Passive Filters

90.7

WSDL

Ocean City

90.3

WHID

Salisbury

Frequency

(MHz)

90.5

WKHS

Worton

91.3

WMLU

Farmville

90.9

WETA

Washington

91.1

WHFC

Bel Air

91.5

WBJC

Baltimore

Tuning a Radio

• Consider tuning in an FM radio station.

• What allows your radio to isolate one station from all of

the adjacent stations?

Filters

• A filter is a frequency-selective circuit.

• Filters are designed to pass some frequencies and reject

others.

Frequency

(MHz)90.9

WETA

Washington

Active and Passive Filters

• Filter circuits depend on the fact that the impedance of

capacitors and inductors is a function of frequency

• There are numerous ways to construct filters, but there

are two broad categories of filters:

– Passive filters are composed of only passive

components (resistors, capacitors, inductors) and do

not provide amplification.

– Active filters typically employ RC networks and

amplifiers (opamps) with feedback and offer a number

of advantages.

Passive Filters

• There are four basic kinds of filters:

– Low-pass filter - Passes frequencies below a critical

frequency, called the cutoff frequency, and attenuates

those above.

Passive Filters

• There are four basic kinds of filters:

– High-pass filter - Passes frequencies above the

critical frequency but rejects those below.

Passive Filters

• There are four basic kinds of filters:

– Bandpass filter - Passes only frequencies in a narrow

range between upper and lower cutoff frequencies.

Passive Filters

• There are four basic kinds of filters:

– Band-reject filter - Rejects or stops frequencies in a

narrow range but passes others.

Low Pass Filters

RL low-pass filterRC low-pass filter

• RC low pass filter works based on the principle of

capacitive reactance, while RL low pass filter works on

the principle of inductive reactance

http://www.learningaboutelectronics.com/Articles/Low-pass-filter.php

Capacitive Reactance

• Capacitive Reactance (Xc) varies with the applied

frequency.

– As the frequency applied to the capacitor increases, its effect is

to decrease its reactance (measured in ohms).

– Likewise as the frequency across the capacitor decreases its

reactance value increases.

(Xc = 1 2𝜋𝑓𝑐) ohms

http://www.electronics-tutorials.ws/filter/filter_1.html

Inductive Reactance

• Inductive Reactance (XL) varies with the applied

frequency.

– To high frequency signals, inductors offer high resistance thus

blocks high frequencies

– As frequencies decrease, the inductor offers low resistance so

low frequencies pass

XL = 2𝜋𝑓𝐿 ohmshttp://faculty.kfupm.edu.sa/ee/malek/EE205/pdfslides-205/Lecture%2028_ee205.pdf

RL low-pass filter RL low-pass filter

at low frequencies

𝜔 = 0

RL low-pass filter at

high frequencies

𝜔 =∞

RC Low-Pass Filter – Frequency Response

• The cutoff frequency is the frequency at which capacitive reactance and

resistance are equal (R = Xc), therefore fc = 1 2𝜋𝑅𝑐

• At cutoff, the output voltage amplitude is 70.7% of the input value or –3 dB

(20 log (Vout/Vin))

RC Low-Pass Filter – Phase

• The phase angle ( Φ ) of the output signal LAGS behind that of the

input and at fc, is -45o out of phase. This is due to time taken to

charge the capacitor as input voltage changes.

• The higher the input frequency, the more the capacitor lags and

circuit becomes more out of phase

fc

Application: RC Integrator Circuit

• The integrator converts square wave input signal into a triangular

output as the capacitor charges and discharges.

• The higher the input frequency, the lower will be the amplitude

compared to that of the input

Filters

Notice the placement of the elements in RC and

RL low-pass filters.

What would result if the position of the elements

were switched in each circuit?

RL low-pass filterRC low-pass filter

RC and RL High-Pass Filter Circuits

Switching elements results in a High-Pass Filter.

co co

1 or [Hz]

2 2

Rf f

RC L

f (Hz)fco

actual

passbandreject-band

“ideal”

cutoff frequency

o

s

V

V

0 dB

–3 dB

Impedance vs. Frequency

Calculate the impedance of a resistor, a capacitor

and an inductor at the following frequencies.

1 L CZ j L Z j

C

f 100 Hz 1000 Hz 10,000 Hz

R 100 W 100 W 100 W

ZL j10 W j100 W j1000 W

ZC -j1000 W -j100 W -j10 W

RC Low-Pass Filter

For this circuit, we want to compare the output (Vo)

to the input (Vs):

v

v2

1

1( )

1 1

1( )

1

Co s

C

o

s

o

s

j CH

j RCR

j C

H

RC

ZV V

R Z

V

V

V

V

Example

What is the cutoff frequency for this filter?Given:

8.2

0.0033

R k

C F

W

co

co

or [Hz]2

RC

fRC

co 5.88 kHzf

RL Low-Pass Filter

Comparing the output (Vo) to the input (Vs):

2

1

1

1

1

o s

L

o

s

o

s

R

R

LR j Lj

R

L

R

V VR Z

V

V

V

V

EXAMPLE – RL Low Pass Filter

Design a series RL low-pass filter to filter out any noise above 10 Hz.

R and L cannot be specified independently to generate a value for fco = 10 Hz

or co = 2fco. Therefore, let us choose L=100 mH. Then,

3(2 )(10)(100 10 ) 6.28coR L W

2 2 22

20( )

400

RL

o s sRL

V V V

f(Hz) |Vs| |Vo|

1 1.0 0.995

10 1.0 0.707

60 1.0 0.164

co co2

1 which implies: or [Hz]

21

o

s

R Rf

L LL

R

V

V

Example: Microphone circuit

Example

What resistor value R will produce a cutoff frequency of 3.4 kHz

with a 0.047 mF capacitor? Is this a high-pass or low-pass filter?

co

co

1 [Hz]

2

1R=

2

fRC

C f

1004R W

This is a High-Pass Filter