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ESKISEHIR TECHNICAL UNIVERSITY
FACULTY OF ENGINEERING
DEPARTMENT OF ELECTRICAL AND ELECTRONICS
ENGINEERING
EEM 206 ELECTRICAL CIRCUITS LABORATORY EXPERIMENT#5
PaSSiVe FiLterS
Name Surname:
Group:
A device that selects a interval of frequencies from an input signal, whose amplitudes
and phase can be modified. There are two filter types:
a) Analogical Filters:
i) Passive Filters: They use only passive elements like ๐ , ๐ฟ and ๐ถ. The output
signal has a smaller amplitude than the input signal and have a gain of less
than one.
ii) Active Filters: They include active elements like operational amplifiers,
transistors, FETs e.t.c. to strengthen the signal.
b) Digital Filters:
i) Discrete time signal is used.
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 or equal to the input. In the field of electronics, there
are many practical applications for these filters such as:
1. Radio communications: Filters enable radio receivers to only "see" the desired signal
while rejecting all other signals (assuming that the other signals have different
frequency content).
2. DC power supplies: Filters are used to eliminate undesired high frequencies (i.e., noise)
that are present on AC input lines. Additionally, filters are used on a power supply's
output to reduce ripple.
3. Audio electronics: A crossover network is a network of filters used to channel low-
frequency audio to woofers, mid-range frequencies to midrange speakers, and high-
frequency sounds to tweeters.
4. Analog-to-digital conversion: Filters are placed in front of an ADC input to minimize
aliasing.
However they can handle larger current or voltage levels than their active alternatives.
Assume that for any type of RLC circuit below, the initials conditions ๐๐ฟ(0) = 0, ๐๐ถ(0) = 0 are
zero.
The transfer function of the circuit can be defined as the ratio of the output signal to input signal
in Laplace domain as,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ ),
where ๐๐๐ข๐ก(๐ ) = โ{๐ฃ๐๐ข๐ก(๐ก)}, ๐๐๐(๐ ) = โ{๐ฃ๐๐(๐ก)}, and โ{ . } shows the Laplace transform.
Replacing ๐ = ๐๐, all notation can be transferred in to the Fourier domain.
The passive filters can be categorized as follows:
1. LOW-PASS Filter
Any input signal ๐ฃ๐๐(๐ก) in time-domain is the combination of sinosoids of different
frequencies. The-low pass filter will select only the low-frequencies by passing them
with a high magnitude (ideally one), while extingshing the high frequencies with almost
zero magnitude as shown below.
๐ฃ๐๐ข๐ก(๐ก) ๐ ๐ฟ๐ถ ๐ถ๐๐๐๐ข๐๐ก
(Filter) ๐ฃ๐๐(๐ก)
Figure 1: ๐ฃ๐๐(๐ก) and ๐ฃ๐๐ข๐ก(๐ก) signal propertes and the filter |H(ฯ)|.
However, the ideal response of the low-pass filter is as in Figure 2. The Ideal filters can not
Figure 2: Ideal low-pass filter characteristics.
BP: Pass band, BA: Attenuation band, ๐๐: Cut-off frequency.
be implemented in practice, since the practical filters can not have a constant response in the
pass band and canโt be entirely zero in the attenuation band. Still, they should approach the
ideal-filter chatacteristics by using different methods. The higher the order of the practical filter
the close is his behavior as an ideal filter.
First-order low-pass filter
It can be implemented by using two different combinations.
Low-pass RC Low-pass RL
The general transfer function representation for these two circuits is,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
๐๐
๐ + ๐๐.
The gain of the filter is generally expressed in Decibels and can be calculated by using
the formula 20 log (|๐๐๐ข๐ก
๐๐๐ | ). At cut-off (corner) frequency ๐๐, the output signal is 70.7%
of the input signal, or equivalently
๐ฃ๐๐(๐ก)
๐ฃ๐๐ข๐ก(๐ก)
๐๐๐(๐ ) ๐๐๐ข๐ก(๐ ) ๐๐๐ข๐ก(๐ )
๐๐ =๐๐
2๐ defined in Hertz causes an attenuation of 3 dB at the signal as shown in
Figure 3. After the cut-off frequency point the response of the circuit decreases to zero
at a slope of -20dB/ decade โroll-offโ. All the
Figure 3: Frequency response of a low-pass filter.
frequencies below this cut-off are unaltered with little or no attenuation and are said to be in
the filters โpass band zoneโ. This pass band zone is the Bandwidth of the filter. Any signal
frequencies above this point cut-off point are generally said to be in the filters โstop band zoneโ
and will be greatly attenuated. Note that the ideal filter marked with the red solid line is a
perfect rectangle with no decrease at that frequency. By carefully selecting the correct resistor-
capacitor or resistor-inductor combination, we can create an RC/RL circuit that allows a certain
bandwidth of a length, for the RC circuit as ๐๐ =1
๐ ๐ถ, while for an RL circuit to be equal to
๐๐ =๐
๐ฟ rad/s.
Lastly, a low pass filter can be used as an integrator for Wave Shaping and Wave
Generating circuits because of easy conversion of one type of electrical signal in to another
form as in the diagram below. The triangular wave is generated due to the capacitors action or
simply charging and discharging pattern of the capacitor.
Second-order low-pass filter
However sometimes a single stage of an RC/RL circuits may not enough to remove all
unwanted requencies then second order filters are used as shown in Figure 4a-4b below.
Ideal filter
Low pass RLC circuit
Figure 4a: Second- order RC low-pass filter. Figure 4b: Second- order RLC low-pass filter.
The second order low pass RC filter can be constructed simply by adding one more stage to
the first-order low-pass filter as in Figure 4a or by cascading the RL and RC filters as in as in
Figure 4b. The set up may change, but since the output is again passed through the capacitor
in shunt as in Figure 4b, the remaining AC component if present in the signal will be grounded,
and will allow pure DC at the output. However, the second-order low-pass does basically the
same function as its first-order counterpart, but has much better response, since it has twice
as much slope of 40 db/Decade as shown in Figure 5. So low frequencies can get in, while
the high ones are filtered twice effectively.
Figure 5: First and Second-order low-pass filter.
Specifically, for the RLC circuit in Figure 4b, the transfer function is,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
1๐ฟ๐ถ
๐ 2 +1
๐ ๐ถ๐ +
1๐ฟ๐ถ
,
and for the RC circuit in Fig 4a is,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
1R1C1R2C2
๐ 2 +(R1C1 + R2C2)
R1C1R2C2๐ +
1R1C1R2C2
.
More general representation for the second-order transfer function can be deduced as follows:
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
๐๐2
๐ 2 +๐๐๐
๐ + ๐๐2
๐๐๐ข๐ก(๐ ) ๐๐๐(๐ ) ๐๐๐(๐ ) ๐๐๐ข๐ก(๐ )
๐๐
(1๐ ๐ก ๐๐๐๐๐ ๐๐๐๐ก๐๐)
๐๐(2nd) ๐๐
Here ๐๐ = 2๐๐๐ is the characteristic frequency of the filter where the magnitude is 0 dB, and
๐๐ =1
โ๐ฟ๐ถ or ๐๐ =
1
โR1C1R2C2. ๐ is the quality factor at the second-order filter and is equal to
๐ = ๐ ๐๐๐ถ =๐
๐๐๐ฟ= ๐ โ
๐ถ
๐ฟ. Remember that the bandwith is equal to ๐ต๐ = ๐๐/๐. The filter
circuit will maintain itโs low pass characteristics only if the transfer function have two real
poles. For this ๐ must be smaller to 1/2 in order to have two real roots. If ๐ = 1/2 a
sharper separation between the pass and the stop bands will be provided.
2. HIGH-PASS filter
High Pass filter allows the frequencies which are higher than the cut off frequency โ๐๐โ
and blocks the lower frequency signals as shown in Figure 6. The value of the cut off frequency
depends on the component values chosen for the circuit design.
Figure 6: ๐ฃ๐๐(๐ก) and ๐ฃ๐๐ข๐ก(๐ก) signal propertes and the filter |H(ฯ)|. BP: Pass band, BA: Attenuation band.
They are commonly used to remove the unwanted sounds near to the lower end of
the audible range or are widely preferred in image processing for sharpening the details, by
exaggerating every tiny part of details in an image.
First-order high-pass filter
The high pass filters can be constructed by interchanging the low pass circuitโs
elements as in the figure below. Due to this interchange of components in the circuit, the
High-pass RC High-pass RL
๐ฃ๐๐(๐ก)
๐ฃ๐๐ข๐ก(๐ก)
๐๐๐(๐ ) ๐๐๐(๐ ) ๐๐๐ข๐ก(๐ ) ๐๐๐ข๐ก(๐ )
ideal high-pass charac.
responses delivered by the capacitor or inductor change and these changes are exactly
opposite to the response of the low pass filter. The capacitor at the low frequencies acts like
an open circuit and at higher frequencies (higher than the cut off frequency) capacitor acts like
a short circuit. The capacitor will block the lower frequencies entering into the capacitor due to
the capacitive reactance of the capacitor. For the RL circuit, the inductor at low frequencies
acts like a short circuit, while at high frequencies act like an open circuit, letting the output
voltage to occur only at high frequencies (above cut off frequency).
The general transfer function representation for these two circuits is,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
๐
๐ + ๐๐.
where ๐๐ =1
๐ ๐ถ is for the RC circuit, and ๐๐ =
๐
๐ฟ rad/s for an RL circuit.
The frequency response for this filter is as given below. At low frequencies the output voltage
Figure 7: Frequency response of a high-pass filter.
is very small, resulting in negative dB gain. The output increases with +20 dB/decade until the
cut off frequency ๐๐ =๐๐
2๐. Above this frequency the input signal directly passes to the output
and at very high frequencies the output voltage is same as that of the input voltage that is
๐๐๐ข๐ก = ๐๐๐, resulting in 20 log (|1|)=0 dB gain.
For normal sinusoidal wave inputs the filter output is just like the first-order high-pass
filter. But when a different type of signals are applied rather than the sine waves, for example
square waves then the circuit behaves like a Differentiator circuit as below.
Second-order low-pass filter
By cascading two first-order high pass filters or an RLC circuit as shown below creates
a second order high-pass filter with a steeper final output which has a slope of +40dB/Decade.
Figure 8a: Second- order RC high-pass filter. Figure 8b: Second-order RLC high-pass filter.
Specifically, for the RC filter in Figure 8a, the gain function is
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
1
๐ 2 +(R1C1 + R2C2)
R1C1R2C2๐ +
1R1C1R2C2
,
While for RLC circuit in Figure 8b, the transfer function is,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
๐ 2
๐ 2 +1
๐ ๐ถ๐ +
1๐ฟ๐ถ
.
Second-order high-pass filters are not only restricted with RC or RLC circuits.
Cascading two RL circuits can serve for the same purpose as well. By considering the charging
or discharging properties of the capacitors and their corresponding time constants, or by
taking into account the inducing properties of the inductors, second order or higher-order
circuits with preknown sharpness characteristics can be designed for special purposes. Thus,
a second-order high-pass filter can generally be represented with
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
๐ 2
๐ 2 +๐๐๐
๐ + ๐๐2
by using the same equations for ๐๐, ๐ the derived in the the low-pass filter section above.
๐๐๐ข๐ก(๐ ) ๐๐๐ข๐ก(๐ ) ๐๐๐(๐ ) ๐๐๐(๐ )
3. BAND-PASS Filter
A Filter circuit which allows a set of frequencies that are between two specified
values can be termed as a Band pass filter. Frequency selective filters pass only certain
band of frequencies, commonly termed as Bandwith.
Figure 9: ๐ฃ๐๐(๐ก) and ๐ฃ๐๐ข๐ก(๐ก) signal propertes and the filter |H(ฯ)|. BP: Pass band, BA: Attenuation band.
Band pass filter is obtained by cascading passive low pass and passive high pass filters. This
arrangement will provide a selective filter which passes only certain frequencies.
This creates a second order filter because the circuit will have two reactive components. Many
different circuit element combinations are possible as long as the properties of low pass and
high pass combinations give a band pass filter. For example, as shown below by arranging
one set of RC elements in series and another set of RC elements in parallel the circuit behaves
like a band pass filter; or a circuit designed in a parallel way by using inductor, capacitor and
Figure 10a: Second-order RC band-pass filter. Figure 10b: Second-order RLC band-pass filter.
resistors will again keep the passing band properties. A band pass filter with inductors can be
designed as well, but we know that due to high reactance of the capacitors the band pass filter
design with RC elements is more advantage than RL circuits.
๐ฃ๐๐(๐ก)
๐ฃ๐๐ข๐ก(๐ก)
ideal high-pass charac.
The frequency responses of the band pass filter in Figure 10a-10b can be written as
H(s) = ๐
๐ ๐ 1๐ถ1
๐ 2 +(R1C1 + R2C2)
R1C1R2C2๐ +
1R1C1R2C2
, H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
1๐ ๐ถ
๐
๐ 2 +1
๐ ๐ถ๐ +
1๐ฟ๐ถ
.
However, the widely used representation is,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
๐๐๐
๐
๐ 2 +๐๐๐
๐ + ๐๐2
with ๐ and ๐๐ defined as in the low pass section. Note that, the quality factor of the filter will
depend linearly upon the resistor value ๐ .
The gain of the circuit is calculated by taking 20 log |๐๐๐ข๐ก
๐๐๐| and can be plotted as in Figure 11.
Figure 11: Frequency response of a band-pass filter.
The center frequency is the resonant frequency that was covered in the previous
labs and can easily be calculated by taking the Geometric mean of lower and upper corner
frequencies ๐๐ฟ and ๐๐ป, respectively,
๐๐๐๐๐ก๐๐ =1
2๐โ๐ฟ๐ถ= โ๐๐ฟ๐๐ป .
The transfer function has an attenuation at low frequencies with the output increasing at a
slope of +20 dB/decade until the frequency reaches to lower cut off frequency ๐๐ฟ. After ๐๐ฟ the
output will increase to attain the maximum gain at resonant frequency and this gain is constant
until the higher cut off frequency ๐๐ป is reached. After ๐๐ป the output will decreas at a slope of
-20 dB/decade. Recall that the bandwidth ๐ต๐ is the difference
๐ต๐ = ๐๐ปโ๐๐ฟ ,
and for a pass band filter to function correctly, the cut-off frequency of the low pass filter must
be higher than the cut-off frequency for the high pass filter.The tabularized summary of the
properties of RLC circuits is given in Table 1.
Table 1: RLC properties at resonance.
Property SERIES RLC PARALLEL RLC
Resonant freq.
๐๐๐๐๐ก๐๐ =1
2๐โ๐ฟ๐ถ ๐๐๐๐๐ก๐๐ =
1
2๐โ๐ฟ๐ถ
Voltage across ๐ Maximum at ๐๐๐๐๐ก๐๐ Constant= ๐๐๐ข๐ก
Current through ๐ Constant=๐๐๐ข๐ก
๐ Maximum at ๐๐๐๐๐ก๐๐
๐ ๐ =
2๐๐๐๐๐๐ก๐๐๐ฟ
๐ ๐ =
๐
2๐๐๐๐๐๐ก๐๐๐ฟ
Bandwidth ๐ต๐ =
๐๐๐๐๐ก๐๐
๐ ๐ต๐ =
๐๐๐๐๐ก๐๐
๐
Impedance below ๐๐๐๐๐ก๐๐ Capacitive Inductive
Impedance above ๐๐๐๐๐ก๐๐ Inductive Capacitive
Effect of changing ๐ ๐ ๐ต๐ ๐ ๐ต๐
Effect of changing ๐ณ/๐ช ๐ณ/๐ช ๐ต๐ ๐ณ/๐ช ๐ต๐
If more filters are cascaded together the resulting circuit will be known as an โnth-orderโ filter
where the โ๐โ stands for the number of individual reactive components within the filter circuit.
For example, filters can be a 2nd-order, 4th-order, 10th-order, etc. The higher the filters order
the steeper will be the slope at ๐ times -20dB/decade.
4. BAND-STOP Filter
A filter circuit shown in Figure 12, which blocks or attenuates a set of frequencies that are
between two specified values can be termed as a Band Stop filter. This filter rejects a band of
frequencies and hence can also be called as Band Reject or Notch filter. The characteristics
of the band stop are exactly opposite of the band-pass filter, formed by the parallel connection
of the low pass and high pass filters instead of the cascading connection represented with the
block diagram below.
Figure 12: ๐ฃ๐๐(๐ก) and ๐ฃ๐๐ข๐ก(๐ก) signal propertes and the filter |H(ฯ)|. BP: Pass band, BA: Attenuation band.
A simple band stop filter circuit with passive components is shown below:
In this circuit at very low and high frequencies, the ๐ฟ๐ถ branch acts like a short circuit because
inductor and capacitor are connected in parallel. At mid frequencies the circuit acts like an
open circuit, not allowing them to pass through the circuit. Like the band pass filter, the band
stop filter is a second-order) filter having the transfer function,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
๐ 2 +1
๐ฟ๐ถ
๐ 2 +1
๐ ๐ถ๐ +
1๐ฟ๐ถ
.
and two -3 dB cut-off frequencies shown as ๐๐ฟ, ๐๐ป in the Figure 13. Since the band pass filter
Figure 13: Frequency response of a band-stop filter.
ideal high-pass charac.
๐ฃ๐๐(๐ก)
๐ฃ๐๐ข๐ก(๐ก)
๐๐๐(๐ ) ๐๐๐ข๐ก(๐ )
is the inverted form of the standard band-pass filter ๐๐ฟ, ๐๐ป , ๐๐ถ are the same as described in
the band pass filter section with ๐๐ถ corresponding to ๐๐๐๐๐ก๐๐. Thus, we can use the same
formulas to calculate the bandwidth ๐ต๐and the quality factor ๐ in the general representation
of the transfer function,
H(s) =๐๐๐ข๐ก(๐ )
๐๐๐(๐ )= ๐
๐ 2 + ๐๐2
๐ 2 +๐๐๐
๐ + ๐๐2
.
If this stop band is very narrow and highly attenuated over a few hertz, then the band
stop filter is more commonly referred to as a notch filter.
Notch Filter
The above circuit shows the Twin โTโ network and is usually used to eliminate a single or a or
very small band of frequencies. The maximum elimination is occurs at the center frequency
๐๐ถ =1
4๐๐ ๐ถ. If the quality factor ๐ is high a very narrow and very deep stop band around the
center frequency of the notch response will occur.
LABORATORY WORK :
1. Suppose that you have to design a filter system for a traffic operating system. The frequency range from up to 3.5 kHz will control the Yellow lights, the frequency range 3.5 kHz-10.5 kHz will control the Green lights, and the frequency components above 10.5 kHz will control the Red lights.
i. Design your filter system by using the ๐ , ๐ถ, ๐ฟ components you have at hand. If the filtering requirements donโt match exactly with your component values you can adjust the requirements. But donโt go for dramatic changes like changing 2 times the cut off frequencies, and try to buy as little (or not ) new components as possible.( If you are fussy about accuracy you can parallel (or cascade) your capacitors/resistors/inductors to get more exact values.)
ii. Obtain the transfer functions for the chosen components in part (i). iii. Before making the Proteus(or Pspice ) simulation you can check the sharpness
of your filters and your cut off frequencies by using a very simle MATLAB code as below;
If your transfer function is of the form ๐ป(๐ ) =๐1 ๐ 3+๐2 ๐ 2+๐3 ๐ +๐4
๐1 ๐ 3+๐2 ๐ 2+๐3 ๐ +๐4. Some
coefficients can be zero and the order of your filter may change. Write,
clear;
num=[ ๐1 ๐2 ๐3 ๐4]; % in descending sort order of the coefficients of the numerator
den=[๐1 ๐2 ๐3 ๐4]; % in descending sort order of the coefficients of the denominator
H=tf(num,den) % the transfer function with numerator and denominator defined
bode(H); % frequency response of the transfer function in logarithmic scale
grid; % gridding for the frequencies
title(โThis is a ...โ);
iv. Check the sharpness of your filters around the pass bands. v. Make your Proteus/Pspice simulations vi. Test your circuit by using the Led lights (the cheapest ones). vii. Increase the order of your filters to have better sharpness and good isolation.
You can modify your MATLAB code for the new transfer function as;
Let the new sharper transfer function be ๐บ(๐ ) = ๐พ(๐ )๐ป(๐ ) with ๐พ(๐ ) =๐1 ๐ 3+๐2 ๐ 2+๐3 ๐ +๐4
๐1 ๐ 3+๐2 ๐ 2+๐3 ๐ +๐4
clear;
num=[ ๐1 ๐2 ๐3 ๐4];
den=[๐1 ๐2 ๐3 ๐4];
H=tf(num,den);
numk=[ ๐1 ๐2 ๐3 ๐4];
denk=[๐1 ๐2 ๐3 ๐4];
K=tf(numk,denk); G=K*H
bode(G); grid on; title(โThis is a ...โ);