Equipment:
Name
Function Generator
Analog Oscilloscope
BNC T Connector
The George Washington University
School of Engineering and Applied Science
Department of Electrical and Computer Engineering
ECE 2110 – LAB
Experiment:
AC Circuits: Passive Filter Design
Tektronix AFG3021B
Instek GOS-620FG
NONE
Specifications
BNC Cables
(2) BNC-to-mini-grabber test leads
NONE
NONE
Prototype Breadboard
Banana-to-mini-grabber test leads
NONE
NONE
Table P1: Equipment
Components:
Kit Part # Spice Part
Name
Part Description Symbol name
(used in schematics throughout this lab manual)
R1 Resistor
Resistor
R
R
3.3 k Ω resistor
510 Ω resistor R2
Capacitor C 820 pF capacitor C1
Inductor L 4700 μ H inductor
Table 2: Components
L1
Objectives:

Find the frequency response of a series RC and RL circuit

Plot the magnitude and phase response of a series RC and RL circuit

Design, build, and test a low-pass filter

Design, build, and test a high-pass filter

Find the frequency response of a series and parallel resonance circuit

Plot the magnitude and phase response of a series and parallel resonance circuit

Design, build, and test a bandpass filter
GWU SEAS ECE Department ©2010 Page 1 of 7
ECE 2110 – Experiment: AC Circuits: Passive Filter Design

Introduction
An electric filter modifies the frequency content of a signal. Figure P1 shows four types of filters: highpass (HPF), low-pass (LPF), bandpass, and bandstop (notch). A LPF allows low frequencies to pass to the load while attenuating high frequencies. A HPF allows high frequencies to pass while attenuating low frequencies. A bandpass filter allows a range of frequencies to pass while attenuating frequencies outside of that range. A notch filter attenuates a range of frequencies while passing frequencies outside of that range. In Figure P1, the x-axes represent frequency ( ω ) in radians per second. By convention, frequency is represented by the variable ω when its units are radians per second and f when its units are Hertz. The y-axes represent the gain of each filter. In this instance, gain is defined as the voltage across the load divided by the input voltage. As is shown in the figure, the gain of a filter is different at different frequencies.
Figure P1: Gain Responses (Thomas et al., page 602)
The range of frequencies that are attenudated is the stopband . The range of frequencies that pass to the load is the passband . The cutoff frequency ( ω c
or f c
) is the frequency at the transition between the stopband and passband (bandpass and notch filters will have two cutoff frequencies). An ideal filter passes frequencies in the passband without modifying their magnitude (gain = 1) and completely attenuates frequencies in the stopband (gain = 0). However, ideal filters do not exist in practice, so one convention is to define cutoff frequencies as the frequency at which the magnitude of the voltage at the load is decreased by 3 dB (
𝑉 𝑚𝑎𝑥
) from its maximum value (there are other ways to define the cutoff
√2 frequency, so when reading or specifying ω c
, make sure that you understand which definition is being
GWU SEAS ECE Department ©2010 Page 2 of 7
ECE 2110 – Experiment: AC Circuits: Passive Filter Design

used). With respect to a filter, a decibel (dB) is defined as 10 times the number of powers of 10 of the ratio of the output power to the input power (deci = 1/10). When the input and output powers are delivered to an equal resistance, a decibel can be defined with respect to the voltage gain of the filter. This derivation is shown in Equation P1. Using this definition, it can be shown that a 3 dB reduction in voltage
1 is approximately equal to a reduction of
√2
in voltage or a reduction of half the power.
# 𝒐𝒇 𝒅𝑩 = 10 log
10
𝑃 𝑜𝑢𝑡
𝑃 𝑖𝑛
= 10 log
10
𝑉 2 𝑜𝑢𝑡
𝑅
𝑉 2 𝑖𝑛
= 10 log
10
𝑉 𝑜𝑢𝑡
𝑉 𝑖𝑛
𝑉 𝑜𝑢𝑡
𝑉 𝑖𝑛
= 10 log
10
𝑉 𝑜𝑢𝑡
𝑉 𝑖𝑛
+ 10 log
10
𝑅
Equation P1: Decibel Derivation (Thomas et al., page 603)
𝑉 𝑜𝑢𝑡
𝑉 𝑖𝑛
= 𝟐𝟎 𝐥𝐨𝐠
𝟏𝟎
𝑽
𝑽 𝒐𝒖𝒕 𝒊𝒏
The center frequency ( ω
0
or f
0
) is the frequency where the voltage at the load is at its maximum value.
The bandwidth (B) of a filter is the difference between the two cutoff frequencies. The quality factor (Q) is the ratio of the center frequency to the bandwidth ( 𝑄 = 𝜔
0
𝐵
). The gain function of a filter is the ratio of the magnitude of the frequency response of the filter at the load to the magnitude of the frequency response of the source. For passive circuits, the gain must be less that one.
As an example, the magnitude and phase of the voltage at the load of a series RL circuit, given a source voltage of 1 V 0°, is shown in Figure P1. The red line illustrates the voltage across the inductor, and the blue line is the voltage across the resistor (load). The magnitude plot shows that the circuit has a cutoff frequency (-3 dB frequency) of approximately 30 kHz, a passband from 0 to 30 kHz, and a stopband from 30 kHz to Infinity. f c
Figure P1: Low-Pass Filter - magnitude (top), phase (bottom)
GWU SEAS ECE Department ©2010 Page 3 of 7
ECE 2110 – Experiment: AC Circuits: Passive Filter Design

Prelab: (Submit electronically prior to lab meeting. Also have a printed copy for yourself during lab.)
Review the provided Excel spreadsheet (lab10_example.xlsx). It demonstrates a solution to Part 1 of the prelab. Using this spreadsheet as an example, produce similar spreadsheets for Parts 2-4.
Part 1 - Series RC Circuit

Compute the equivalent impedance for the circuit illustrated in Figure 1 .

Establish the general equations for the phasor voltages ( V
C1
, V
R1
) associated with C1 and R1
(leave in rectangular form).

Use Excel to calculate the amplitudes and phase differences of the phasor voltages ( V
C1
, V
R1
) for frequencies from 1KHz to 10 MHz. Use the following increments: 1k, 2k ,…, 9k,10k, 20k,…, 90k,
100k, 200k,..., 900k,1M, 2M,…,10M. Verify that V
C1
+ V
R1
= Vin for all frequencies (using Excel).
( Hint: review the Excel help files for the commands COMPLEX, IMDIV, IMPRODUCT, IMREAL,
IMAGINARY, IMABS, IMTAN2, PI ). o Vin = 1 V 0 ° o C1 = 820 pF o R1 = 3.3 k Ω

Plot a graph of amplitudes versus frequency (use a legend to identify the different curves). Find the -3dB frequency for the R1 curve (Vout).

Plot a graph of phase differences versus frequency (use a legend to identify the different curves).
Figure 1: Series RC Circuit Figure 2: Series RL Circuit
Part 2 - Series RL Circuit

Compute the equivalent impedance for the circuit illustrated in Figure 2 .

Establish the general equations for the phasor voltages ( V
L1
, V
R1
) and associated with L1 and R1
(leave in rectangular form).

Use Excel to calculate the amplitudes and phase differences of the phasor voltages ( V
L1
, V
R1
) for frequencies from 1KHz to 10 MHz. Verify that V
L1
+ V
R1
= Vin for all frequencies (using Excel). o Vin = 1 V 0 ° o L1 = 4700 μ H o R1 = 3.3 k Ω

Plot a graph of amplitudes versus frequency (use a legend to identify the different curves). Find the -3dB frequency for the R1 curve (Vout).

Plot a graph of phase differences versus frequency (use a legend to identify the different curves).
GWU SEAS ECE Department ©2010 Page 4 of 7
ECE 2110 – Experiment: AC Circuits: Passive Filter Design

Part 3 – Series Resonant Circuit

Compute the equivalent impedance for the circuit illustrated in Figure 3 .

Establish the general equations for the phasor voltages ( V
L1
, V
C1
, V
R1
) associated with L1, C1, and R1 (leave in rectangular form).

Use Excel to calculate the amplitudes and phase differences of the phasor voltages ( V
L1
, V
C1
, V
R1
) using for frequencies from 1KHz to 10 MHz. Verify that V
L1
+ V
C1
+ V
R1
= Vin for all frequencies
(using Excel). o Vin = 1 V 0 ° o L1 = 4700 μ H o C1 = 820 pF o R1 = 3.3 k Ω

Plot a graph of amplitudes versus frequency (use a legend to identify the different curves). Find cutoff frequencies ( ωc
1
and ωc
2
), Bandwidth ( B ), center frequency ( ω
0
) and Quality factor ( Q ).

Plot a graph of phase differences versus frequency (use a legend to identify the different curves).
Figure 3: RLC Circuit 1 Figure 4: RLC Circuit 2
Part 4 – Parallel Resonant Circuit

Compute the equivalent impedance for the circuit illustrated in Figure 4 .

Establish the general equations for the phasor voltages ( V
L1
, V
C1
, V
R2
) associated with L1, C1, and R1 (leave in rectangular form).

Use Excel to calculate the amplitudes and phase differences of the phasor voltages ( V
L1
, V
C1
, V
R2
) using for frequencies from 1KHz to 10 MHz. Verify that V
C1
(or V
L1
) + V
R2
= Vin for all frequencies
(using Excel). o Vin = 1 V 0 ° o L1 = 4700 μ H o C1 = 820 pF o R2 = 510 Ω

Plot a graph of amplitudes versus frequency (use a legend to identify the different curves). Find cutoff frequencies ( ωc
1
and ωc
2
), Bandwidth ( B ), center frequency ( ω
0
) and Quality factor ( Q ).

Plot a graph of phase differences versus frequency (use a legend to identify the different curves).
GWU SEAS ECE Department ©2010 Page 5 of 7
ECE 2110 – Experiment: AC Circuits: Passive Filter Design

Lab :
Part 1 - Series RC Circuit - Verification a) Construct the circuit in Figure 1 using the following components:

Vin = 1.0 V peak

C1 = 820 pF

R1 = 3.3 k Ω b) Measure Vout (magnitude and phase) for different frequencies. Start with the frequency = 1 kHz and raise it up to 10 MHz. c) Plot magnitude versus frequency using the data collected. Find the -3dB frequency. d) Plot the phase difference versus frequency using the data collected.
Part 2 - Series RL Circuit - Verification a) Construct the circuit in Figure 2 using the following components:

Vin = 1 V peak

L1 = 4700 uH

R1 = 3.3 k Ω b) Measure Vout (magnitude and phase) for different frequencies. Start with the frequency = 1 kHz and raise it to 10 MHz. c) Plot magnitude versus frequency using the data collected. Find the -3dB frequency. d) Plot the phase difference versus frequency using the data collected.
Part 3 - Design of a High Pass Filter a) Use ORCAD to design and simulate a high pass filter that meets the following specifications
(show all steps of your design).

Applied Voltage: 1 V rms

-3dB Frequency Point: 20 kHz

All tolerances are 5%. b) Build, test, and demonstrate this circuit to your instructor.
Part 4 - Design of a Low Pass Filter a) Use ORCAD to design and simulate a low pass filter that meets the following specifications (show all steps of your design).

Applied Voltage: 1 V rms

-3dB Frequency point: 500 Hz

All tolerances are 5%. b) Build, test, and demonstrate this circuit to your instructor.
Part 5 – Series Resonant Circuit - Verification a) Construct the circuit in Figure 3 , using the following components:

Vin = 1 V peak

L1 = 4700 μ H

C1 = 820 pF

R1 = 3.3 k Ω b) Measure Vout (magnitude and phase) for the range of frequencies from 1 kHz to 10 MHz. c) Plot magnitude versus frequency using the data collected. d) Find cutoff frequencies ( ω c1
and ω c2
), Bandwidth ( B ), center frequency ( ω
0
) and Quality factor
( Q ). e) Plot phase difference versus frequency using the data collected.
GWU SEAS ECE Department ©2010 Page 6 of 7
ECE 2110 – Experiment: AC Circuits: Passive Filter Design

Part 6 – Series Parallel Circuit - Verification a) Construct the circuit in Figure 4 using the following components:

Vin = 1 V peak

L1 = 4700 μ H

C1 = 820 pF

R2 = 510 Ω b) Measure Vout (magnitude and phase) for different frequencies. Start with the frequency = 1 kHz and increase it to 10 MHz. c) Plot magnitude versus frequency using the data collected. d) Find cutoff frequencies ( ω c1
and ω c2
), bandwidth ( B ), center frequency ( ω
0
) and Quality factor ( Q ). e) Plot phase difference versus frequency using the data collected.
Part 7 - Design of a Band Pass Filter a) Use ORCAD to design and simulate a band pass filter that meets the following specifications
(show all steps of your design):

Applied Voltage: 1 V rms

Quality Factor: 1

Bandwidth: 15 kHz

All tolerances are 5%. b) Build, test and demonstrate this circuit.
Analysis to consider in Lab write-up:

Compare the calculated results to the measured results and explain any and all differences.

Describe the motivation behind defining the cutoff frequency at the point where the gain is -3 dB as opposed to -4 dB or -5 dB.

What type of filter would you want to implement if you observed high frequency noise in your voltage signal?

Describe a situation where a bandpass filter would be desired.

Does it make sense to define the bandwidth of a HPF? Explain.

Describe the relationship between ω c
and f c
.

Hum noise is a common phenomenon in electronic devices especially hi-fi equipment. The noise comes from the line (110 Vac rms
@ 60 Hz). Using the information you have learned so far, how could you eliminate this noise?

Describe how the quality factor (Q) is used to distinguish between narrow-band and wide-band filters.
References:
Thomas, R. E., A. J. Rosa, and G. Toussaint, The Analysis and Design of Linear Circuits (Prentice Hall:
2009).
GWU SEAS ECE Department ©2010 Page 7 of 7
ECE 2110 – Experiment: AC Circuits: Passive Filter Design