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電子回路論第2回Electric Circuits for Physicists #2
東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo Katsumoto
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Review
Metals: super-screening material
(but not superconducting. The difference is important in designing
superconducting circuits.)
Local electromagnetic field
→ Lumped constant circuits (集中定数回路)
local magnetic fields (parts) are connected by metallic wires
→ Circuit diagrams
Resistors, Capacitors and Inductors
Ch.1 Electromagnetic field and electric circuits
Ch.2 Introduction to linear response systems
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Outline
1. Transfer function (伝達関数) (continued)
2. Representative passive devices in the linear treatment
3. Impedance, admittance and other parameters in the linear treatment
4. Power sources
5. Circuit networks
6. Four terminal (two terminal-pair) circuits
7. Circuit theorems
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Linear response: Transfer function
Transfer function
output input
impulse response
This is a consequence of linear response: No mode-mode coupling
: impulse response
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Extension to the complex plane
Laplace transform:
We can expect this integral should converge for the responses because
of the causality.
Ξ(𝑠) is also called transfer function.
Expansion to the complex plane:
On the imaginary axis (the frequency space)
This style is often adopted in the field of electronics.
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Impedance
Assignment of physical quantities to input and output for three representative passive elements.
input : current 𝐼12 output: voltage 𝑉12
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Impedance as a transfer function
Transfer function
in w space
impedance
Impedance can be extended to the complex plane.
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Admittance
input : voltage 𝑉12 output: current 𝐼12
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Example of equivalent circuit
Spring pendulumk
m
F(t)
x
𝑅
L
C
V(t)
q
RLC circuit with
electromotive force
Electric circuits can simulate a wide range of classical
dynamical systems.
Parallelism
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2.2 Power sources
An active device: electric power source, electromotive force
Realistic power source can be simulated with
ideal power source + non-ideal factors (simulated by circuit elements)
Ex) ideal voltage source: keeps the voltage to the specified value whatever
load is connected.
Simulated as limits of passive elements:
𝑉
Power (energy) sources:
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2.2 Power sources
There are some voltage drops in real voltage sources when finite currents
are extracted.
Ideal voltage source
+resistor𝐽 =
𝑉
𝑅 + 𝑅𝐿𝑅𝑜𝑢𝑡
𝑅𝐿𝑉𝑜𝑢𝑡 =
𝑅𝐿𝑅𝐿 + 𝑅
𝑉
𝑅𝐿: Load resistance, 𝑅𝑜𝑢𝑡: Output resistance (impedance)
Ideal current source
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Power consumption
𝑉0 𝜔′𝑍(𝜔′)
Alternating current (AC)
voltage source:
Energy dissipation per unit time: 𝑃 = 𝑉𝐽 Power consumption
J𝑉0𝑒
𝑖𝜔′𝑡
complex instantaneous power
time average
Effective power (有効電力)
Reactive power (無効電力)
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Power consumption (2)
Apparent power (皮相電力)
Moment (力率)
𝜙: Phase shift between voltage and current
: generally holds
: effective value
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2.3 Circuit network
2.3.1 Kirchhoff’s law
At all nodes
Kirchhoff’s first law
↑ Charge conservation
=0
For all looping paths
↑ Single-valuedness of electric potential
Kirchhoff’s second law
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Kirchhoff’s law (2)
From Kirchhoff’s law, synthetic admittance and impedance are
for parallel connection:
for series connection:
𝑉0 𝜔
𝐶 𝑅
Ex.)
: Dissipation factor
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2.3.3 Superposition theorem
Network: node, (directional) branch : directional graph (digraph)
All the branches: electromotive force 𝐸𝑖, resistance 𝑅𝑖
Superposition theorem:
The total current distribution is the superposition of those for
single electromotive forces.
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2.3.4 Ho (鳳) – Tevenin’s theorem
Pick up two nodes in the network under consideration.
The voltage between these two nodes is 𝑉0.
Set all the electromotive forces to zero and measure the resistance
between the two nodes. Assume that the result is 𝑅0.
Now connect the two nodes with resistance 𝑅 and reset the electromotive
forces to the original values. Then the current through resistance 𝑅 is
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2.3.5 Tellegen’s theorem
𝑖 = 1,⋯ , 𝑛: index of nodes, 𝑗 = 1,⋯ ,𝑚: index of branches
incidence matrix
: redundancy in 𝑎𝑖𝑗 → (𝑛 − 1) × 𝑚 matrix 𝐷 :
irreducible incidence matrix
Consider network with n nodes, m branches.
Kirchhoff’s first law: 𝐷𝑱 = 0 𝐽𝑗: current along branch j
vector 𝑾: 𝑊𝑖 electrostatic potential of node i,
vector 𝑽: 𝑉𝑗 voltage across branch j
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2.3.5 Tellegen’s theorem (2)
𝑗𝑖 𝑘
(Kirchhoff’s second law)
Tellegen’s theorem
No assumption on • Species of elements
• Linearity of the circuits
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4-terminal (2-terminal pair) circuits
Terminal pair (端子対)
Current: circulation, no net current
J
J
2-terminal pair (4-terminal) circuit
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F-matrix of 4-terminal circuit
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Impedance matrix, Admittance matrix
Impedance matrix
Admittance matrix
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Examples with impedances
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Connections of 4-terminal circuits
Cascade
Series Parallel
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Theorems for terminal-pair circuits
Superposition theorem
Ho-Thevenin’s theorem
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Norton’s theorem
𝑉 =𝐽𝑠
𝑌 + 𝑌𝑖
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Duality 双対性
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Duality
Series Parallel
Open Short
Voltage Current
Impedance Admittance
Capacitance Inductance
Electric field Magnetic field
Resistance Conductance
Ho-Thevenin Norton
Kirchhoff’s 2nd law Kirchhoff’s 1st law
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Power Sources in Lab. 電源の雑知識
AC Power from distribution board 配電盤からの電力供給
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AC Power from distribution board 配電盤からの電力供給
Single-phase 2-wire
Single-phase 3-wire
Three-phase 3-wire
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Japanese outlet tap definition 日本式コンセント
Ground 接地線
Cold line 中性線
Hot line 電圧線
検電ドライバーElectroscopic
Screwdriver
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DC Stabilized Power Supply 直流安定化電源Series (Dropper) regulation
From TDK web page
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Series regulator power supply
Uni-polar Dual tracking
High precision Bi-polar current source
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Switching regulation
From TDK
web page
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Switching regulator power supply
Molecular beam epitaxy
Control panel
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Bin 電源ビン
Complicated power lines Bin
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Exercise A-1
Express J1, J2, J3 with other parameters.
𝑅1 𝑅2 𝑅3
𝐽1 𝐽2 𝐽3
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Exercise A-2
R
All the resistors have the
same resistance R.
Obtain the combined
resistance.
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Exercise A-3
V(t)
t
Obtain the effective value of voltage for the saw tooth wave.
0
V0
……