Lecture 30
• Review:
• First order highpass and lowpass filters
• Cutoff frequency
• Checking frequency response results
• Time-to-frequency domain relations for first
order filters
• Bode plots
•Related educational modules:
–Section 2.8.2, 2.8.3
Filters
• Circuits categorized by their amplitude response
– Filters “pass” some frequencies and “stop” others
– Lowpass filters pass low frequencies
– Highpass filters pass high frequencies
• Filters are also categorized by their order
– The filter order corresponds to the order of the circuit’s
governing differential equation
• Circuits I: only first order low- and highpass filters
First order filters
• Review: Low- and highpass filter response examples
• Cutoff frequency separates the pass- and stopbands
– Where the amplitude response is
1
2
times the max amplitude
• Annotate previous slide to show cutoff
frequencies
– Note that cutoff frequency is related to maximum
gain of filter
• Change max gain on figure (write in) => gain at corner
frequency changes
Time vs. frequency domain characterization
• In the time domain, we characterized first order
circuits by their step response
– DC gain, time constant ( )
• We can likewise characterize first order circuits by
their frequency response
– DC gain, cutoff frequency (c)
• On previous slide, sketch step response, show
DC gain
• Then show that cosine input becomes a step
input as frequency -> zero
Time-to-Frequency domain relations – cont’d
• Governing differential equation for first order system:
• Converting to frequency domain:
1
Þ
Y
a1
H ( j ) 

U
j  1

1
a1
H ( j ) 

2
  2
 1
• On previous slide, show:
– (1) time domain DC gain, tau
– (2) conversion to frequency response =>
u(t)=Ue^(jwt), etc.
– (3) freqency response -> magnitude response
– (4) DC gain and cutoff frequency from mag. Resp.
Checking amplitude responses
• We can (fairly) easily check our amplitude
responses at very low and very high frequencies
• Capacitors, inductors replaced by open, short
circuits
– Results in purely resistive network
– Analyze resistive network, and compare result to
amplitude response
– Provides “physical insight” into low, high frequency
operation
Inductors at low, high frequencies
• Inductor impedance:
Z L  j L
• =0Þ
–
ZL  0
Inductor behaves like short circuit at low frequencies
• Þ
ZL  
– Inductor behaves like open circuit at high frequencies
Capacitors at low, high frequencies
• Capacitor impedance:
ZC 
• =0Þ
–
1
j C
ZC  
Capacitor behaves like open circuit at low frequencies
• Þ
ZC  0
– Capacitor behaves like short circuit at high frequencies
Example – checking amplitude response
• What is the amplitude response of the circuit below as 0
and  ?
Bode plots – introduction
• We have used linear scales to plot frequency responses
• Selective use of logarithmic scales has a number of
advantages:
– Amplitudes and frequencies tend to span large ranges
– Logarithms convert multiplication and division to addition and
subtraction
– Human senses work logarithmically
• Bode plots use logarithmic scales to simplify plotting of
frequency responses and interpretation of plots
• Do demos on previous slide
– Note large ranges of frequencies; sensitivity to
frequency ranges
– Note human senses work logarithmically
Nomenclature relative to log scales
• Logarithmic scales convert multiplicative factors to
linear differences
• Some of these multiplicative factors have special
names
– A factor of 10 change in frequency is a decade difference
on a log scale
– A factor of 2 change in frequency is an octave difference
on a log scale
Bode Plots
• Bode plots are a specific format for plotting
frequency responses
– Frequencies are on logarithmic scales
– Amplitudes are on a decibel (dB) scale
H ( j )
dB
 20 log 10  H ( j  ) 
– Phases are on a linear scale
Decibels
• Named for Alexander Graham Bell
• Common gains and their decibel values:
10
1
0.1
0.5
1
2

0dB


 -3dB
20dB
-20dB
-6dB
Bode plots of first order filters
• Frequency response for typical first order lowpass
filter:
H ( j ) 
Y

U
c
j   c
• Magnitude, phase responses:
H ( j ) 
c

2
 H ( j  )   tan
2
 c
1  



 c 
Bode plot – magnitude response
H ( j ) 
c

2
2
 c
• Annotate previous slide to show asymptotic
behavior, -3dB point at wc, intersection of
asymptotes is at wc
• Note that I can muliply this by a gain, without
affecting the shape of the curve – it simply
moves up and down
Bode plot – phase response
 H ( j  )   tan
1  



 c 
• Annotate previous slide to show asymptotic
behavior, -45 degrees at wc
Example
• Sketch a Bode plot for the circuit below.