V s () 1 V s() 2ssawyer/CircuitsSpring2020... · units of Decibels (dB). The magnitude of the transfer function can be converted to dB by using the relationship ( ) ( ) (H (s)) V

Alpha Laboratories

ECSE-2010 Spring 2020

Written by J. Braunstein Revised by S. Sawyer Spring 2020: 1/28/2020

Rensselaer Polytechnic Institute Troy, New York, USA

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LABORATORY 6: Transfer Functions, Filters, and Bode Plots

Material covered:

• First Order Filters

• Multi-stage Circuits

• Bode Plots

• Design Problem

Part A: First Order Filters

Overview Notes:

Decibels: When considering the magnitude of the transfer function, a log-log plot

is useful for gaining physical insight to the circuit. Typically, the vertical axis has

units of Decibels (dB). The magnitude of the transfer function can be converted to

dB by using the relationship ( )

( )( )( )sH

sV

sV

in

outlog20log20 =

[dB]. Using this formula,

when we plot the transfer function of a first order circuit, we can make some

observations.

Cutoff frequency: The cutoff frequency of a first order circuit is defined as the

frequency at which the amplitude of the output voltage is 2

1the amplitude of the

input voltage. In other words the ratio of output to input is ( )

( ) 2

1=

sV

sV

in

out .

Substituting that value into the above dB expression, at the cutoff frequency the

expression evaluates to -3dB. The cutoff frequency is often referred to as the 3dB

point. In first order circuits, the cutoff frequency occurs when the magnitudes of

the real part and the imaginary part of transfer function denominator are the same.

Passband: The frequency range where the output signal has approximately a

constant amplitude.

Stopband: The frequency range where the output signals are attenuated relative to

passband . In the stopband of a first order circuit, a 20dB difference occurs every

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time the frequency changes one order of magnitude. For example, in a low pass

filter with a cutoff frequency of 1kHz, the dB value of |H(jω)| at 50kHz would

have a value 20dB lower than the the dB value of |H(jω)| at 5kHz. This attribute is

called a reduction of 20dB/decade. (We will discuss these concepts in class.)

We will be performing frequency sweeps in the upcoming labs. We will need to

configure the voltage component to an AC voltage source by right clicking and

putting 1V in the AC analysis section as shown in the figure below.

You must choose the AC Analysis tab in the simulation profile.

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To perform frequency sweeps;

1) Edit the simulation command and select the AC Analysis Tab

2) Change the “Type of sweep:” to Decade

3) To generate smoother, more accurate plots, choose 100 points per decade.

4) You will then need to select the frequency limits. These limits will likely

depend on your circuit. For the first order circuits, we want the lower limit to

be well below the cutoff frequency and the upper limit to be well above the

cutoff frequency. For the above circuit, we will use a range of 1Hz-1E6Hz

5) LTSpice will output a graph with both amplitude and phase plotted on the

right and left hand axes respectively.

a. The amplitude is expressed in dB.

b. Phase is expressed in degrees.

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A.1. RL Filter Circuit Calculations

Determine the transfer function for the voltage across the resistor,

HVR(s) = ______________________

Determine the transfer function for the voltage across the inductor,

HVL(s) = ______________________

What is the cutoff frequency for this circuit? _________________

As the frequency approaches zero (DC), the voltage across the inductor

approaches _________

As the frequency approaches infinity, the voltage across the inductor

approaches _________

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When measuring the voltage across the inductor, is this circuit a low pass filter or a

high pass filter?

A.1. RL Filter Circuit Simulation and Experiment

Build the circuit. Set your source amplitude to 2Vpp (1V amplitude) with a 0V

offset and adjust the frequency as indicated in the table. Measure the output

voltage amplitude and phase across the inductor for various frequencies and fill in

the following table with your calculations, LTSpice, Analog Discovery Board

measurements.

Inductor

Magnitude Phase

[Degrees]

Freq.

[Hz]

Rad.

Freq.

[rad/s]

Calculated LTSpice Measured Calculated LTSpice Measured

47.7

159

477

fc

1590

4770

15.9k

Include this table of results in your Proof of Concept Report. There are

multiple steps here. Arrange and label with headings in an easy to understand

way for grading TAs!

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If there are any significant differences between measured and calculated results,

comment on possible reasons. (Recall the characteristics of a real inductor.)

For your above measured results, determine the magnitude in dB and the phase of

the transfer function. When calculating the transfer function magnitude, remember

to divide to divide the output magnitude by the source amplitude (which in this

case is 1). The phase can be obtained directly from your measured results in the

previous table.

Radial

Frequency

[rad/s]

log(ω) 20log|H(s)|

[dB]

Phase ( )sH

300

1E3

3E3

ωc

1E4

3E4

1E5

Include this table of results in your Proof of Concept Report. There are

multiple steps here. Arrange and label with headings in an easy to understand

way for grading TAs!

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Using your measured results, generate a log-log plot (dB-log (ω)) of the magnitude

of the transfer function vs. frequency. Also, plot the angle (phase) of the transfer

function against log(ω). For the same circuit, perform an AC Sweep in LTSpice

and compare your results. Can you find a way to get this information using the

Analog Discovery Board directly? How?

Magnitude

Is the cutoff frequency a -3dB point?

Does your stopband display a 20dB/decade drop?

Include screen shots of your results in your Proof of Concept Report. There

are multiple steps here. Arrange and label with headings in an easy to

understand way for grading TAs!

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Phase

What is the change of phase between ω = 300 [rad/s] and ω = 5E4 [rad/s]? Is your

result consistent with expectations from the transfer function?




PART A: Proof of Concepts list

A.1: Prove that gain and phase change with frequency for an RL circuit.

A.1: Prove that the Bode Plot for gain and phase reflects this change of frequency

for an RL circuit.

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Part B: Second Order Filters and Bode Plots

Overview Notes:

Two stage circuits:

H2(s)H1(s)

+++

----

+

Vout2Vout1 Vin2Vin1

The above figure represents a two stage circuit. Recall, the transfer function relates

the ratio the input to the output of a stage, ( )( )

( )sHsV

sV

in

out = . In the above figure,

recognize that Vout1 = Vin2. The transfer function can be applied to each stage.

Applying the transfer function to each stage we can derive the equation,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )sVsHsHsVsHsVsHsV inoutinout 11212222 === . Finally, the relationship

between Vout2 and Vin1 can be written as ( )( )

( ) ( )sHsHsV

sV

in

out

12

1

2 = . This equation is the

product of the two transfer functions. By designing each stage to produce a

particular circuit response, the final output can be designed to meet a specific goal.

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Filter types:

Lowpass filter

Highpass Filter

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Bandpass Filter

Notch/Bandstop Filter

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B1: Multi-stage Filter Circuit Calculations

Vin

R1

1k

R21k

R3

~9k

R4

1k

0

U1

OPAMP

+

-

OUT

0

C11E-6 C2

1E-6

Vout

0

Determine the transfer function for the voltage across the capacitor, C2

H(s) = ________________________________ (symbolically)

This circuit is which of the following?

highpass filter, lowpass filter, bandpass filter, notch filter

What are the zeros of the transfer function? __________________

What are the poles of the transfer function? ________________

What is the gain of the passband, in dB?__________________

What is the slope of the stopband in dB/decade? __________________

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Multi-stage Filter Circuit Simulation and Experiment

Build the circuit and measure the output voltage for various frequencies. Your

source amplitude should 200mV. Your Oscilloscope V/div should be 0.5V/div or

less. Fill in the following table with your calculations and Analog Discovery Board

measurements. Remember to scale your measured output voltage with your input

voltage to obtain the transfer function.

Reminder: The amplifier requires +9/-9 voltage connections. You should double

check your batteries and make sure they still have close to a 9V supply.

Sometimes, storing them in the kits without a cover results in a short circuit

connection across the leads, which discharges the battery.

Magnitude, |H(s)|

Frequency

[Hz] Calculated Measured

20log|H(s)|

(use measured

value)

15.9

48

159

488

1590

Using your measured results, generate a Bode plot of the magnitude. Compare the

measured results to your analytic expression and a LTSpice simulation.




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B2: Multi-stage Filter Circuit Calculations

Vin1

R1

100

R21k

R3

~9k0

U2

OPAMP

+

-

OUT

0

C11E-6

VoutC2

1E-6

R41k

0

Determine the transfer function for the voltage across the resistor, R4.

H(s) = ________________________________ (symbolically)

This circuit is which of the following?

highpass filter, lowpass filter, bandpass filter, notch filter

What are the zeros of the transfer function? __________________

What are the poles of the transfer function? ________________

What is the gain of the passband, in dB?__________________

What is the slope of the stopband(s) in dB/decade? __________________

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Multi-stage Filter Circuit Simulation and Experiment

Build the circuit and measure the output voltage for various frequencies. Your

source amplitude should 200mV. Your Oscilloscope V/div should be 0.5V/div or

less. Fill in the following table with your calculations and Analog Discovery Board

measurements. Remember to scale your measured output voltage with your input

voltage to obtain the transfer function.

Magnitude, |H(s)|

Frequency

[Hz] Calculated Measured

20log|H(s)|

(use measured

value)

15.9

48

159

488

1590

Using your measured results, generate a Bode plot of the magnitude. Compare the

measured results to your analytic expression and LTSpice simulation.




PART B: Proof of Concepts list

B.1: Prove the expected Magnitude Bode plot for the first multistage circuit.

B.2: Prove the expected Magnitude Bode plot for the second multistage circuit.

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Part C: Filter Design Problem

Design a lowpass filter that meets the following specifications.

1. Cutoff frequency: 1.59kHz

2. 60dB rolloff in the stopband

3. A single unity gain opamp

4. |H(jω)| →-3dB relative to the passband at the cutoff frequency. (Remember

a triple pole has a -9dB correction relative to the straight line approximation.

You will need to underdamp a second order circuit.)

5. Use L and C component values found in your kit. You can use resistors

found on the center table in the laboratory room.

Some flexibility exists in meeting the specifications. In design problems,

perfection is not usually possible, but deviations should be small. If you don’t quite

meet specification, explain why and explain what you would do to fix the problem.

C.1. Filter Design Calculations

Determine the transfer function that meets the above specifications

H(s) = ________________________________ (symbolically)

Draw the circuit, labeling the component values

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C.2. Filter Design Simulation and Calculations

Build the circuit and verify that the specifications are met by taking measurements

at appropriate frequencies. You should base your Vin amplitude around that

setting.

Magnitude, |H(s)|

Radial

Frequency

[rad/s]

Calculated Measured

|Vout|/|Vin|

20log|H(s)|

(use measured

value)

100

300

1E3

3E3

9E3

10E3

15E3

20E3

30E3

100E3

Due to the steep rolloff and noise effects, you may have difficulty obtaining data as

you move further into the stopband.

For the same circuit, perform an AC Sweep in LTSpice and compare your results.

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Part D. Alpha Lab Applications (Extra Credit….)

Passive filter vs. Active filter

1. What is the difference between a passive filter and an active filter?

2. Find three different active filter configurations. How are they different?

3. Describe any conditions for which an active filter is much more useful than

a passive filter.

4. Simulate any active filter configuration. Calculate and compare.

Include screen shots of your results in your Proof of Concept Report. You

do not need to build the circuit. It is optional.

SUMMARY of Concepts Concept List that must be accounted for in your Proof of Concepts

PART A:

1. Prove that gain and phase change with frequency for an RL circuit.

2. Prove that the Bode Plot for gain and phase reflects this change of frequency for an

RL circuit.

PART B:

1. Prove the expected Magnitude Bode plot for the first multistage circuit.

2. Prove the expected Magnitude Bode plot for the second multistage circuit.

PART C:

1. Prove that your designed filter meets specifications.

PART D: (Extra Credit)

1. Prove/discuss how an active filter is different from a passive filter.

PART C: Proof of Concepts list

Prove that your designed filter meets specifications.

PART D: Proof of Concepts list

Prove/discuss how an active filter is different from a passive filter.

Documents

V s () 1 V s() 2ssawyer/CircuitsSpring2020... · units of Decibels (dB). The magnitude of the transfer function can be converted to dB by using the relationship ( ) ( ) (H (s)) V