Design of Passive and Active Multistage Filters

ECE 303L: Signals and Systems Laboratory
Design of Passive and Active Multistage Filters
Purpose:
 Analysis and design of multistage passive filters.
 Analysis and design of multistage active filters.
 Filter specification and realization – a preliminary design approach.
Background: The design of LTIC systems, broadly speaking, focuses on either transient or time-domain
performance and/or sinusoidal steady-state or frequency-domain performance. Our previous
experimental investigation of the synthesis of PID compensators is one example where the driving issue
from a system design perspective is the necessity for pre-specified transient performance for a given
system. Design approaches seek to exploit the effect of integral and derivative action on the output
error of the system to try to guarantee specific transient and steady-state performance metrics or
specifications. This is one of the primary approaches to system design that you will encounter later in
the course ECE 415: Control Systems.
In contrast, there are many applications where frequency-domain performance is more important and
this laboratory exercise focuses on the design and implementation of multistage, passive and active
filters. A filter is simply a circuit that selectively amplifies or attenuates input signals differently based
on their frequency content. A typical example is one in which a circuit is designed extract information
content within a noisy input signal, generally within a specific range of frequencies, and to reject or
attenuate the noise on the input signal. The design of LTIC filters focuses on realizing a predefined inputoutput model or transfer function that exhibits specific gain and phase characteristics in specified
frequency ranges. Although we can pose this design problem in a very general framework (something
you will likely encounter in ECE 445: Signal Processing), many filter design problems can be approached
using common, well-understood circuit models.
Given a candidate circuit, we can develop a differential equation description of the circuit. Using various
approaches, the input-output transfer function can be developed in terms of the circuit parameters: R, L
and C typically. These parameters then determine the polynomial coefficients of the numerator and
denominator of the transfer function and hence determine circuit the poles (characteristic roots) and
zeros. Thus, for a candidate circuit, we define:
 
Y s P s K s z s z s z X s Q s s p s p s p      H s            0 1 0 1  ….. …. NM 

H j H s H j H j           s j       where the system zeros are    z z z 0 1 , ….. M ; the system poles are    p p p 0 1 , ….. N ; and the system
frequency response is defined in terms of two characteristics, the system gain H j   and the system
phase H j  . The poles and zeros are often shown graphically as a pole-zero map or plot, and the
system gain and phase are plotted versus frequency and are collectively known as the system Bode plot.
Within this framework the filter design problem involves four sequential tasks. (i) Design specification
based on an technical analysis of the problem being addressed; (ii) selection and/or design of candidate
circuit solutions to the design problem; (iii) determination of the circuit parameters to achieve the
design filter specifications; and, (iv) computational and experimental verification of the design solution.
Ideally, the simplest design solution is sought and simple first- and second-order passive filters provide a
range of circuit models that can be employed with some utility. Such design solutions, whilst often
suboptimal in terms of performance, can provide low component count (and low cost) solutions. As the
design specifications become more rigorous, multistage passive or active filters are often required to
synthesize more complex transfer functions. Multistage filters are composed of several filters or filter
stages, cascaded in series or connected in parallel. The design of these circuits using candidate circuit
models becomes problematic as the complexity of the desired transfer function increases since it
becomes difficult to relate the selection of individual circuit components (R, L & C) to pre-specified poles
and zeros. In contrast, we can realize desired transfer functions using active filters (similar to the
development of the PID controller earlier) that can be constructed from standard “building blocks”.
Passive Filter Design: Three-stage, passive low-pass RC filter.
Analytical Preliminaries: As mentioned earlier, a first approach to filter design often employs well
understood, candidate circuits to solve design problems. If we require a low-pass filter to resolve an LTIC
design problem, then the simple RC filter can sometimes be used. This candidate circuit is shown in
Figure 1 below and the corresponding transfer function is readily developed as Equation (3). Here the
values of R1 and C1 are simply “placeholders” and need to be selected as part of the design process to
meet pre-determined design specifications.
   
 
   
 
1 1 1
1 1 1 1
1
1
1 1
1 1 1 1
1
1 1/
1 1 1/
1 1/
1 1/
sC R C
H s
sR C s R C
R
sC
R C
H j
j R C j R C

 
 
 
 
  
   
  
 
 
 

The circuit frequency response exhibits the following features.

 
0 0
1 1
1
lim lim 1 0
1
H j
  j R C

  
 
     
    
0 0
1 1
1
lim lim 1 0
1
H j
  j R C

  
 
     
    
 
1 1
1 1
1 1
1 1 1
12 2 1
0
1
45
c 2
c
H j when
j R C
R C or
R C
and H j
 


  


 

   
   
R1
1.0kΩ
C1
1µF
x(t) y(t)
Hence, the filter is said to pass input frequency components below the cut-off frequency to the output
and attenuates frequencies above the cut-off frequency. The cut-off frequency is determined by
selection of R1 and C1, which in turn, determines the location of the pole or characteristic root of the
circuit. The cut-off frequency defines the location of the “transition band” where the filter gain changes
from the pass band value of 1 to the stop band value of 0.
One of the problems with passive filters is that the transition band is often too wide for many
applications. To address this problem, passive filters are cascaded as shown in Figure 2 below. Again, the
component values shown are simply “placeholders” and specific values of all R and C components are
selected to meet the design specifications.
Figure 2: Three-stage, passive RC low-pass filter.
 Using the MATLAB Symbolic toolbox, determine the input-output transfer function the circuit
shown in Figure 3. Use either a node analysis or a mesh analysis approach, whichever your
group favors.
Your analysis reveals many things however the more apparent observations are as follows: the
polynomial coefficients of the transfer function are functions of multiple R and C values; the cut-off
frequency is more difficult to determine from a simple analysis of H j  , and the circuit poles and
zeros are no longer simple functions of specific R and C values. This makes the circuit more difficult to
use as a design prototype.
 Repeat your analysis assuming R1=R2=R3=R, and C1=C2=C3=C. Calculate the low- and highfrequency characteristics. Is there a simple relationship between R and C, and the filter cut-off
frequency? Is the circuit any easier to use as a design prototype? Explain your answers.
To address the challenges of selecting the R and C values, a common approximation is employed. The
problem with this circuit is the each filter state “bleeds” current from the proceeding stage (essentially
because of the parallel loading effect), shifting the filter cut-off frequency. To minimize this effect, the
impedance of each filter stage can be increased at each stage.
Two approaches to the design of the circuit are apparent. (i) Select R1 and C1 to meet a pre-specifiedc ,
and then simply design three identical filter stages. This approach ignores the parallel loading effect.
And (ii), select R1 and C1 to meet a pre-specifiedc , and then select R2C2 and R3C3 to meet the same prespecifiedc , but increasing the effective impedance of each stage by selecting higher resistance values.
To minimize this loading effect, we increase the impedance of each subsequent filter stage by
approximately 10 times to improve the overall performance. Thus, an approximate filter transfer
function is synthesized as,
R1
1.0kΩ
C1
x(t) 1µF
R2
1.0kΩ
R3
1.0kΩ
C2
1µF
C3
1µF
   
   
   
1
1 1
2
2 2 1 1
3
3 3 2 2 1 1
1
1
1 1
1 1
1 1 1
1 1 1
Y j X j
j R C
Y j X j
j R C j R C
Y j X j
j R C j R C j R C
 

 
 
 
  
 
  
  
  
   
    
   
    
       

Circuit Design and Evaluation:
The performance of low-pass filters can be specified in terms of an ideal “brick wall” filter characteristic
such as that shown below in Figure 3. The common specifications are the low-frequency gain, typically
specified to be equal to 1, and the cut-off frequency, f or c c  .

 For a simple, first-order filter, select the values of the R1 and C1 appropriately for a cut-off
frequency of 2 kHz. Model the circuit in Multisim and determine (i) the circuit poles and zeros,
and (ii) plot the circuit frequency response. Validate your design solution by confirming the lowfrequency and high-frequency gain and phase, the filter cut-off frequency, and the filter gain
and phase at the cut-off frequency.
 Using the first multi-stage filter design approach above, where all R values and C values are
equal, model the three-stage filter in Multisim and (i) determine the circuit poles and zeros, and
(ii) plot the circuit frequency response. Validate your design solution by confirming the lowfrequency and high-frequency gain and phase. Determine the filter cut-off frequency, and
compare the realized cut-off frequency with the design value. Note the gain and phase of the
filter at both frequencies.
 Using the second design approach above, select the values of R2 and C2, and R3 and C3, for the
same cut-off frequency. An approximate impedance increase of 10 is recommended for each
successive filter stage. Model the three-stage filter in Multisim and (i) determine the circuit
poles and zeros, and (ii) plot the circuit frequency response. Validate your design solution by
confirming the low-frequency and high-frequency gain and phase. Determine the filter cut-off
frequency, and compare the realized cut-off frequency with the design value. Note the gain and
phase of the filter at both frequencies.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
-0.2
0
0.2
0.4
0.6
0.8
1
f (Hz)
H(w) V/V
Ideal Low-Pass Filter Characteristic
Experimental Evaluation:
 Construct both three-stage filters designed above using the closest available component values.
Use single components for each R and C in the circuit to keep the total component count as low
as possible. Measure the filter gain (output amplitude divided by the input amplitude) and
phase characteristic of each filter over a range of frequencies (100 Hz to 10 kHz is probably
sufficient) and plot the transfer characteristics of each of the filters (Bode plot). The more
samples you take the better the plot will be. Typically you will need more data points where the
gain and phase characteristics are changing rapidly (i.e. in the vicinity of c .)
Computational Evaluation:
 Model the filter circuit shown in Figure 2 in MATLAB. For each of the practical filter circuits
constructed, determine (i) the pole-zero map of the filter and (ii) the Bode plot of the filter.
 Model the approximate filter transfer function (Equation 7) in MATLAB. For each of the practical
filter circuits constructed, determine (i) the pole-zero map of the filter and (ii) the Bode plot of
the filter.
[Note: The generation of the filter frequency response using MATLAB may appear redundant given the
Multisim results generated earlier. However, the latter results are coupled fundamentally to an
analytical approach to filter design and have utility beyond the scope of this study.] Active Filter Design: Three stage, active low-pass Butterworth1 filter.
The particular characteristics of operational amplifiers to synthesize more complex transfer functions,
such as the PID compensator investigated earlier, can also be exploited to synthesize filter transfer
functions or arbitrary order. The passive filter investigated above has a transfer function that is thirdorder, with the circuit poles effectively being set individually by selection of the R and C values. Similarly,
the active circuit shown below in figure 4 has a second-order transfer function. Cascading a simple RC
filter to the output provides an overall transfer function that is third-order. In contrast to the passive
filter case, the passive RC filter added to the output of the active circuit does not change the circuit
transfer function since the operational amplifier output voltage is load independent providing the
output current rating is not exceeded.
One specific class of filters, known as Butterworth Filters, specifies the poles of filter in a particular way.
Butterworth filters are known as maximally flat filters, meaning that the 2N-1 derivatives of H j  
with respect to  are zero at   0 for an n-th order filter. For a third-order, low-pass Butterworth
filter with a cut-off frequency of c , the three filter poles are designed to lie on a semi-circle in the lefthalf complex s-place of radius c . This means that one pole is located on the negative real axis and the
remaining pair of poles is selected as a complex conjugate pole-pair.
Analytical Preliminaries: Verify by appropriate analysis, the transfer function of the circuit shown in
Figure 4. Assuming the circuit poles are complex – p j 1,2 1 1      , develop design equations for the
1
The Butterworth type filter was first described by the British engineer Stephen Butterworth in his paper “On the Theory of Filter Amplifiers”,
Wireless Engineer (also called Experimental Wireless and the Wireless Engineer), vol. 7, 1930, pp. 536-541.
[http://www.rfcafe.com/references/electrical/butter-poles.htm]
filter by equating the coefficients of the denominator of the filter transfer function with the following
characteristic equation, Q s s j s j s s                 1 1 2 1   2 2 c c .
   
   
   
     
1 2
2 2
1 2 2
2
1 2
2 2
1 1 2
Assuming ,
( )
1
2 1
1/
2 / 1/
out in
out in
out in
R R R
V s H s V s
V s V s
R C C s RC s
R C C
V s V s
s RC s R C C
 
 
 

 

Cascading a simple passive RC low-pass filter at the output of the operational amplifier yields an overall
transfer function that is 3-rd order and expressed as,
   
    
 
    
2
1 2 3 3
2 2
1 1 2 3 3
1/ 1/
out in 2 / 1/ 1/
R C C R C
Y s V s
s RC s R C C s R C

  

where R3 and C3 are the parameters of the passive filter.
For a 3-rd order low-pass Butterworth filter, the circuit poles must be selected to satisfy to yield the
following characteristic equation:
Q s s s s           c c c  2 2

Circuit Design and Evaluation:
 Assuming a design specification for a cut-off frequency of 2 kHz, select the values of the R3 and
C3 to ensure that the real pole is appropriately located at s  c . Next, using the design
equations developed above, select component values for R, C1 and C2 to locate the complexconjugate pole-pair at the desired locations. [A good initial strategy is to select R-values of 1 k
to limit the OpAmp currents to reasonable levels, and then determine the necessary
capacitances.] Model the circuit in Multisim and determine (i) the circuit poles and zeros, and (ii)
plot the circuit frequency response. Validate your design solution by confirming the lowfrequency and high-frequency gain and phase, the filter cut-off frequency, and the filter gain
and phase at the cut-off frequency.
Experimental Evaluation:
 Construct the third-order Butterworth filter designed above using the closest available
component values. Use single components for each R and C in the circuit to keep the total
component count as low as possible. If necessary, adjust the component design values to yield R
U1
741
2 3
4 7
6
1 5
R1
1.0kΩ
R2
1.0kΩ
C1
1µF
C2
Vin 1µF Vout
and C values as close as possible to the available components. Measure the filter gain (output
amplitude divided by the input amplitude) and phase characteristic of each filter over a range of
frequencies (100 Hz to 10 kHz is probably sufficient) and plot the transfer characteristics of each
of the filters (Bode plot). The more samples you take the better the plot will be. Typically you
will need more data points where the gain and phase characteristics are changing rapidly (i.e. in
the vicinity of c .)
Computational Evaluation:
 Model the Butterworth filter circuit in MATLAB. Determine (i) the pole-zero map of the filter
and (ii) the Bode plot of the filter.
Report: Write an extended abstract report for this experiment using the guidelines provided. Your
report should include sufficient analytical, computational and experimental detail to address the stated
purposes of this experiment. Tabulate the key metrics for each of the filter circuits developed and
measured above such as the low-frequency and high-frequency gains, the cut-off frequencies and the
filter poles and zeros. Include a comparative evaluation of the passive and active filter circuits.
As always, work as a group to prepare your experimental and/or computational results in final form.
Discuss the results together to clarify your own understanding and to critique the results in light of what
you know about linear, time-invariant systems. Make notes from these discussions. Ensure that you are
confident in your answers to all questions embedded within the experimental procedure regarding
measurements of the circuit behavior. Write your own report using notes prepared from your team
discussions along with supplementary information from the course text.