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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