電子回路論第4回

Electric Circuits for Physicists #4

東京大学理学部・理学系研究科物性研究所勝本信吾

Shingo Katsumoto

Review

Last week we introduced

Resonance: Represented as a zero-point in impedance. A pole in admittance.

Bode diagram: Plots of the absolute value and the argument of a transfer

function as a function of frequency.

A Pole on the Real Axis

Circuit example: Z of

R

C

Lear system with

transfer function

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 100 1000

0

0.5

1

1.5

𝑊(𝑖𝜔)

−arg[𝑊

𝑖𝜔]

𝜋/2

𝜔

𝜔 ≫ 1

−1

s𝑖𝜔

i

𝜔 = 0

𝜔 = 1

arg(𝑊)

~𝑠−1

(dimensionless)

arrows: s +1

A Pole with Finite Imaginary Part

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 100 1000

-1.5

-1

-0.5

0

0.5

1

1.5

𝑊(𝑖𝜔)

−arg[𝑊

𝑖𝜔]

𝜔

𝜔0 = 1

𝜔0 = 1

10

100

10

100 −1

s 𝑖𝜔

𝜔0

Corresponds to resonance.

(This approximately holds for 𝜔~𝜔0)

(*1)

(*1 Complex poles in transfer function always

appear as conjugate pairs.)

Zeros and Poles of Transfer Function

Bode plot

(generally represented in rational formula)

Bode plots are sum of those for single poles and zeros.

3.2 Terminal pair circuits as signal

transfer interfaces

3.2.1 Impedance matching

𝑉𝑜𝑢𝑡

𝑍𝑜𝑢𝑡

𝑉0

𝑍

𝐽

Hence the maximum power transfer

can be obtained under the impedance matching condition:

Impedance matching condition

Efficient transfer of energy from electro-motiveforce

When Z is real: 𝑍 = 𝑍out

3.2.3 Image parameters (matching with 4-terminal circuits)

𝑍1 𝑍2𝑍1 𝑍2

If the 4-terminal parameters can be tuned to give the situation as

𝑍1 𝑍1 𝑍2𝑍2

That is

𝑍1 and 𝑍2 are called image impedances.

Conditions for image parameters

𝐴 𝐵𝐶 𝐷

𝑉1 𝑉2

𝐽1 𝐽2

𝑍2

𝑍1

From definition:

Image (mirror)

condition

When 𝐴𝐵𝐶𝐷 ≠ 0,

Image parameters

𝜃: Image propagation constant

The 4-terminal interface costs some power loss:

Image parameters

𝑍1, 𝑍2, 𝜃: Image parameters

3.2.4 Impedance matching with two terminal-pair circuits

R

𝐴 = 𝐷 = 0

𝐴𝐵𝐶𝐷 ≠ 0

𝐵 = 𝐶 = 0

R

n

Matching transformer

3.2.5 Fidelity and distortion in wave transformation

When a linear response is just a time delay with 𝜏0, that is

No distortion conditions are:

(1) 𝐴0 does not depend on frequency: No filter effect

In other words the conditions can be described as

(2) :group delay

No dispersion in group delay

Effect of distortion in amplitude

(1) Sinusoidal amplitude distortion (amplitude modulation)

Paired echo

Effect of distortion in group delay

Sinusoidal group delay distortion

Paired echo with inverted sign

Distortion (paired echo)

Cosine Amplitude

Distortion

Sine Delay Distortion

3.2.6 Filter Circuit

Transmission

wLow pass filter

Transmission

w

Band pass

filter

Transmission

w

High pass filter

Transmission

w

Notch filter

Terms for Filters

Tra

nsm

issi

on o

r G

ain

𝐴(𝜔)

Pass Band

Stop Band

(Transient Band)

Cut-off Frequency

𝐴(𝜔) is sometimes

called as “gain”

Transmission

Can an ideal filter exist?tr

ansm

issi

on

𝐴0𝜔

𝜔0

Ideal low pass filter

Transfer function is written as Heaviside function

-10 -5 0 5 10

0

0.5

1

x

sinc(x)

w(t): output to d-function input at t =0

The output begins 𝑡 < 0 hence breaks the causality.

No ideal low pass filter can exist.

Transmission

𝐴 𝐵𝐶 𝐷

𝑉1 𝑉2

𝐽1 𝐽2

𝑍2

Voltage transmission

coefficient:

Square root power transmission coefficient

attenuation phase shift

Sometimes called as “gain”

An example: Constant K type filter

0 0.5 1 1.5 2

:constant

L

C𝑍1 𝑍2

definition

Image parameters are

Butterworth Filter

10-3 10-2 10-1 100 101 102 10310-3

10-2

10-1

100

𝜔

Tra

nsm

issi

on

𝑛 = 1

24

9

16

Bessel Filter

Inverse Bessel Polynomial

Chebyshev Filter

0 102 4 6 81 3 5 7 9

0

1

0.2

0.4

0.6

0.8

0.1

0.3

0.5

0.7

0.9

n =13

5

3

n =11e = 0.2

0.40.6

e = 0.2

𝜀: Ripple coefficeint

𝑇𝑛: n-th order Chebyshev polynomial

Packaged Filters

Mini-Circuits

Band Pass

19.2 – 23.6MHz 50Ohm

Web selection page

https://ww3.minicircuits.com/WebStore/RF-Filters.html

3.3 Transient response of circuits

composed of passive elements

3.3.1 Classification with the number of storages

(a) Single energy storage

(b) Double energy storage

Number of storages: order of polynomial in the

denominator of rational expression for transfer

function

Ex) Transient response to step function input

𝑉0

t

Heaviside function

Fourier transform

𝑢(𝑡)

R

C 𝑤(𝑡)

Responses are obtained as

Ex)

Ex) Transient response to step function input

From the initial condition and

renormalization of t we obtain

input

output

𝑉0

t

Introduction of freeware: Scilab

https://www.scilab.org/

MATLAB clone but free!

Graphical linear response

analysis: Scicos

Convenient set of commands

for linear response analysis

Transfer function analysis with Scilab

Transient response: Use of Scilab

Transient response: Use of Scilab

R L