Department of EECS University of California, Berkeley

EECS 105 Fall 2003, Lecture 2

Lecture 3: Bode Plots

Prof. Niknejad

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Get to know your logs!

Engineers are very conservative. A “margin” of 3dB is a factor of 2 (power)!

Knowing a few logs by memory can help you calculate logs of different ratios by employing properties of log. For instance, knowing that the ratio of 2 is 3 dB, what’s the ratio of 4?

dB ratio dB ratio-20 0.100 20 10.000-10 0.316 10 3.162

-5 0.562 5 1.778-3 0.708 3 1.413-2 0.794 2 1.259-1 0.891 1 1.122

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Bode Plot Overview

Technique for estimating a complicated transfer function (several poles and zeros) quickly

Break frequencies :

)1()1)(1(

)1()1)(1()()(

22

210

pmpp

znzzK

jjj

jjjjGH

ii

1

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Summary of Individual Factors

Simple Pole:

Simple Zero:

DC Zero:

DC Pole:

j1

1

j1

j

j1

1

dB0

dB0

dB0

dB0

90

90

90

90

1

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Example

Consider the following transfer function

Break frequencies: invert time constants

)1)(1(

)1(10)(

31

25

jj

jjjH

ps100

ns10

ns100

3

2

1

Grad/s10Mrad/s100Mrad/s10 321

)1)(1(

)1(10

)(

31

25

jj

jj

jH

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Breaking Down the Magnitude

Recall log of products is sum of logs

Let’s plot each factor separately and add them graphically

)1)(1(

)1(10

log20)(

31

25

dB

jj

jj

jH

31

25

1log201log20

1log2010

log20

jj

jj

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Breaking Down the Phase

Since

Let’s plot each factor separately and add them graphically

)1)(1(

)1(10)(

31

25

jj

jjjH

baba

31

25

11

110

)(

jj

jj

jH

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Magnitude Bode Plot: DC Zero

80

20

60

40

-20

-60

-80

-40

104 105 106 107 108 109 1010 1011

510

j

0 dB

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Phase Bode Plot: DC Zero

180

45

135

90

-45

-135

-180

-90

104 105 106 107 108 109 1010 1011

510

j

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Magnitude Bode Plot: Add First Pole

80

20

60

40

-20

-60

-80

-40

104 105 106 107 108 109 1010 1011

dB510

j

dB710

1

1

j

Mrad/s101

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Phase Bode Plot: Add First Pole

180

45

135

90

-45

-135

-180

-90

104 105 106 107 108 109 1010 1011

510

j

7101

1

j

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Magnitude Bode Plot: Add 2nd Zero

80

20

60

40

-20

-60

-80

-40

104 105 106 107 108 109 1010 1011

dB810

1j

Mrad/s1002

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Phase Bode Plot: Add 2nd Zero

180

45

135

90

-45

-135

-180

-90

104 105 106 107 108 109 1010 1011

8101

j

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Magnitude Bode Plot: Add 2nd Pole

80

20

60

40

-20

-60

-80

-40

104 105 106 107 108 109 1010 1011

dB1010

1

1

j

Grad/s103

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Phase Bode Plot: Add 2nd Pole

180

45

135

90

-45

-135

-180

-90

104 105 106 107 108 109 1010 1011

10101

j

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Comparison to “Actual” Mag Plot

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Comparison to “Actual” Phase Plot

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Why do I say “actual”?

I plotted the transfer characteristics with Mathematica

The range of frequency for the plot is 6 orders of magnitude. The program has to find the “hot spots” in order to plot the function. Near the hot spots, more points are plotted. In between hot spots, the function is interpolated. If you pick the wrong points, you’ll end up with the wrong plot:

mag = LogLinearPlot[20*Log[10, Abs[H[x]]], {x, 10^4, 10^11},PlotPoints -> 10000, Frame -> True,PlotStyle -> Thickness[.005], ImageSize -> 600,GridLines -> Automatic, PlotRange -> {{10^4, 10^11}, {-20, 100}} ]

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Don’t always believe a computer!

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Second Order Transfer Function

The series resonant circuit is one of the most important elementary circuits:

The physics describes not only physical LCR circuits, but also approximates mechanical resonance (mass-spring, pendulum, molecular resonance, microwave cavities, transmission lines, buildings, bridges, …)

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Series LCR Analysis

With phasor analysis, this circuit is readily analyzed

RICj

ILjIVs

1

RR

CjLj

VRIV

RCj

LjIV

s

s

1

1

0

+Vo

−

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Second Order Transfer Function So we have:

To find the poles/zeros, let’s put the H in canonical form:

One zero at DC frequency can’t conduct DC due to capacitor

RCj

Lj

R

V

VjH

s

1

)( 0+Vo

−

RCjLC

CRj

V

VjH

s

20

1)(

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Poles of 2nd Order Transfer Function

Denominator is a quadratic polynomial:

LR

jjLC

LR

j

RCjLC

CRj

V

VjH

s

22

0

)(11

)(

LR

jj

LR

jjH

220 )(

)(LC

120

Qjj

Qj

jH022

0

0

)()(

R

LQ 0

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Finding the poles…

Let’s factor the denominator:

Poles are complex conjugate frequencies The Q parameter is called the

“quality-factor” or Q-factor This parameters is an important

parameter:

Re

Im

0)( 20

02 Q

jj

Q

jQQQ 4

11

242 002

0

200

0RQ

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Resonance without Loss

The transfer function can parameterized in terms of loss. First, take the lossless case, R=0:

When the circuit is lossless, the poles are at real frequencies, so the transfer function blows up!

At this resonance frequency, the circuit has zero imaginary impedance

Even if we set the source equal to zero, the circuit can have a steady-state response

Re

Im

020

200

42 j

QQQ

EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad

University of California, BerkeleyDepartment of EECS

Magnitude Response

The response peakiness depends on Q

Qj

Qj

LR

j

LR

j

jH022

0

0

0

0220

0

0

)(

1Q

10Q

100Q

0

1)(0

020

20

20

0

Qj

Qj

jH

1)( 0 jH

0)0( H