Passive Low Pass and High Pass Filter

Lab 08: Passive First Order Low Pass & High Pass Filter

Experiment 08  To design RC low pass & high pass filter circuit with cutof frequency of 5KHZ, also draw a frequency response: (a) magnitude response (b) phase response.  To design RL low pass & high pass filter circuit with cutof frequency of 5KHZ, also draw a frequency response: (a) magnitude response (b) phase response. Introduction: This laboratory studies the use of passive components to create filters to separate portions of time-dependent waveforms. Filters are an essential tool in our complex world of mixed signals — both electronic and otherwise. Passive components (resistors, capacitors, and inductors) have long served as filter components for everything from selecting radio stations to filtering out electrical noise. EDUCATIONAL OBJECTIVES: (1) Learn the general filter types: High-pass, Low-pass. (2) Learn to alter filter type by changing contacts for output voltage. (3) Learn phase angle at cutoff for simple RC and RL filters. (4) Design simple filter. (5) Frequency response (amplitude and phase). EXPERIMENTAL OBJECTIVES: (1) Calculate and measure cutoff frequency for series RC and RL filters. (2) Design simple RC low-pass & high pass filter. (3) Design simple RL low-pass & high pass filter. (4) Bode plots for series filters.

PRE-LAB: Reading: (1) Study the Background section of this Laboratory. (2) Study textbook Chapter 14 (14.1 -14.4). Written:

Note: Before the starts of simulation make sure that the calculated values of R, L and C are available in Lab. Otherwise Design your circuits according to the available values.

Lab 08: Passive First Order Low Pass & High Pass Filter (1)Simulate the circuit shown in Fig (5), Fig (6), Fig (7) and Fig (8) in LTspice and Record the simulation result in Table 1, Table 2, Table 3 and Table 4. (2)Sketch the Bode Plots of the simulation results recorded in Table 1, Table 2, Table 3 and Table 4.

Lab 08: Passive First Order Low Pass & High Pass Filter EQUIPMENT AND MATERIALS  Digital Function Generator  Digital Oscilloscope with Probes.  Bread Board  Capacitors  Inductors  Resistors

Background: Frequency Response The frequency response of a circuit is the variation in its behavior with change in signal frequency. Transfer Function The transfer function

H ( ω)

of a circuit is the frequency dependent ratio of a

phasor output y ( ω ) (an element voltage or current) to a phasor input X ( ω ) (source voltage or current). H ( ω )=

Y (ω) X ( ω)

Bode Plots Bode plots are semi-log plots of the magnitude (in decibels) and phase (in degrees) of a transfer function versus frequency. Cutof Frequency The frequency at which gain becomes 0.707 times of the maximum gain is called cutoff frequency. It is also called the half power frequency because at this frequency the average power delivered by the circuit is one half the maximum average power. ¿ H ( ω )∨¿ MAX |H ( ωC )|=0.707 ¿

Lab 08: Passive First Order Low Pass & High Pass Filter P(ω c )=

Pmax 2

Filter A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others. Passive Filters Elements used in passive filters are resistors, capacitors and inductors. The gain of passive filters is unity. Low Pass Filter A low pass filter passes low frequencies and rejects high frequencies.

Magnitude First-Order

Bode Plot for Low Pass Filter

Lab 08: Passive First Order Low Pass & High Pass Filter

Phase Bode Plot for First-Order Low Pass Filter

High Pass Filter A high pass filter passes high frequencies and rejects low frequencies.

Magnitude Bode Plot for First-Order High Pass Filter

Lab 08: Passive First Order Low Pass & High Pass Filter

Phase Bode Plot for First-Order High Pass Filter

Lab 08: Passive First Order Low Pass & High Pass Filter Design of RC Low Pass Filter The cutoff frequency for RC circuit is given below

ωc=

1 1 , f c= −−−−−−−(1) RC 2 πRC

The derivation of equation (1) is given in Appendix A

Fig (5) RC

Low Pass Filter Circuit Let C=10 n F , C=9.08 nF (By Capacitance Meter ) Put C=9.08 n F ,

f c =5 KHz ∈ Equation(1)

5∗10 ¿ 2 π ¿(¿ 3¿)( 9.08∗10−9 ) ¿ 1 R= ¿ R=3.5 KΩ Make sure that the Capacitor∧Resistorof this valusis available ∈Lab; otherwise do the

calaculations according ¿ the available values . The transfer function of circuit shown in fig (3) is

H ( ω )=

1 1+ jωRC

¿ Equation ( 1 ) ωc = H ( ω )=

//Derivation is given in Appendix A

1 RC

1 jω 1+ ωc

|H ( ω )|=

1

√

ω ( 1) + ωc 2

2

( )

−−−−−−−−(2)

Lab 08: Passive First Order Low Pass & High Pass Filter ϕ=−tan−1

( ωω )−−−−−−−−−−−−(3) c

It can be seen¿ equations (2 )∧( 3 ) that at ω=0,| H ( ω )|=1∧ϕ=0

It can be seen¿ equations (2 )∧( 3 ) that at ω=∞ ,|H ( ω )|=0∧ϕ=−90 0

Lab 08: Passive First Order Low Pass & High Pass Filter Design of RC High Pass Filter

C=10 nF

The values of R and C are same as found in RC low pass filter circuit. The transfer function of circuit shown in fig (6) is

H ( ω )=

1 1 1+ jωRC

−−−−−−−−−( 4)

//Derivation is given in Appendix B

Pass Filter Circuit ¿ Equation ( 1 ) ωc =

Put RC =

1 1+

|H ( ω )|=

1 RC

1 ∈the Equation(4) ωc

H ( ω )=

1 jωRC

1

=

1− j

1

√

ϕ=−tan−1

2

ωc ω

−−−−−−−−(5)

( )

−ωc 2 ( 1) + ω

−ω c ω =tan−1 c −−−−−−(6) ω ω

( )

R=3.5 K Ω

( )

It can be seen¿ equations (5 )∧( 6 ) that at ω=0,|H ( ω )|=0∧ϕ=90 0

It can be seen¿ equations (5 )∧( 6 ) that at ω=∞ ,|H ( ω )|=1∧ϕ=00

Fig (6) RC High

Lab 08: Passive First Order Low Pass & High Pass Filter Design of RL Low Pass Circuit The cutoff frequency for RC circuit is given below

ωc=

R R , f c= −−−−−−−−(7) L 2 πL

The derivation of equation (7) is given in Appendix C

Let L=10 mH

Fig (7) RL Low Pass Filter

Circuit Make sure that the inductor f this valus is available ∈Lab Put L=1 mH ∧¿

f c =5 KHz∈the Equation(7)

R=2 πf c L R=2 π∗5∗103∗10∗10−3 R=314.1 Ω

Make sure that the inductor∧resistorof this valus is available∈Lab ; otherwise do the calaculations according ¿ the available inductor value.

The transfer function of circuit shown in fig (3) is H ( ω )=

1 −−−−−−−−−(8) jωL 1+ R

¿ Equation ( 7 )

Put

L 1 = R ωc

ωc=

R L

¿ the Equation(8)

//Derivation is given in Appendix C

Lab 08: Passive First Order Low Pass & High Pass Filter H ( ω )=

1 jω 1+ ωc

|H ( ω )|=

1

√

ϕ=−tan−1

ω ( 1) + ωc 2

2

−−−−−−−−(9)

( )

ω −−−−−−−−−−−−(10) ωc

( )

It can be seen¿ equations ( 9 )∧( 10 ) that at ω=0,|H ( ω )|=1∧ϕ=0

It can be seen¿ equations ( 9 )∧( 10 ) that at ω=∞ ,|H ( ω )|=0∧ϕ=−900

Design of RL High Pass Circuit: The values of R and L is same as found in RL low pass filter circuit. The transfer function of circuit shown in fig (8) is

Fig (8) RL High Pass Filter Circuit H ( ω )=

1 R 1+ jωL

−−−−−−(11)

¿ Equation ( 7 ) ω c = Put

R L

R =ωc ∈the Equation(11) L

Lab 08: Passive First Order Low Pass & High Pass Filter H ( ω )=

1 R 1+ jωL

|H ( ω )|=

1

=

1− j 1

√

ϕ=−tan−1

2

ωc ω −−−−−−−−(12)

( )

−ωc 2 ( 1) + ω

(−ωω )=tan ( ωω )−−−−−−(13) c

−1

c

It can be seen¿ equations (12 ) ∧( 13 ) that at ω=0,|H ( ω )|=0∧ϕ=900

It can be seen¿ equations (12 ) ∧( 13 ) that at ω=∞ ,|H ( ω )|=1∧ϕ=00

Lab 08: Passive First Order Low Pass & High Pass Filter Task 1 RC Low Pass Filter: 1) Set up the circuit shown in Fig (5). Use the function generator FGEN for the supply voltage vin 10 VP-P. 2) Connect channels 1 and 2 of the oscilloscope to measure Vs and Vout simultaneously. 3) Vary the frequency from 500 Hz to 10 kHz in steps indicated in Table 1, and record the indicated value. With each frequency change, make sure that V in is still 10Vpp. a) Using the data of Table 1, sketch a Bode plots of the of the filter’s output voltage.

Task 2 RC High Pass Filter: Repeat the Task 1 for the circuit shown in Fig (6), and record your result in Table 2

Task 3 RL Low Pass Filter: Repeat the Task 1 for the circuit shown in Fig (7), and record your result in Table 3

Task 4 RL High Pass Filter: Repeat the Task 1 for the circuit shown in Fig (8), and record your result in Table 4

Post Lab: 1) Why are capacitors preferred over inductors in filter design? 2) Create Bode plots of the magnitude transfer functions of your low-pass and high-pass filters. The theoretical plots (using measured values of resistors and capacitors) should be drawn as lines. Include your data (taken at 1/10, ½, 1, 2, and 10 times the cutoff frequency) as points. 3) Compare the measured results and simulation results.

Lab 08: Passive First Order Low Pass & High Pass Filter Calculated Results: Task 1: Passive RC Low Pass Filter Frequency (KHz)

ω(rad /s)

|H ( ω )|=

0.01fc= 0.1fC = 0.5fc = fc = 2fc = 4fc = 6fc = 8fc = 10fc = 100fc =

1

√

ω ωc

H ( ω )=20 log

2

( )

( 1 )2 +

Vo (dB) V¿

ϕ=−tan−1

Vo (dB) V¿

ϕ=tan−1

( ωω ) c

Table 1 (RC Low Pass Filter) Task 2: Passive RC High Pass Filter Frequency (KHz)

0.01fc= 0.1fC = 0.5fc = fc = 2fc = 4fc = 6fc = 8fc = 10fc = 100fc =

ω(rad /s)

|H ( ω )|=

1

√

2

2

( −ωω )

(1) +

H ( ω )=20 log

c

Table 2 (RC High Pass Filter)

( ωω ) c

Lab 08: Passive First Order Low Pass & High Pass Filter Calculated Results: Task 3: Passive RL Low Pass Filter Frequency (KHz)

ω(rad /s)

|H ( ω )|=

0.01fc= 0.1fC = 0.5fc = fc = 2fc = 4fc = 6fc = 8fc = 10fc = 100fc =

1

√

ω ωc

H ( ω )=20 log

2

( )

( 1 )2 +

Vo (dB) V¿

ϕ=−tan−1

Vo (dB) V¿

ϕ=tan−1

( ωω ) c

Table 3 (RL Low Pass Filter) Task 4: Passive RL High Pass Filter Frequency (KHz)

0.01fc= 0.1fC = 0.5fc = fc = 2fc = 4fc = 6fc = 8fc = 10fc = 100fc =

ω(rad /s)

|H ( ω )|=

1

√

2

( )

−ωc 2 (1) + ω

H ( ω )=20 log

Table 4 (RL High Pass Filter)

( ωω ) c

Lab 08: Passive First Order Low Pass & High Pass Filter Section 4 – Measurement Tables Name: __________________________ _______________________

Reg. No

Task 1: Passive RC Low Pass Filter Vin (p-p) = 10 VP-P Frequency (KHz)

ω(rad /s)

Vout(P-P)

H (ω)=

Vo V¿

H ( ω )=20 log

Vo (dB) V¿

ϕ (degrees)

Vo (dB) V¿

ϕ (degrees)

0.01fc= 0.1fC = 0.5fc = fc = 2fc = 4fc = 6fc = 8fc = 10fc = 100fc = Table 1 (RC Low Pass Filter) Task 2: Passive RC High Pass Filter Vin (p-p) = 10 VP-P Frequency (KHz)

0.01fc= 0.1fC = 0.5fc = fc = 2fc = 4fc = 6fc = 8fc = 10fc = 100fc =

ω(rad /s)

Vout(P-P)

H (ω)=

Vo V¿

H ( ω )=20 log

Lab 08: Passive First Order Low Pass & High Pass Filter Table 2 (RC High Pass Filter)

Section 4 – Measurement Tables Name: __________________________ _______________________

Reg. No

Task 3: Passive RL Low Pass Filter Vin (p-p) = 10 VP-P Frequency (KHz)

ω(rad /s)

Vout(P-P)

H (ω)=

Vo V¿

H ( ω )=20 log

Vo (dB) V¿

ϕ (degrees)

0.01fc= 0.1fC = 0.5fc = fc = 2fc = 4fc = 6fc = 8fc = 10fc = 100fc = Table 3 (RL Low Pass Filter) Task 4: Passive RL High Pass Filter Vin (p-p) = 10 VP-P V V Frequency ω(rad /s) Vout(P-P) H (ω)= o H ( ω )=20 log o (dB) (KHz) V¿ V¿

0.01fc= 0.1fC = 0.5fc = fc = 2fc = 4fc = 6fc = 8fc = 10fc =

ϕ (degrees)

Lab 08: Passive First Order Low Pass & High Pass Filter 100fc = Table 4 (RL High Pass Filter)

Name: __________________________ _______________________

Reg. No

Bode Plot for RC Low Pass Filter: Magnitude (dB)

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Frequency (rad/Sec)

Lab 08: Passive First Order Low Pass & High Pass Filter

Phase (degree)

Frequency Name: (rad/Sec)

__________________________ _______________________

Reg. No

Bode Plot for RC High Pass Filter: Magnitude (dB)

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Frequency (rad/Sec)

Lab 08: Passive First Order Low Pass & High Pass Filter

Phase (degree)

Frequency (rad/Sec) Name:

__________________________ _______________________

Reg. No

Bode Plot for RL Low Pass Filter: Magnitude (dB)

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Frequency (rad/Sec)

Lab 08: Passive First Order Low Pass & High Pass Filter

Phase (degree)

Frequency (rad/Sec) Name:

__________________________ _______________________

Reg. No

Bode Plot for RL High Pass Filter: Magnitude (dB)

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Frequency (rad/Sec)

Lab 08: Passive First Order Low Pass & High Pass Filter

Phase (degree)

Frequency (rad/Sec) Appendix

A (RC Low Pass Filter)

From the circuit diagram,

1 jωC V o= V¿ 1 R+ jωC

V o is given by

Vo 1 = V ¿ 1+ jωRC

As we know that

Vo =H (ω) V¿ H ( ω )=

1 −−−−−−−−−−−−(1) 1+ jωRC

Lab 08: Passive First Order Low Pass & High Pass Filter |H ( ω )|=

At

ω=¿

ωc

|H ( ω )|

1 √1+ ( ωRC )

↔

= 0.707

2

0.707 =

0.5=

0.707=

1 √ 1+( ωc RC )

1 2 1+ ( ω c RC )

1 1+ ( ω c RC ) 2 2

0.5+0.5 ( ω c RC ) =1 0.5 ( ω c RC )2=1−0.5 0.5 2 ( ω c RC ) = 0.5 2 ( ω c RC ) =1

ω c2=

ωc= f c=

1 ( RC)2

1 −−−−−−−−−−−−−(2) RC

1 2 πRC

Appendix B (RC High Pass Filter) From the circuit diagram,

V o is given by

Lab 08: Passive First Order Low Pass & High Pass Filter V o=

R 1 R+ jωC

Vo = V¿

1 1+

H ( ω )=

1 1 1+ jωRC

|H ( ω )|=

ω=¿

1 jωRC

Vo =H (ω) V¿

As we know that

At

V¿

ωc

|H ( ω )|

1

√

1+

= 0.707

ω

√ Squaring both sides

−−−−−−−−−−(5)

1 ( ¿¿ c RC ) 1 0.707= ¿

1+

ω 1 1+ (¿¿ c RC) 1 0.7072 = ¿

1 (ωRC )

Lab 08: Passive First Order Low Pass & High Pass Filter ω 1 1+ (¿¿ c RC ) 1 0.5= ¿ ω (¿¿ c RC) 1 1+ ¿ ¿ 0.5 ¿

ω (¿¿ c RC )2 =1 0.5 0.5+ ¿

ω (¿¿ c RC )2 =1−0.5 0.5 ¿ 0.5 2 ( ω c RC ) = 0.5 2 ( ω c RC ) =1

2

ωc =

ωc=

f c=

1 ( RC)2

1 RC

1 2 πRC

Lab 08: Passive First Order Low Pass & High Pass Filter

Lab 08: Passive First Order Low Pass & High Pass Filter Appendix C (RL Low Pass Filter) From the circuit diagram,

Vo is given by

R V R+ jωL ¿

V o=

Vo = V¿

R jωL R(1+ ) R

As we know that

Vo =H (ω) V¿

H ( ω )=

1 jωL 1+ R

|H ( ω )|=

At

ω=¿

ωc

|H ( ω )|

1

√

= 0.707

0.707=

1

√

( ωRL )

1+

c

( ωLR )

1+

Lab 08: Passive First Order Low Pass & High Pass Filter 0.7072=

1 ω L 1+ c R

2

( )

0.5=

1 ω L 1+ c R

2

( )

ωc L 2 0.5+0.5 =1 R

( )

0.5

ωc L 2 =1−0.5 R

( )

ωc L 2 0.5 = R 0.5

( )

ωc L 2 =1 R

( ) 2

R ω = 2 L 2 c

ωc=

f=

R L

R 2 πL

Lab 08: Passive First Order Low Pass & High Pass Filter Appendix D (RL High Pass Filter) From the circuit diagram,

Vo is given by

V o=

jωL V R+ jωL ¿

Vo = V¿

jωL R jωL( +1) jωL

As we know that Vo =H (ω) V¿ 1

H ( ω )= 1+

|H ( ω )|=

At

ω=¿

ωc

R jωL 1

√

( ωLR )

1+

|H ( ω )| = 0.707 0.707=

1

√

R ωc L

( )

1+

Lab 08: Passive First Order Low Pass & High Pass Filter 0.7072=

0.5=

1 R 1+ ωc L

1 R 1+ ωc L

( )

2

( )

R 2 0.5+0.5 =1 ωc L

( ) 2

0.5

R =1−0.5 ωc L

( )

R 2 0.5 = ωc L 0.5

( )

2

R =1 ωc L

( ) 2

ωc =

ωc=

f=

R2 L2

R L

R 2 πL

2