Essay Available:

You are here: Home → Math Problem → Engineering

Pages:

4 pages/≈1100 words

Sources:

3 Sources

Level:

APA

Subject:

Engineering

Type:

Math Problem

Language:

English (U.S.)

Document:

MS Word

Date:

Total cost:

$ 21.6

Topic:

# Math Problem Assignment: Circuit and Network Theory III (Math Problem Sample)

Instructions:

THE TASK IS ABOUT FILTER DESIGN AND THE QUESTION IS :

a). Design a two stage low pass filter with a d.c gain of 8, a bandwidth of 2000 Hz and a suitable roll- off rate (dB/dec).

b). Draw the circuit and analyse how the specs will be achieved.

c). Calculate the practical values of the components using the standard values of resistors and capacitors.

d). Calculate the voltages at each node as output for an input sinusoidal signal of 10V.

e). Draw the Bode plots and discuss whether the requirements have been achieved.

NB: Ensure that the cut-off gain is not below -3dB point.

Content:

Name:

Instructor:

Course Title: Electrical Engineering

Unit Title: Circuit and Network Theory III

Assignment: Filter Design

Date:

QUESTION

a). Design a two stage low pass filter with a d.c gain of 8, a bandwidth of 2000 Hz and a suitable roll- off rate (dB/dec).

b). Draw the circuit and analyse how the specs will be achieved.

c). Calculate the practical values of the components using the standard values of resistors and capacitors.

d). Calculate the voltages at each node as output for an input sinusoidal signal of 10V.

e). Draw the Bode plots and discuss whether the requirements have been achieved.

NB: Ensure that the cut-off gain is not below -3dB point.

SOLUTIONS

a). Designing a two stage low pass filter with a d.c gain of 8, a bandwidth of 2000 Hz and a suitable roll- off rate (dB/dec);

* Choose two low pass filters with a roll-off rate of -20 dB/dec. These two low pass filters will be cascaded to give a roll-off rate of -40dB/dec, each contributing 20dB/dec.

* The Op-Amp, an active component, is used to ensure that a cut-off of -3dB is achieved.

* The passive components of the circuit will consist of resistors and capacitors. To obtain a d.c gain of 8, the d.c gain of filter A is set to 2 and that of filter B to 4. The two d.c gains are multiplied to give the desired d.c gain of 8.

b). Drawing the circuit;

Analysing how the specifications will be achieved;

A). Analysis of the first filter (Filter A);

The transfer function T(s) will be given by:

T

T

Where K = 1R2C1 and P = 1R1C1

Since K = 1CR=2,000, the two poles will be both at 2,000Hz

B). Analysis of the second filter (Filter B);

The transfer function T(s) will be given by:

T(s) = R1'//C1'R2'= VoVi

T(s) = 1R2'C1'R1'C1'+1= 1R2'C1'S+1R1'C1' = KS+P

Where P = 1R1'C1' and K = 1R2'C1'

When we cascade the two filters A and B, the overall transfer function will be given by;

T(s) overall = (T(s of Filter A) Ã— (T(s) of Filter B) which is expressed as

c). Calculating the practical values of the components using the standard values of resistors and capacitors;

To obtain the parameters of the first filter, we let K = 4, C1 = 1, Kf = 104, and Km = 105 .

Therefore, the transfer function will be; T(s) = KS+P= KS+1.26

The resistances will be; R1' = 1PC1'= 11.26Ã—1=0.794Î©

R2' = 1PC1'= 14Ã—1=0.25Î©

Applying the scaling factor Km = 105, we obtain the real values as

New R1' = KmR1'Old = 105Ã—0.794=79.4 kÎ©

New R2' = KmR2'Old= 105Ã—0.25=25 kÎ©

New C1' = 1KmKfC1'Old = 1105Ã—104Ã—1=10-9F=1 nano Farads

From the expression of the transfer function of filter A, we evaluate its transfer function to be

T(s) = 1R2'C1'S+1R1'C1' = 1(25 Ã— 103)(10-9)S+1(79.4 Ã— 103)(10-9) = 4 Ã—103S+(1.259 Ã— 104)

The frequency of operation is given by Ï‰c=2Ï€fc=2Ï€ Ã—2000=1.26 Ã— 104 rad/s

Next we evaluate the parameters of the second filter (Filter B).

Let K = 2, C1 = 1, Kf = 104, and Km = 105

The transfer function will reduce toT(s) = KS+P= KS+1.26

The values of the resistances will be; R1' = 1PC1'= 11.26Ã—1=0.794Î©

R2' = 1PC1'= 12Ã—1=0.5Î©

Applying the scaling factor Km = 105, we obtain the real values as

New R1 = KmR1Old= 105Ã—0.794=79.4kÎ©

New R2 = KmR2Old= 105Ã—0.5=25kÎ©

New C1 = 1KmKfC1Old= 1104Ã—105Ã—1=10-9F

From the expression of the transfer function of filter B, we evaluate the transfer function of filter B to be T(s) = 1R2C1S+1R1C1 = 1(50 Ã— 103)(10-9)S+1(79.4 Ã— 103)(10-9) = 2 Ã—104S+(1.259 Ã— 104)

Lastly, from the overall transfer function expression T(s) = T(s) of Filter A Ã— T(s) of Filter B, we obtain the overall transfer function as T(s) 2 Ã—104S+(1.259 Ã— 104) Ã— 4 Ã—104S+(1.259 Ã— 104) =8 Ã—108S2+25200s+1.59 Ã—108)

d). Calculating the voltages at each node as output for an input sinusoidal signal of 10V;

Voltage at Node A;

V(a) = R1 R1 +R2 Ã— Vin= 79.4 79.4 +25 Ã—10=6.136 Volts

Voltage at Node B;

V(b) = R1' R1' +R2' Ã— Vin= 79.4 79.4 +25 Ã—6.136 = 4.667 Volts

e). Drawing the Bode plots and discussing whether the requirements have been achieved;

We use the expression for magnitude T(s) = 8 Ã—108Ï‰2 + 25200Ï‰ + 1.59Ã—108) to obtain values for plotting the magnitude Bode plot by substituting with different values of frequency.

For example, frequency, w = (0, 100, 400, 1000, 5000) rads/s and so on.

We get the plotting points to be; Point 1 = 14....

Get the Whole Paper!

Not exactly what you need?

Do you need a custom essay? Order right now:

### Other Topics:

- Measurements And InstrumentationDescription: Measurements And Instrumentation Engineering Math Problem...2 pages/≈550 words| 2 Sources | APA | Engineering | Math Problem |
- Analysis of suction mechanics in aquatics. Engineering Math ProblemDescription: The different ways in which aquatic organisms feed have been under constant debate because it has not yet been actually proven the main dominant feeding technique....5 pages/≈1375 words| 2 Sources | APA | Engineering | Math Problem |
- Beam Design. Beam Design to Eurocode 2. Engineering Math ProblemDescription: Beam is a structural component that carries laterally applied loads. It primarily deflects through bending. When loads are applied on a beam, reactions forces are produced at its supports (Wight & MacGregor, 2005)....6 pages/≈1650 words| 5 Sources | APA | Engineering | Math Problem |