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 materials:

– Chapter 11.3, 11.4

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 times the max amplitude2

1

• 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

11

j

aUY

)j(H Þ 22

1

1

1

a)j(H

• 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:

• = 0 Þ– Inductor behaves like short circuit at low frequencies

• Þ– Inductor behaves like open circuit at high frequencies

LjZL

0LZ

LZ

Capacitors at low, high frequencies• Capacitor impedance:

• = 0 Þ– Capacitor behaves like open circuit at low frequencies

• Þ– Capacitor behaves like short circuit at high frequencies

CjZC

1

CZ

0CZ

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

– Phases are on a linear scale

)j(Hlog)j(HdB

1020

Decibels

• Named for Alexander Graham Bell

• Common gains and their decibel values:10 20dB1 0dB0.1 -20dB0.5 -6dB

-3dB21

Bode plots of first order filters

• Frequency response for typical first order lowpass filter:

• Magnitude, phase responses:

ctan)j(H

1

c

c

jUY

)j(H

22c

c)j(H

Bode plot – magnitude response

22c

c)j(H

• 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

ctan)j(H

1

• Annotate previous slide to show asymptotic behavior, -45 degrees at wc

Example• Sketch a Bode plot for the circuit below.