A low-pass filter passes signals with frequencies lower than a cutoff frequency and attenuates higher frequencies. A first order filter has one reactive component like a capacitor or inductor affecting its frequency response, while a second order filter has two reactive components. The document discusses low-pass filters including their transfer functions, Bode plots, pole-zero diagrams, and key terminology like Laplace transform, quality factor and decaying exponential response.

1. Unit 2 Electronic Circuits
Lesson 5 Low-pass Filters

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A low-pass filter is a filter that passes signals with a frequency
lower than a certain cutoff frequency and attenuates signals
with frequencies higher than the cutoff frequency.
Low-pass Filters
A cutoff frequency is a boundary in a system's frequency response
at which energy flowing through the system begins to be reduced
(attenuated or reflected) rather than passing through.

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Lesson 5 Low-pass Filters
• Backgrounds
• Text tour
• Language in use
– Vocabulary

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• Terminology
– Laplace transform
– Bode plot
– transfer function
– complex plane
– decaying-exponential response
– quality factor
Backgrounds

6. Laplace Transform
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The Laplace transform is a frequency-domain approach for
continuous time signals irrespective of whether the system is
stable or unstable.

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Text tour
• Outline
– First-Order Filters
• A simple RC integrator
• A simple RC low-pass filter
– Second-Order Low-Pass Filters
• An RLC low-pass filter
• The variables

8. • A first order filter would have one capacitor
or one inductor, that affects the filters
frequency response.
• A second order filter would have two
capacitors or two inductors or one capacitor
and one inductor, that affects the filter's
frequency response.
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First-Order Filters (1)
• A simple RC integrator
• Transfer function
Figure 2.1a A simple RC integrator
0
1
OUT C
IN
V X sC
V R R s

  

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First-Order Filters (1)
Figure 2.1b A Bode plot of a simple integrator
• Bode plot

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First-Order Filters (2)
Figure 2. 2a. A simple RC low-pass filter
0
0
1 1
1 1
OUT
IN
V sC
V R sC sRC s


  
  
• A simple RC low-pass filter
• Transfer function
Example:

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First-Order Filters (2)
Figure 2.2b. The complex function of an RC low-pass filter
• The complex function

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First-Order Filters (2)
Figure 2.2c. A Bode plot of a low-pass filter
(gain- frequency magnitude response)
• Bode plot
Power gain is shown in decibels (i.e. a 3 dB decline reflects an
additional half-power attenuation).

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Second-Order Low-Pass Filters
Figure2. 3a An RLC low-pass filter
 
2
1
1
1
1
C
L C
X
f s
R X X
sC
R sL sC
LCs RCs

 

 

 
 
2
0
2 2
0 0
f s
s s Q

 

 
• A simple RLC low-pass filter
• Transfer function

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Second-Order Low-Pass Filters
Figure 2.3b A pole-zero diagram of an RLC low-pass filter
• Pole-zero diagram: A pole-zero plot shows the location in
the complex plane of the poles and zeros of the transfer function of a
dynamic system, such as a controller, compensator, sensor, equalizer,
filter or communications channel.

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Pole zero diagram example

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Second-Order Low-Pass Filters
Figure 2.4a. The complex function of a second-order low-pass filter (Q = 0.707)

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Second-Order Low-Pass Filters
Figure 2.4b The complex function of a second-order low-pass filter (Q = 2)

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• Vocabulary
– variable, transient , function , factor, origin
Language in use

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Vocabulary

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Variable in dictionary
• A quantity that can assume any of a given set of values.
• Any factor or condition which can be measured, altered, or
controlled (e.g. temperature, pressure, flow, liquid level,
humidity, weight, chemical composition, color etc.).
• In programming, a character or group of characters that
refers to a value and in the execution of a computer
program, corresponds to an address.

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Variable in text
• You can easily derive the transfer function as
where s is the complex-frequency variable σ+ jω and ω0
is 1/RC.
0
1
OUT C
IN
V X sC
V R R s

  

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Variable in use
• complex variable
• n-dimensional random variable
• normal random variable
• real variable
• scalar variable
• time variable

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Transient in dictionary
• Lasting only for a short time; impermanent:
• Remaining in a place only a brief time:
• Decaying with time, especially as a simple exponential
function of time.
• A transient phenomenon or property, especially a transient
electric current.

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Transient in text
• While the complex frequency's imaginary part (j) helps
describe a response to AC signals, the real part () helps
describe a circuit's transient response.

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Transient in use
• damping transient
• forward switching transient
• high-speed transients
• initial transient
• starting transient

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• A relationship or expression involving one or more variables
• A specific purpose of an entity, or its characteristic action.
• In data communication, a machine action such as carriage return or
line feed.
Function in dictionary

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Function in text
• As before, Figure 2.4a shows the function as a curved
surface, depicted in the three-dimensional space formed by
the complex plane and a vertical magnitude vector.

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• normal probability function
• orthonormal function
• self-correlation function
• trigonometrical function
• unbounded function
• unit step function
• zero Bessel function
Function in use

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Factor in dictionary
• One that actively contributes to an accomplishment, a
result or a process
• One who acts for someone else; an agent.
• A person or firm that accepts accounts receivable as
security for short-term loans
• One of two or more quantities that divides a given quantity
without a remainder
– 2 and 3 are factors of 6; a and b are factors of ab.
• A quantity by which a stated quantity is multiplied or
divided, so as to indicate an increase or decrease in a
measurement
– The rate increased by a factor of ten.

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Factor in text
• The second-order function decreases twice as fast,
however, because of the s2 factor in the denominator.

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Factor in use
• Duty actor
• Equation factor
• Fidelity factor
• Greatest common factor
• Prime factor
• Scale factor

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Origin in dictionary
• The point at which something comes into existence or from
which it derives or is derived.
• Ancestry:
• The fact of originating; rise or derivation:
• The point of intersection of coordinate axes, as in the
Cartesian coordinate system.

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Origin in text
• Varying 0 changes the poles' distance from the origin.

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Origin in use
• antenna origin
• band origin
• coal origin
• grid origin
• infrared origin
• noise origin
• non-domestic origin
• program origin

38. Thank You
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