The Digital Modeling of a Simple
RC Low Pass Filter
Ralph J. Coppola (r_j_coppola<at>hotmail.com)
Introducing the Basic RC Low Pass
Filter
Filters play an important role in many kinds of analog electronic
circuits.
Figure 1 illustrates an example of the exaggerated
output signal from a noisy analog sensor.
Figure 1. Exaggerated raw signal.
It is generally desirable to smooth or filter
the clutter of the short-term noise so that the signal's long-term
trends may be more readily visualized. This is especially
true if we are using the signal to control a process, such
as temperature, where we may want the process to respond to
the trend and not the short term spikes.
When dealing with analog signals, this filtering
can be easily accomplished by using a simple resistor/capacitor
(RC) low pass filter, such as the one shown in Figure 2.
Figure 2. RC low pass filter.
It is not my intention to go too deeply into
the theory of RC Filters. It is sufficient to say that the
operation of an RC low pass filter is based on the time that
it takes for the capacitor to charge or discharge. Those who
may not be familiar with the subject are encouraged to refer
to a text on basic electronics or refer to the Web links for
“ RC Filters ” and/or RC Time Constants ”
that I have provided at the end of this article.
Figure 3 shows a RC Low Pass Filter's DC response
to a positive step input.
Note: 1 Time Constant (seconds) = R (ohms) x
C (farads)
Figure 3. RC low pass filter's response to a step input
signal.
As a rule of thumb, the output of the circuit
is considered to have reached a steady state level after 5
RC time constants.
Table 1 shows the percentage of the steady state
value at each time constant (TC) interval.
Time
|
%
of Max |
0 TC |
0 |
1 TC |
63.2% |
2 TC |
86.5% |
3 TC |
95.0% |
4 TC |
98.2% |
5 TC |
99.3 % |
Table1. Percentage of the steady
state.
The effect of the simple RC low pass filter
is governed by the value of the RC time constant. In other
words, the longer the time constant, the greater the filtering.
Filter Simulation
If the signals of interest are in a digital
format, such as the output from an analog/digital converter
or a computer text file, some method other than an RC filter
must be used.
The following equation provides a simple method
to use software to closely simulate the characteristics of
an RC low pass filter. The smoothing is accomplished by taking
a fraction of the new input and adding it to a percentage
of the previous filtered result.
S = (f * R) + (( 1 – f ) * P)
Where S is the smoothed or filtered value, f is the filter
factor, R is the raw data input, and P is the previous value
of S. 0 = maximum filtering (no output: S = 0 ), and 1 = minimum
filtering ( S = R ).
Note that this simulation equation uses only
simple arithmetic. This allows it to be implemented in computer
languages that do not support higher math functions.
The Code
The following code solves for the equation given
above. It is written in a generic form and should be easily
translated into your favorite language.
| # ***********************************
# * Example of Filter Program
*
#
*
# ***********************************
# S = (f*R) + ((1-f) * P)
S = 0 # Smoothed
or filtered value
f = 0.2 # Filter
factor
D = 0 # Raw data
input
P = 0 # Previous
Smoothed value
Start input D
# Get the raw (D)ata sample
S
= (f*D) + ((1-f) * P) # Compute the new (S)moothed
value
output
S # Output (S)moothed
value to the application
P
= S # Save the new
(S)moothed value as (P)revious
goto
Start # Loop for the next
sample
End
|
Filter Response
Table 2 illustrates that if a filter factor
of 0.2 is used, the simulation mimics an actual RC low pass
filter very closely.
RC
Low Pass Filter |
Simulation
(f=0.2) |
Time
|
%
of Max |
Samples
|
%
of Max |
0 RC |
0 |
0 |
0 |
1 RC |
63.2% |
4 |
59.0% |
2 RC |
86.5% |
8 |
83.2% |
3 RC |
95.0% |
12 |
93.1% |
4 RC |
98.2% |
16 |
97.2% |
5 RC |
99.3 % |
20 |
98.8% |
Table 2. Comparison of an RC low pass filter
to the simulation.
It should be noted that unlike the RC filter,
which is related to time, the simulation's output is related,
instead, to the number of samples.
Figure 4 shows the simulation's response to
a square pulse, while Figure 5 illustrates its response to
a cyclic or random input.
Figure 4. Simulation's response to
a square pulse.
Figure 5. Simulation's response to a random function.
Tables 2 and 3 show a Microsoft S Excel setup
that is similar to the program that was used to produce the
example shown in Fig. 9.
|
A
|
B
|
C
|
E
|
1
|
= RAND ()
|
0 |
0 |
0 |
2
|
= RAND ()
|
=0.2*A1 |
=0.8*E1 |
=B2 + C2 |
3
|
= RAND ()
|
=0.2*A2 |
=0.8*E2 |
=B3 + C3 |
4
|
= RAND ()
|
=0.2*A3 |
=0.8*E3 |
=B4 + C4 |
|
|
|
|
|
|
n
|
=
RAND () |
=0.2*A(n-1)
|
=0.8*E(n-1)
|
=Bn + Cn |
Table 3. Microsoft Excel programming
for Fig. 5.
|
Column
|
Description
|
|
A
|
Random
Input |
|
B
|
Use 20%
of Input |
|
C
|
Use 80%
of Old Value |
|
E
|
New Value
= B + C |
|
-
- - |
Net Old
Value = New Value |
Table 4. Values used in Fig.
5.
Conclusion
Besides its simplicity, the main advantages
of this model is that it can be used in real time, while the
data is being collected or at some later time from stored
data.
Similar techniques can be found in many industrial
process control applications, and it is hoped that you may
find it of some value in your applications.
References
|
