filter can be found, as shown below.
Frequency Response of a 1st Order Low Pass Filter.
The Bode Plot shows the Frequency Response of the filter to be nearly flat for low frequencies
and all of the input signal is passed directly to the output, resulting in a gain of nearly 1,
unity until it reaches the Cut-off Frequency point ( ƒc ). This is because the reactance
of the capacitor is high at low frequencies and blocks any current flow through the capacitor. After this point the
response of the circuit decreases giving a slope of -20dB/ Decade or (-6dB/Octave) "roll-off"
as signals above this frequency become greatly attenuated, until at very high frequencies the reactance of the capacitor
becomes so low that it gives the effect of a short circuit condition on the output terminals resulting in zero output.
For this type of Low Pass Filter circuit, all the frequencies below this cut-off,
ƒc point that are unaltered with little or no attenuation and are said to be in the filters
Passband zone. This passband zone also represents the Bandwidth of the filter. Any signal frequencies above
this point cut-off point are generally said to be in the filters Stopband zone and they will be greatly attenuated.
This "Cut-off", "Corner" or "Breakpoint" frequency is defined as being the frequency point where the
capacitive reactance and resistance are equal, R = Xc = 4k7Ω. When this occurs the output
signal is attenuated to 70.7% of the input signal value or -3dB (20 log (Vout/Vin)) of
the input. Although R = Xc, the output is not half of the input signal. This is because
it is equal to the vector sum of the two and is therefore 0.707 of the input. As the filter contains a capacitor, the
Phase Angle ( Φ ) of the output signal LAGS behind that of the input and at the
-3dB cut-off frequency ( ƒc ) and is -45o out of phase. This is due to the time
taken to charge the plates of the capacitor as the input voltage changes, resulting in the output voltage (the voltage
across the capacitor) "lagging" behind that of the input signal. The higher the input frequency applied to the filter the
more the capacitor lags and the circuit becomes more and more "out of phase".
The cut-off frequency point and phase shift angle can be found by using the following equation:
Cut-off Frequency and Phase Shift
Then for our simple example of a "Low Pass Filter" circuit above, the cut-off frequency
(ƒc) is given as 720Hz with an output voltage of 70.7% of the input
voltage value and a phase shift angle of -45o.
Low Pass Filter Summary
So to summarize, the Low Pass Filter has a constant output voltage from D.C. (0Hz),
up to a specified Cut-off frequency, ( ƒc ) point. This cut-off frequency point is 0.707
or -3dB (dB = -20log Vout/Vin) of the voltage gain allowed to pass. The frequency range "below" this cut-off point
ƒc is generally known as the Pass Band as the input signal is allowed to pass
through the filter. The frequency range "above" this cut-off point is generally known as the Stop Band as the
input signal is blocked or stopped from passing through. A simple 1st order low pass filter can be made using a single
resistor in series with a single non-polarized capacitor (or any single reactive component) across an input signal
Vin, whilst the output signal Vout is taken from across the capacitor.
The cut-off frequency or -3dB point, can be found using the formula, ƒc = 1/(2πRC).
The phase angle of the output signal at ƒc and is -45o
for a Low Pass Filter.
The gain of the filter or any filter for that matter, is generally expressed in Decibels and is
a function of the output value divided by its corresponding input value and is given as:
Applications of passive Low Pass Filters are in audio amplifiers and speaker systems to direct the lower
frequency bass signals to the larger bass speakers or to reduce any high frequency noise or "hiss" type distortion. When
used like this in audio applications the low pass filter is sometimes called a "high-cut", or "treble cut" filter.
If we were to reverse the positions of the resistor and capacitor in the circuit so that the output
voltage is now taken from across the resistor, we would have a circuit that produces an output frequency response curve
similar to that of a High Pass Filter,
and this is discussed in the next tutorial.
Time Constant
We know from above, that the filters cut-off frequency (ƒc) is the product
of the resistance (R) and the capacitance (C) in the circuit with respect
to some specified frequency point and that by altering any one of the two components alters this cut-off frequency point by
either increasing it or decreasing it. We also know that the phase shift of the circuit lags behind that of the input signal
due to the time required to charge and then discharge the capacitor as the sine wave changes. This combination of
R and C produces a charging and discharging effect on the capacitor known
as its Time Constant (τ) of the circuit as seen in the
RC Circuit tutorials.
This time constant, tau (τ), is related to the cut-off frequency
ƒc as.
or expressed in terms of the cut-off frequency, ƒc as.
The output voltage, Vout depends upon the time constant and the frequency of
the input signal. With an AC sinusoidal signal the circuit behaves as a simple 1st order low pass filter. But what if we
where to change the input signal to that of a "square wave" shaped signal that has an almost vertical step input, the
response of the circuit changes dramatically and produces a circuit known commonly as an Integrator.
The RC Integrator
The Integrator is basically a low pass filter circuit that converts a square wave step response
input signal into a triangular shaped waveform output as the capacitor charges and discharges. A Triangular waveform
consists of alternate but equal positive and negative ramps. As seen below, if the RC time constant
is long compared to the time period of the input waveform the resultant output waveform will be triangular in shape and the
higher the input frequency the lower will be the output amplitude compared to that of the input.
The RC Integrator Circuit
This then makes this type of circuit ideal for converting one type of electronic signal to another for
use in wave-generating or wave-shaping circuits.
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