Passive Low Pass Filter
Low Pass Filter Introduction
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 other words they "filter-out" unwanted signals and an ideal filter will separate and pass sinusoidal input signals
based upon their frequency.
In low frequency applications (up to 100kHz), passive filters are generally constructed using simple
RC (Resistor-Capacitor) networks, while higher frequency filters (above 100kHz) are usually
made from RLC (Resistor-Inductor-Capacitor) components. Passive filters are made up of
passive components such as resistors, capacitors and inductors and have no amplifying elements (transistors, op-amps, etc)
so have no signal gain, therefore their output level is always less than the input.
Filters are so named according to the frequency range of signals that they allow to pass through them,
while blocking or "attenuating" the rest. The most commonly used filter designs are the:
1. The Low Pass Filter the low pass filter only allows low frequency
signals from 0Hz to its cut-off frequency, ƒc point to pass while blocking those any higher.
- 2. The High Pass Filter the high pass filter only allows high frequency
signals from its cut-off frequency, ƒc point and higher to infinity to pass through while
blocking those any lower.
- 3. The Band Pass Filter the band pass filter allows signals falling within
a certain frequency band setup between two points to pass through while blocking both the lower and higher frequencies
either side of this frequency band.
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 of the filter, ( 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.
As the function of any filter is to allow signals of a given band of frequencies to pass unaltered while
attenuating or weakening all others that are not wanted, we can define the amplitude response characteristics of an ideal
filter by using an ideal frequency response curve of the four basic filter types as shown.
Ideal Filter Response Curves
Filters can be divided into two distinct types: active filters and passive filters. Active filters
contain amplifying devices to increase signal strength while passive do not contain amplifying devices to strengthen the
signal. As there are two passive components within a passive 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).
A Low Pass Filter can be a combination of capacitance, inductance or resistance intended to produce high
attenuation above a specified frequency and little or no attenuation below that frequency. The frequency at which the transition
occurs is called the "cutoff" frequency. The simplest low pass filters consist of a resistor and capacitor but more sophisticated
low pass filters have a combination of series inductors and parallel capacitors. In this tutorial we will look at the simplest
type, a passive two component RC low pass filter.
The Low Pass Filter
A simple passive RC 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 "first-order filter" or "one-pole filter", why first-order or single-pole?, because it
has only "one" reactive component, the capacitor, in the circuit.
RC 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
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.
A Low Pass Filter circuit consisting of a resistor of 4k7Ω
in series with a capacitor of 47nF is connected across a 10v sinusoidal
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.
At a frequency of 10kHz.
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,
called unity, until it reaches its 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 cut-off frequency 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
Pass band zone. This pass band 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 Stop band 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.
Second-order Low Pass Filter
Thus far we have seen that simple first-order RC low pass filters can be made by connecting a single
resistor in series with a single capacitor. This single-pole arrangement gives us a roll-off slope of -20dB/decade attenuation
of frequencies above the cut-off point at ƒ3dB . However, sometimes in filter
circuits this -20dB/decade (-6dB/octave) angle of the slope may not be enough to remove an unwanted signal then two stages
of filtering can be used as shown.
Second-order Low Pass Filter
The above circuit uses two passive first-order low pass filters connected or "cascaded" together
to form a second-order or two-pole filter network. Therefore we can see that a first-order low pass filter can be converted
into a second-order type by simply adding an additional RC network to it and the more RC stages
we add the higher becomes the order of the filter. If a number ( n ) of such RC stages
are cascaded together, the resulting RC filter circuit would be known as an "nth-order"
filter with a roll-off slope of "n x -20dB/decade".
So for example, a second-order filter would have a slope of -40dB/decade (-12dB/octave), a fourth-order
filter would have a slope of -80dB/decade (-24dB/octave) and so on. This means that, as the order of the filter is increased,
the roll-off slope becomes steeper and the actual stop band response of the filter approaches its ideal stop band characteristics.
Second-order filters are important and widely used in filter designs because when combined with first-order
filters any higher-order nth-value filters can be designed using them. For example, a third order
low-pass filter is formed by connecting in series or cascading together a first and a second-order low pass filter.
But there is a downside too cascading together RC filter stages. Although there is no limit to the order
of the filter that can be formed, as the order increases, the gain and accuracy of the final filter declines. When identical
RC filter stages are cascaded together, the output gain at the required cut-off frequency ( ƒc )
is reduced (attenuated) by an amount in relation to the number of filter stages used as the roll-off slope increases. We can
define the amount of attenuation at the selected cut-off frequency using the following formula.
Passive Low Pass Filter Gain at ƒc
where "n" is the number of filter stages.
So for a second-order passive low pass filter the gain at the corner frequency
ƒc will be equal to 0.7071 x 0.7071 = 0.5Vin (-6dB), a third-order passive low pass filter
will be equal to 0.353Vin (-9dB), fourth-order will be 0.25Vin (-12dB) and so on. The corner frequency, ƒc
for a second-order passive low pass filter is determined by the resistor/capacitor (RC) combination and is given as.
2nd-Order Filter Corner Frequency
In reality as the filter stage and therefore its roll-off slope increases, the low pass filters
-3dB corner frequency point and therefore its pass band frequency changes from its original calculated value above by an
amount determined by the following equation.
2nd-Order Low Pass Filter -3dB Frequency
where ƒc is the calculated cut-off frequency, n
is the filter order and ƒ-3dB is the new -3dB pass band frequency as a result
in the increase of the filters order.
Then the frequency response (bode plot) for a second-order low pass filter assuming the same -3dB
cut-off point would look like:
Frequency Response of a 2nd-order Low Pass Filter
In practice, cascading passive filters together to produce larger-order filters is difficult
to implement accurately as the dynamic impedance of each filter order affects its neighbouring network. However, to
reduce the loading effect we can make the impedance of each following stage 10x the previous stage, so
R2 = 10 x R1 and C2 = 1/10th C1. Second-order and above filter
networks are generally used in the feedback circuits of op-amps, making what are commonly known as
Active Filters or as a phase-shift
network in RC Oscillator
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.
Until now we have been interested in the frequency response of a low pass filter and 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 giving the filter a response
in the time domain.
The 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 a sinusoidal signal that changes smoothly over time, the circuit behaves as a simple 1st order
low pass filter as we have seen above. But what if we were to change the input signal to that of a "square wave" shaped
ON/OFF type signal that has an almost vertical step input, what would happen to our filter circuit now. The output
response of the circuit would change dramatically and produce another type of circuit known commonly as an
The RC Integrator
The Integrator is basically a low pass filter circuit operating in the time domain
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.