ESKISEHIR TECHNICAL UNIVERSITY FACULTY OF ENGINEERING

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 𝑅, 𝐿 and 𝐶. 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 𝑖𝐿(0) = 0, 𝑉𝐶(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) =𝑉𝑜𝑢𝑡(𝑠)

𝑉𝑖𝑛(𝑠),

where 𝑉𝑜𝑢𝑡(𝑠) = ℒ{𝑣𝑜𝑢𝑡(𝑡)}, 𝑉𝑖𝑛(𝑠) = ℒ{𝑣𝑖𝑛(𝑡)}, and ℒ{ . } shows the Laplace transform.

Replacing 𝑠 = 𝑗𝜔, 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 𝑣𝑖𝑛(𝑡) 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.

𝑣𝑜𝑢𝑡(𝑡) 𝑅𝐿𝐶 𝐶𝑖𝑟𝑐𝑢𝑖𝑡

(Filter) 𝑣𝑖𝑛(𝑡)

Figure 1: 𝑣𝑖𝑛(𝑡) and 𝑣𝑜𝑢𝑡(𝑡) signal propertes and the filter |H(ω)|.

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, 𝜔𝑐: Cut-off frequency.

be implemented in practice, since the practical filters can not have a constant response in the

pass band and can’t 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,


𝑉𝑖𝑛(𝑠)= 𝑘

𝜔𝑐

𝑠 + 𝜔𝑐.

The gain of the filter is generally expressed in Decibels and can be calculated by using

the formula 20 log (|𝑉𝑜𝑢𝑡

𝑉𝑖𝑛 | ). At cut-off (corner) frequency 𝜔𝑐, the output signal is 70.7%

of the input signal, or equivalently

𝑣𝑖𝑛(𝑡)

𝑣𝑜𝑢𝑡(𝑡)

𝑉𝑖𝑛(𝑠) 𝑉𝑜𝑢𝑡(𝑠) 𝑉𝑜𝑢𝑡(𝑠)

𝑓𝑐 =𝜔𝑐

2𝜋 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 “roll-off”. 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 ‘pass band zone’. 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 ‘stop band zone’

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 𝜔𝑐 =1

𝑅𝐶, while for an RL circuit to be equal to

𝜔𝑐 =𝑅

𝐿 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,



1𝐿𝐶

𝑠2 +1

𝑅𝐶𝑠 +

1𝐿𝐶

,

and for the RC circuit in Fig 4a is,



1R1C1R2C2

𝑠2 +(R1C1 + R2C2)

R1C1R2C2𝑠 +

1R1C1R2C2

.

More general representation for the second-order transfer function can be deduced as follows:



𝜔𝑜2

𝑠2 +𝜔𝑜𝑄

𝑠 + 𝜔𝑜2

𝑉𝑜𝑢𝑡(𝑠) 𝑉𝑖𝑛(𝑠) 𝑉𝑖𝑛(𝑠) 𝑉𝑜𝑢𝑡(𝑠)

𝑓𝑜

(1𝑠𝑡 𝑜𝑟𝑑𝑒𝑟 𝑓𝑖𝑙𝑡𝑒𝑟)

𝑓𝑜(2nd) 𝑓𝑐

Here 𝜔𝑜 = 2𝜋𝑓𝑜 is the characteristic frequency of the filter where the magnitude is 0 dB, and

𝜔𝑜 =1

√𝐿𝐶 or 𝜔𝑜 =

1

√R1C1R2C2. 𝑄 is the quality factor at the second-order filter and is equal to

𝑄 = 𝑅𝜔𝑜𝐶 =𝑅

𝜔𝑜𝐿= 𝑅√

𝐶

𝐿. Remember that the bandwith is equal to 𝐵𝑊 = 𝜔𝑜/𝑄. The filter

circuit will maintain it’s low pass characteristics only if the transfer function have two real

poles. For this 𝑄 must be smaller to 1/2 in order to have two real roots. If 𝑄 = 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 ‘𝜔𝑐’

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: 𝑣𝑖𝑛(𝑡) and 𝑣𝑜𝑢𝑡(𝑡) signal propertes and the filter |H(ω)|. 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 circuit’s

elements as in the figure below. Due to this interchange of components in the circuit, the

High-pass RC High-pass RL

𝑣𝑖𝑛(𝑡)


𝑉𝑖𝑛(𝑠) 𝑉𝑖𝑛(𝑠) 𝑉𝑜𝑢𝑡(𝑠) 𝑉𝑜𝑢𝑡(𝑠)

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,



𝑠

𝑠 + 𝜔𝑐.

where 𝜔𝑐 =1

𝑅𝐶 is for the RC circuit, and 𝜔𝑐 =

𝑅

𝐿 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 𝑓𝑐 =𝜔𝑐

2𝜋. 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

𝑉𝑜𝑢𝑡 = 𝑉𝑖𝑛, 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



1

𝑠2 +(R1C1 + R2C2)

R1C1R2C2𝑠 +

1R1C1R2C2

,

While for RLC circuit in Figure 8b, the transfer function is,



𝑠2

𝑠2 +1

𝑅𝐶𝑠 +

1𝐿𝐶

.

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



𝑠2

𝑠2 +𝜔𝑜𝑄

𝑠 + 𝜔𝑜2

by using the same equations for 𝜔𝑜, 𝑄 the derived in the the low-pass filter section above.

𝑉𝑜𝑢𝑡(𝑠) 𝑉𝑜𝑢𝑡(𝑠) 𝑉𝑖𝑛(𝑠) 𝑉𝑖𝑛(𝑠)

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.


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.

𝑣𝑖𝑛(𝑡)



The frequency responses of the band pass filter in Figure 10a-10b can be written as

H(s) = 𝑘

𝑠𝑅1𝐶1

𝑠2 +(R1C1 + R2C2)

R1C1R2C2𝑠 +

1R1C1R2C2

, H(s) =𝑉𝑜𝑢𝑡(𝑠)


1𝑅𝐶

𝑠

𝑠2 +1

𝑅𝐶𝑠 +

1𝐿𝐶

.

However, the widely used representation is,



𝜔𝑜𝑄

𝑠

𝑠2 +𝜔𝑜𝑄

𝑠 + 𝜔𝑜2

with 𝑄 and 𝜔𝑜 defined as in the low pass section. Note that, the quality factor of the filter will

depend linearly upon the resistor value 𝑅.

The gain of the circuit is calculated by taking 20 log |𝑉𝑜𝑢𝑡

𝑉𝑖𝑛| 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 𝑓𝐿 and 𝑓𝐻, respectively,

𝑓𝑐𝑒𝑛𝑡𝑒𝑟 =1

2𝜋√𝐿𝐶= √𝑓𝐿𝑓𝐻 .

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 𝑓𝐿. After 𝑓𝐿 the

output will increase to attain the maximum gain at resonant frequency and this gain is constant

until the higher cut off frequency 𝑓𝐻 is reached. After 𝑓𝐻 the output will decreas at a slope of

-20 dB/decade. Recall that the bandwidth 𝐵𝑊 is the difference

𝐵𝑊 = 𝑓𝐻−𝑓𝐿 ,

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.

𝑓𝑐𝑒𝑛𝑡𝑒𝑟 =1

2𝜋√𝐿𝐶 𝑓𝑐𝑒𝑛𝑡𝑒𝑟 =

1

2𝜋√𝐿𝐶

Voltage across 𝑅 Maximum at 𝑓𝑐𝑒𝑛𝑡𝑒𝑟 Constant= 𝑉𝑜𝑢𝑡

Current through 𝑅 Constant=𝑉𝑜𝑢𝑡

𝑅 Maximum at 𝑓𝑐𝑒𝑛𝑡𝑒𝑟

𝑄 𝑄 =

2𝜋𝑓𝑐𝑒𝑛𝑡𝑒𝑟𝐿

𝑅 𝑄 =

𝑅

2𝜋𝑓𝑐𝑒𝑛𝑡𝑒𝑟𝐿

Bandwidth 𝐵𝑊 =

𝑓𝑐𝑒𝑛𝑡𝑒𝑟

𝑄 𝐵𝑊 =

𝑓𝑐𝑒𝑛𝑡𝑒𝑟

𝑄

Impedance below 𝑓𝑐𝑒𝑛𝑡𝑒𝑟 Capacitive Inductive

Impedance above 𝑓𝑐𝑒𝑛𝑡𝑒𝑟 Inductive Capacitive

Effect of changing 𝑅 𝑅 𝐵𝑊 𝑅 𝐵𝑊

Effect of changing 𝑳/𝑪 𝑳/𝑪 𝐵𝑊 𝑳/𝑪 𝐵𝑊

If more filters are cascaded together the resulting circuit will be known as an “nth-order” filter

where the “𝑛” 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 𝑛 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.


A simple band stop filter circuit with passive components is shown below:

In this circuit at very low and high frequencies, the 𝐿𝐶 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,



𝑠2 +1

𝐿𝐶

𝑠2 +1

𝑅𝐶𝑠 +

1𝐿𝐶

.

and two -3 dB cut-off frequencies shown as 𝑓𝐿, 𝑓𝐻 in the Figure 13. Since the band pass filter

Figure 13: Frequency response of a band-stop filter.


𝑣𝑖𝑛(𝑡)


𝑉𝑖𝑛(𝑠) 𝑉𝑜𝑢𝑡(𝑠)

is the inverted form of the standard band-pass filter 𝑓𝐿, 𝑓𝐻 , 𝑓𝐶 are the same as described in

the band pass filter section with 𝑓𝐶 corresponding to 𝑓𝑐𝑒𝑛𝑡𝑒𝑟. Thus, we can use the same

formulas to calculate the bandwidth 𝐵𝑊and the quality factor 𝑄 in the general representation

of the transfer function,



𝑠2 + 𝜔𝑜2

𝑠2 +𝜔𝑜𝑄

𝑠 + 𝜔𝑜2

.

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 ‘T’ 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

𝑓𝐶 =1

4𝜋𝑅𝐶. If the quality factor 𝑄 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 𝑅, 𝐶, 𝐿 components you have at hand. If the filtering requirements don’t match exactly with your component values you can adjust the requirements. But don’t 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 𝐻(𝑠) =𝑎1 𝑠3+𝑎2 𝑠2+𝑎3 𝑠+𝑎4

𝑏1 𝑠3+𝑏2 𝑠2+𝑏3 𝑠+𝑏4. Some

coefficients can be zero and the order of your filter may change. Write,

clear;

num=[ 𝑎1 𝑎2 𝑎3 𝑎4]; % in descending sort order of the coefficients of the numerator

den=[𝑏1 𝑏2 𝑏3 𝑏4]; % 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(‘This is a ...’);

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 𝐺(𝑠) = 𝐾(𝑠)𝐻(𝑠) with 𝐾(𝑠) =𝑐1 𝑠3+𝑐2 𝑠2+𝑐3 𝑠+𝑐4

𝑑1 𝑠3+𝑑2 𝑠2+𝑑3 𝑠+𝑑4

clear;

num=[ 𝑎1 𝑎2 𝑎3 𝑎4];

den=[𝑏1 𝑏2 𝑏3 𝑏4];

H=tf(num,den);

numk=[ 𝑐1 𝑐2 𝑐3 𝑐4];

denk=[𝑑1 𝑑2 𝑑3 𝑑4];

K=tf(numk,denk); G=K*H

bode(G); grid on; title(‘This is a ...’);

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