Electric Circuits for Physicists
2016
Shingo Katsumoto

2016年度
電子回路論 第１回

東京大学
理学部・理学系研究科
物理学専攻
物性研究所
勝本信吾

ノート・資料等の置き場
http://kats.issp.u-tokyo.ac.jp/kats/

2週に１回簡単な練習問題を出題→

2週間以内に解答を提出
TAが採点してコメントをメール送付します
試験は期末レポート．練習問題と合わせて採点します

シラバス
1.

2.

電磁場と電子回路
1.1 この講義について
1.2 電子回路とは
1.3 ２端子素子
1.4 回路図
1.5 抵抗器1.6 キャパシタ
1.7 インダクタ
線形回路序論2.1 線形システムと電子回路
2.2 電源2.3 回路網
2.4 4端子(2端子対)回路
2.5 端子対回路の諸定理
2.6 双対性
2.7 受動素子と能動素子

3.

伝達関数と周波数応答・過渡応答
3.1 受動素子2端子回路の伝達関数
3.2 ２端子受動素子回路
3.3 受動素子回路の過渡応答

4.

増幅回路
4.1 増幅回路と系の制御4.2 OPアンプ
4.3 トランジスタ
4.4 電場効果トランジスタ

５．分布定数回路
5.1 伝送路
5.2 伝送路の伝播現象
5.3 S行列(Sパラメタ)
5.4 シュレディンガー方程式とLC伝送路

シラバス２6.

信号，雑音，波形解析
6.1 ゆらぎ
6.2 増幅器の雑音
6.3 変調とアナログ信号伝送
6.4 離散化信号

7.

ディジタル信号とディジタル回路
7.1 ディジタル信号序論
7.2 論理ゲート
7.3 論理ゲートの実装
7.4 論理演算の回路化と簡単化
7.5 A-D/D-A コンバータ
7.6 ディジタルフィルター
7.7 ハードウェア記述言語：HDL

Syllabus
1. Electromagnetic field and electric circuits
1.1 About this lecture
1.2 What is electric circuit?
1.3 Two-terminal devices
1.4 Circuit diagrams
1.5 Resistors
1.6 Capacitors
1.7 Inductors
2. Introduction to linear circuits
2.1 Linear systems and electric circuits
2.2 Power sources
2.3 Networks
2.4 4-terminal (2-terminal pair) circuits
2.5 Theorems in terminal pair circuits
2.6 Duality
2.7 Passive devices, active devices

Syllabus
3. Transfer function and transient response
3.1 Transfer function of passive two-terminal pair circuits
3.2 Two-terminal passive circuits
3.3 Transient response of passive circuits
4. Amplifiers
4.1 System control and amplifiers
4.2 Operational amplifiers
4.3 Transistors
4.4 Field effect transistors
5. Distributed constant circuits
5.1 Transmission lines
5.2 Propagation through transmission lines
5.3 S matrix (S parameters)
5.4 Schrodinger equation and LC transmission circuit

Syllabus
6. Signal, noise, waveform analysis
6.1 Fluctuation
6.2 Noise from amplifiers
6.3 Modulation and analog signal transfer
6.4 Discrete signal
7. Digital signal and digital circuits
7.1 Introduction to digital signal
7.2 Logic gates
7.3 Logic circuits implementation
7.4 Circuit implementation and simplification of logic
operation
7.5 AD/DA converters
7.6 Digital filters
7.7 Language to describe hardware: HDL

Outline Today
1. Electromagnetic field and electric circuits
1.1 About this lecture
1.2 What is electric circuit?
1.3 Circuit diagrams
1.4Two-terminal devices
1.5 Resistors
1.6 Capacitors
1.7 Inductors
2. Introduction to linear circuits
2.1 Linear systems and electric circuits

Ch.1 Electromagnetic
field and electric circuits

For what this lecture is?
Experimentalists:
Knowledges on electric circuit are indispensable.
Purposes:
Understand how circuits work.
Design circuits along research plans
General physicists:
Meta physics

Coupled pendulum and neutrino oscillation

Pendulum oscillationtimeElectric Circuit: A treasure house of concept
and language

Electric
CircuitElectromagnetic
Field

SignalNoise

Lumped constantCircuitDistributed
constant
Circuit

Modulation
Discrete signal

Material
Science
Metal
Semiconductor
Ferromagnet

Linear response

System stability

Analog and digital

Transfer function

AmplifierFourier tr.- z-tr.

Resonance

Feedback

Transient response

Nyquist diagam

Analog filter- Digital filter1.2 What is electric circuits?: Thunderbolt struck a plane!

Your answer?
1.

2.

Fell down and crashedDamagedNothing happened3.

Plasma frequency
+
+
+

-

𝑥

Electric polarization:: Plasma frequency
Cu: 𝑛 = 8.5 × 1022 /cc𝑚∗ = 1.3𝑚0
l = 130nm Near ultra-violet

Metals are super-screening materials!

Equipotential lines1.3 Various circuit diagrams

Picture diagram

三田無線研究所
DELICA DX-CS-7

Various circuit diagrams
Parts + Wiring= Wiring (Circuit) diagram

Block diagramWirings in electric circuits

Connected

Not connected

Not connected

Violate electromagnetism theory
Concept of local electromagnetic field= Lumped constant circuits (集中定数回路)

Circuit symbols for two-terminal devices

Basic concepts in electric circuits

What is current-voltage characteristics?

Resistor

What is current-voltage characteristics?
Curve tracer

Variable resistors

Slide
(a)

(b)

(d)

Helicalpotentiometer

Rotary switch
potentiometer

Resistance

Carbon helical(c)

Rotation angle
Cermet trimmer
trimmerFixed resistors

Chip resistors

Carbon film resistors

Metallic film resistors (meander)

Metallic film resistors (spiral)

High power type

抵抗器のカラーコード

色

第１数字 第２数字 第３数字 第４数字 第５数字

黒

0

0

0

100

X

茶

1

1

1

101

±1%

赤

2

2

2

102

±2%

橙

3

3

3

103

X

黄

4

4

4

104

X

緑

5

5

5

105

X

青

6

6

6

106

X

紫

7

7

7

107

X

灰

8

8

8

108

X

白

9

9

9

109

X

金

X

X

X

10-1

±5%

銀

X

X

X

10-2

±10%

無色

X

X

X

X

±20%

Color code for resistors

Big
boys
race
our
young
girls
but
Violetgenerallywins.

Variable capacitors

Steatite

TandemAir capacitor

Poly-Ethylene

Ceramic

Fixed capacitors
Ceramic capacitors
Stacked type

Single disk pair type

Film capacitors

誘導

無誘導

蒸着無誘導

Chemical capacitors

タンタル電解Surface mount chip capacitors

Variable inductors

Fixed inductors

Open flux path

Closed flux path

Toroidal coil
Chip inductor

Ch.2 Introduction tolinear circuits

2.1 Linear system and electric circuit
2.1.1 What is linear system?

Input(Cause)SystemOutput(Result)

Linear system: definition: Response
Requirements
Invariance:
Causality:
Principle of superposition:

Transfer function
1
Variable 𝑞 → time t

𝑐(𝑡)

−𝛿𝑡/2

𝛿𝑡/2

t

Transfer function (Impulse response): Impulse response, weight function

Fourier transform:

Convolution

Transfer function

Summary

1. Electromagnetic field and electric circuits
Metals: super screening material
Circuit diagrams
Local electromagnetic field
2. Introduction to linear circuits
Transfer function

電子回路論 第2回
理学系研究科・物理専攻 (物性研究所) 勝本信吾

2016.10.5

Electric Circuits No.2
Shingo Katumotoノート・資料等の置き場
http://kats.issp.u-tokyo.ac.jp/kats/

2週に１回簡単な練習問題を出題→ 2週間以内に解答を提出

TA
試験は期末レポート．練習問題と合わせて採点しますReview
Ch.1 Electromagnetic field and electric circuits
Metals: super-screening material(but not superconducting. The difference is important indesigning superconducting circuits.)

Local electromagnetic field→ Lumped constant circuits (集中定数回路)
local magnetic fields (parts) are connected bymetallic wires → Circuit diagrams

Resistors, Capacitors and Inductors
Ch.2 Introduction to linear response systems

Outline Today
1. Transfer function (伝達関数) (continued)
2. Representative passive devices in the linear

treatment
3. Impedance, admittance and otherparameters in the linear treatment
4. Power sources
5. Circuit networks

6. Four terminal (two terminal-pair) circuits
7. Circuit theorems

Linear response: Transfer function
response

input

impulse response

Transfer function
Laplace:Expansion to the complex plane:
On the imaginary axis (the frequency space)Impedance
Current to voltage

resistor
capacitor
inductor

Impedance 𝑍(𝑖𝜔)Admittance
Voltage to current

Admittance 𝑌(𝑖𝜔)Example of equivalent circuit 「等価回路」の例
Spring pendulum (ばね振り子)

x

k

𝑑 2𝑥
𝑑𝑥
𝑚 2 +𝑏+ 𝑘𝑥 = 𝐹(𝑡)
𝑑𝑡
𝑑𝑡

m
F(t)
𝑅

RLC circuit with electromotive force(電源を接続したRLC回路)

V(t)

𝑑2𝑞
𝑑𝑞 1
𝐿 2 +𝑅+ 𝑞 = 𝑉(𝑡)
𝑑𝑡
𝑑𝑡 𝐶

L
C

Parallelismq

並行論2.2 Power sources
An active device: electric power source, electromotive force
Realistic power source: ideal power source + non-ideal factors
Ideal voltage source

𝑉

Voltage source+resistor

Ideal current source

R

𝑅𝐿

𝑉
𝐽=
𝑅 + 𝑅𝐿

𝑅𝐿
𝑉𝑜𝑢𝑡 =𝑉
𝑅𝐿 + 𝑅Power consumption
Energy dissipation per unit time: 𝑃 = 𝑉𝐽

𝑉0 𝜔′
𝑍(𝜔′)

Electric power consumption

Complex instantaneous power
Effective power(有効電力)
Reactive power(無効電力)Power consumption (2)Apparent power (皮相電力）

Moment (力率)𝜙: Phase shift between voltage and current: generally holds: effective value2.3 Circuit network

2.3.1 Kirchhoff’s law

At all nodes

Kirchhoff’s first law↑ Charge conservation=0
For all looping paths

Kichhoff’s second law↑ Single-valuedness of electric potential2.3 Circuit network (2)From Kirchhoff’s law, synthetic admittance and impedance arefor parallel connection:
for series connection:

Ex.)

𝑉0 𝜔′

𝐶

𝑅
: Dissipation factor

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 forsingle electromotive forces.

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 theresistance between the two nodes. The result is 𝑅0 .
Now connect the two nodes with resistance 𝑅 and reset theelectromotive forces to the original values. Then the current
through resistance 𝑅 is
𝐽 = 𝑉0/(𝑅 + 𝑅0 )

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
Kirchhoff’s first law: 𝐷𝑱 = 0 𝐽𝑗 : current along branch j

𝑾: 𝑊𝑖 electrostatic potential of node i, 𝑽: 𝑉𝑗 voltage across branch j𝑗𝑘𝑖(Kirchhoff’s second law)

4-terminal (2-terminal pair) circuits 4端子回路

Terminal pair (端子対)Current: circulation, no net current

JJ
2-terminal pair (4-terminal) circuit

F-matrix of 4-terminal circuitImpedance matrix, Admittance matrix

Examples with impedancesConnections of 4-terminal circuits
𝑁

Cascade𝐹tot =

𝐹𝑖
𝑖=1

Series𝑁

𝑍tot =

𝑍𝑖
𝑖=1

Parallel𝑁

𝑌tot =𝑌𝑖
𝑖=1

Theorems for terminal-pair circuits
Superposition theorem

Ho-Thevenin’s theorem

Norton’s theorem

𝐽𝑠
𝑉=
𝑌 + 𝑌𝑖Duality 双対性

Duality

SeriesOpen
Voltage
Impedance
CapacitanceElectric field
Resistance
Ho-Thevenin
Kirchhoff’s 2nd law

Parallel
Short
Current
AdmittanceInductance
Magnetic field
Conductance
NortonKirchhoff’s 1st law

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

AC Power from distribution board 配電盤からの電力供給
Single-phase 2-wireSingle-phase 3-wireThree-phase 3-wireJapanese outlet tap definition 日本式コンセント
Cold line 中性線

Hot line 電圧線

Ground 接地線
検電ドライバーElectroscopic
ScrewdriverDC Stabilized Power Supply 直流安定化電源
Series (Dropper) regulation

From TDK web pageSeries regulator power supply

Uni-polar

Dual tracking

High precision

Bi-polar current source

Switching regulation

From TDK
web pageSwitching regulator power supply
Molecular beam epitaxy
Control panel

Bin 電源ビン

Complicated power lines

BinExercise A-1
Express J1, J2, J3 with other parameters.

Exercise A-2

All the resistors have thesame resistance R. Obtainthe combined resistance.

RExercise A-3

Obtain the effective value of voltage for the saw tooth wave.

V(t)
V0

……

t
0

電子回路論第3回
Electric Circuits for Physicists
東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo Katsumoto電源の雑知識 (続き）
Miscellaneous knowledge
on power supplies (continued)

DC Stabilized Power Supply 直流安定化電源
Series (Dropper) regulation

From TDK web pageSeries regulator power supply

Uni-polar

Dual tracking

High precision

Bi-polar current source

Switching regulation

From TDK
web pageSwitching regulator power supply
Molecular beam epitaxy
Control panel

Bin 電源ビン

Complicated power lines

BinOutline Today
2.5 Theorems for paired terminal circuits
Superposition, Ho-Tevenin, Reciprocity
2.6 Duality
2.7 Passive devices (elements) and active devices
Ch.3 Transfer function and transient response
3.1 Transfer function of single-pair terminal circuits
Resonance circuit
Bode plot
General properties
Appendix B Bridges and balance circuits
Appendix C General properties of resonance circuits

Theorems for terminal-pair circuits
Superposition theorem

𝐽𝑖 : The current caused by i-th power source.

Ho-Thevenin’s theorem
3Consider a circuit with an open terminal pair (No.3). Obtain current J when the
open pair is connected with impedance Z.

1.
2.

Because:

𝑍
+
-

Then

Measure the open terminal voltage 𝑉0 .
Turn off all the power sources (voltage sources: short, current
sources: open). Measure the open circuit impedance 𝑍𝑖 .

𝐽𝑡 = 0+-𝑉0𝐽0 = −
𝑍 + 𝑍𝑖

𝑍
+
-

𝑉0

𝑉0

Norton’s theorem

𝐽𝑠
𝑉=
𝑌 + 𝑌𝑖
Dual theorem for Ho-Tevenin.

Comments on Tellegen’s theorem
𝑖 = 1, ⋯ , 𝑛: index of nodes, 𝑗 = 1, ⋯ , 𝑚: index of branches
incidence matrix
redundancy → (𝑛 − 1) × 𝑚 matrix 𝐷 : irreducible incidence matrix
𝐽𝑗 , 𝑉𝑗 : current and voltage along branch j, 𝑊𝑖 : potential of node i.
Kirchhoff’s first law: 𝐷𝑱 = 0

Second law:Power of i-th branchComments
1. Power conservation law
2. Holds for any kind of circuit (irrespective of linear, or non-linear)
3. Holds for two independent circuit conditions (as long as D is the same)

Reciprocity theorem
An 𝑛-terminal pair linear circuit
At one state
𝑉1 , 𝐽1 , 𝑉2 , 𝐽2 , ⋯ , (𝑉𝑛 , 𝐽𝑛 ),at another state
𝑉′1 , 𝐽′1 , 𝑉′2 , 𝐽′2 , ⋯ , (𝑉′𝑛 , 𝐽′𝑛 )Proof: Consider a two terminal-pair circuit with 𝑚 branches.

𝐽2

𝐽′1

𝑉2

𝑉′1

𝐽1
𝑉1

𝑉𝑘 = 𝑍𝑘 𝐽𝑘𝐽′2
𝑉′𝑘 = 𝑍𝑘 𝐽′𝑘Tellegen’s theorem(and comment no.3)//(This also holds for circuits with mutual inductances.)𝑉′2

2.6 Duality 双対性

2.6 Duality

SeriesOpen
Voltage
Impedance
CapacitanceElectric field
Resistance
Ho-Thevenin
Kirchhoff’s 2nd law

Parallel
Short
Current
AdmittanceInductance
Magnetic field
Conductance
NortonKirchhoff’s 1st law

2.7 Definition: Passive elements and active elements
Two terminal: current J, voltage V

𝐽𝑉 ≥ 0: passive element
𝐽𝑉 < 0: active element

Locally active two-terminal element

More than three-terminal: treat as a terminal pair circuit𝑃 ≥ 0: passive element
𝑃 < 0: active element

Ch.3 Transfer function andtransient response3.1 General Properties of Resonance and Resonance Circuits
3.1.1 Resonance Phenomena
R

Harmonic oscillator:
Kirchhoff's law

Resonant (angular) frequency

CLTransfer function, resonance and phase shift

Resonance: Reactance =0
Total Phase Shift Change: p
Im Z

Im Y

1

w

Z(iw)

w =0
R
w0

0
w2

(a)

w0

Re Z

Re Y

0

w

w

(b)

(c)

w
w1 0 w2

Bode diagram
104

R
𝑍/𝑅

102

100

L

10-2

1.5
1

arg (Z)

C

Q=10
5
2
1

Q=20
10
5

0.5
0

2

-0.5

1

-1
-1.5
0.001

0.01

0.1

1

10

100

Transient response of resonant circuit
R

C
V

L

V
𝑉0
t

Damped oscillation with time constant 𝛾 −1 , frequency 𝜔𝑠Transient response of resonance circuit (transfer function)
Synthesized impedance, admittanceZero (pole) of 𝑍tot (𝑠) (𝑌tot (𝑠))Time constant: Re(𝑠0 ) Frequency: Im(𝑠0 )Laplace transformation of voltage: 𝑉(𝑠)Natural current

𝑠𝑖 : poles of 𝑌(𝑠)Driving point impedance
R
C

L

R

Open
L

=

Short

Resistance bridge 抵抗ブリッジ
Wheatstone bridge

AVS-47 Resistance bridge
Not a “bridge” circuit!

Schering Bridge

Hartshorn bridge
Magnetic moment measurement
Resistance measurement

Capacitance bridge キャパシタンス ブリッジ

General Radio
3-terminal
Capacitance bridge

Agilent E4981A

Kondo Resonance and Phase shift
E
𝐸F

1
2

𝑠↑ 𝑑 ↓ − |𝑠 ↓ |𝑑 ↑Jun Kondo

Many body resonance.
But still has the phase shift of p/2 !
Co atoms on Ag (111) surface
Co (magnetic)

Defect (non-magnetic)

Schneider et al., Phys. Rev. B65, 121406 (2002).

Quartz crystal filter

Response (dB)

0

(a)

-20

-40

(b)

(c)

-60
4433.254433.54433.75 4434
Frequency (Hz)4434.25

Circuit Simulator
Download LTSpice from the web site of Linear Technology

What is Spice?
SPICE: Simulation Program with Integrated Circuit Emphasis
A language which describes electronic circuits (corresponding to circuit diagrams).
ex) a CR circuit and a dc power source

Graphical user interface: Circuit diagram
Linear Technology
web site* 0---R1---1---C1---2---V1---0
R1 0 1 10
C1 1 2 20
V1 2 0 5.END

Operation example

Summary
Theorems for paired terminal circuits
Superposition, Ho-Tevenin, Reciprocity
Duality
Passive devices (elements) and active devices

Transfer function and transient response
Transfer function of single-pair terminal circuits
Resonance circuit
Bode plot
General properties

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo Katsumoto

電子回路論第４回
Electric Circuits for Physicists

Introduction of useful free software
Circuit Simulator
LTSpice (Linear Technology)

・・とはならない
By トランジスタ技術

Circuit Simulator
Download LTSpice from the web site of Linear Technology

What is Spice?
SPICE: Simulation Program with Integrated Circuit Emphasis
A language which describes electronic circuits (corresponding to circuit diagrams).
ex) a CR circuit and a dc power source

Graphical user interface: Circuit diagram
Linear Technology
web site

Newest version
LTSpice XVII !!

* 0---R1---1---C1---2---V1---0
R1 0 1 10
C1 1 2 20
V1 2 0 5.END

Operation example

References
For circuit basics:“Foundations of Analog and Digital Electronic Circuits”
by A. Agarwal and J. H. Lang (Elsevier, 2005)
1000 printed pages!
Ch.1 The Circuit Abstraction“Teach Yourself Electricity and Electronics” 3rd Ed.
by Stan Gibilisco (McGraw-Hill, 2002) 748 printed pages“Schaum’s outlines: Electric Circuits” 6th ed.
by M. Nahvi, J. A. Edminister (McGraw-Hill, 2014) 504pages

For measurement circuits:“Electrical and Electronics Measurements”
by G. K. Banerjee (PHI Learning Private, 2012) 835 pages

Outline today
3.2 Two terminal-pair passive circuits
3.2.1 Impedance matching (concept)
3.2.2 Poles and zeros of transfer function and
Bode diagram
3.2.3 Image impedance
3.2.4 Impedance matching with terminal-paircircuits
3.2.5 Fidelity and distortion
3.2.6 Filter circuits

Impedance matching
𝐽

𝑍𝑜𝑢𝑡
𝑉𝑜𝑢𝑡

𝑍

𝑉0
Maximum power:

Impedance matching condition:

Zeros and Poles of Transfer Functions{𝛼𝑗 }: Poles{𝛽𝑗 }: Zeros
Bode diagramEffect of a Pole on the Real Axis for Bode Diagram

𝑖𝜔
s
−1

1

Effect of a Resonance Pole (Finite Imaginary Part)(fake example)

s

𝑖𝜔
𝜔0

−1

Image parameters
𝑍1

𝑍2

𝑍1
𝑍1

𝐽1
𝑉1

𝑍2

𝑍1 and 𝑍2 are image
impedances
𝐽2

𝐴
𝐶

𝐵
𝐷

𝑉2

𝑍2

Image parameters

𝜃: Image propagation constant

Image attenuation constant

Image phase shift

Image parameters

𝑍1 , 𝑍2 , 𝜃: Image parametersImpedance matching with two terminal-pair circuits
𝑅𝑜𝑢𝑡
𝐴
𝐶

𝐴𝐵𝐶𝐷 ≠ 0
𝐴=𝐷=0
𝐵=𝐶=0

Matching transformer

𝐵
𝐷

𝑅

Fidelity and distortion in wave transformation
Linear response:

No distortion condition:(1) No filter effect(2) No dispersion in group delay
Breaks (1): amplitude distortion, (2): delay distortionEffect of distortion
Sinusoidal amplitude distortion (amplitude modulation)

Paired echoEffect of distortion
Sinusoidal group delay distortion

Paired echo

Distortion(paired echo)
CosineAmplitude
Distortion

Sine
Delay
DistortionFilter Circuit
Transmission

Transmission

High pass filter
Low pass filterw

w

Transmission

Transmission
Band passfilterNotch filterww

Transmission
Voltage transmission coefficient:attenuation
𝐽1

𝑉1

Phase shift
𝐽2

𝐴
𝐶

𝐵
𝐷

Square root power transmission coefficient

𝑉2

𝑍2

Terms for FiltersΞ 𝑖𝜔 = 𝐴 𝜔 𝑒 𝑖𝜙(𝜔)(Transient Band)

Transmission or Gain

𝐴(𝜔)Stop Band

Pass Band

Cut-off Frequency

gain or transmission

Ideal filter (not exist)
𝐴0

Ideal low pass filter𝜔𝜔0

Heaviside functionSinc function
1

𝑃(𝜔)

1

𝜔
−𝜋/𝜏

𝜋/𝜏
0.5

≳
≩
≮
⊼
≸
≳ ≩ ≮ ≣∨ ≸ ∩ ∽ ⊼ ≸

0

-10

-5

0

5

10

Constant K type filter





0

0.5



1

1.5

2

Butterworth Filter≇∲∨≩∡∽∡∰∩ ∽ ≪≈ ∨≩∡∩≪∲ ∽∱∱ ∫ ∨∡∽∡∰∩∲≮

100

10-1

10

n=1

2

-2

4

16

10-3 -3
10

9

10-2

10-1

100

101

102

103

Bessel Filter
Inverse Bessel Polynomial≂ ∰ ∽ ∱ ∻ ≂ ∱ ∨≳ ∩ ∽ ≳ ∫ ∱≂≮∨≳∩ ∽∨∲≮ ⊡ ∱∩≂≮⊡∱∨≳∩ ∫ ≂≮⊡∲∨≳∩≳∲

≂
∨
∰
∩
≮
⊥ ∨≳∩ ∽ ≂≮ ∨≳∩n=3
5

Chebyshev Filter∱≇≮∨≩⊭∩ ∽ ≪≈≮∨≩⊭∩≪ ∽ ≰∱ ∫ ⊲∲≔ ∲∨⊭∩≮

𝜀: Ripple coefficeint

𝑇𝑛 : n-th order Chebyshev polynomial
e = 0.2

e = 0.2

n =11

1

0.9

0.8

0.7

0.6

0.4
0.6

0.5

0.4

3

n =13

0.3

0.2

5

0.1

0
0

1

2

3

4

5

6

7

8

9

10

Quartz crystal filter

Response (dB)

0

(a)

-20

-40

(b)

(c)

-60
4433.254433.54433.75 4434
Frequency (Hz)4434.25

Packaged filters

Web selection page
http://www.minicircuits.com/products/Filters.shtml

Mini-Circuits
Band Pass
19.2 – 23.6MHz 50Ohm

Classification with the number of energy storages(a) Single energy storage(b) Double energy storage過渡応答 (Transient Response)≚∱≷∨≴∩ ∽ ⊡∱

≤
∡
≩
∡
≴
⊥ ∨ ≩∡ ∩ ≕ ∨ ≩∡ ∩ ≥∸∾∼ ∰ ≴ ∼ ∰∻≈ ∨≴∩ ∽ ∾∱∽∲ ≴ ∽ ∰∻∺∱ ≴ ∾ ∰∱≆ ≦ ≈ ∨ ≴∩ ≧ ∽ ≩ ∡ ∫ ⊼ ⊱ ∨ ∡ ∩Heaviside

𝑉0

∲⊼

t

Simple application
R
𝑢(𝑡)

C

𝑤(𝑡)

⊷

⊵ ≴ ⊶⊸≖ ∽ ≖ ∰ ∱ ⊡ ≥≸ ≰ ⊡ ≃ ≒Exercise B-1
Calculate the voltage V over capacitor C2 by using Norton theorem.

L
C1

R
J

V
C2

Exercise B-2
Obtain F-matrices for the circuits below.
(a)

(b)
C

C

L

(c)
R

C

C

C

L

L

C

Exercise B-3

The switch below is turned on at t = 0.
Obtain the time evolution of voltage v henceforth.
R

L

V

C

v

電子回路論第5回
Electric Circuits for Physicists

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoOutline
Introduction of a freeware “Scilab”
Ch.4 Amplification circuit
4.1 Amplification and system stability
4.1.1 What is amplifier?
4.1.2 Feedback
4.1.3 Stability of feedback
4.2 Operational amplifier (OP-amp)
4.2.1 Linear model of OP-amp
4.2.2 Package
4.2.3 Circuit examples
4.2.4 Datasheet
4.2.5 Stability

A convenient freeware: Scilab

Transfer function analysis with ScilabSimple application
R
𝑢(𝑡)

C

𝑤(𝑡)

Transient response: Use of Scilab

Transient response: Use of Scilab
R

LChapter 4
Amplification circuits

Linear amplifier
gain = 1
gain >1

passive filter

amplifier

four terminal circuit model

Circuit symbol
Controlled power source models
𝑗𝑖𝑛

J-J

𝛼𝑗𝑖𝑛

𝑗𝑖𝑛

+
-

J-V

𝑔𝑚 𝑣𝑖𝑛 𝑣𝑖𝑛

𝑟𝑚 𝑗𝑖𝑛 𝑣𝑖𝑛

V-J

+
-

V-V

𝛼𝑣𝑖𝑛

Gain, and “Unit” for gain
𝑣𝑜𝑢𝑡
Voltage gain:
𝑣𝑖𝑛

𝑗𝑜𝑢𝑡
Current gain:
𝑗𝑖𝑛

𝑣𝑜𝑢𝑡 𝑗𝑜𝑢𝑡
Power gain:
𝑣𝑖𝑛 𝑗𝑖𝑛

When we say “the gain of the amplifier …”, the gain means power gain.

quantity 𝑄, unit 𝑄0 : 𝑄 in log scale:cf. deca- 10

dB : (decibel)

1/10

From: G. Bell(B, bel)Alexander Graham Bell1847 - 1922dB units: dBm (1mW: 0dBm), dBv (1V: 0dBv), etc.

Feedback circuit
Feedforward

Feedback
𝑈(𝑠)

𝑊(𝑠)
−

𝑊 𝑠 = Ξ 𝑠 𝑈(𝑠)Ξ(𝑠)𝑊 𝑠 = Ξ 𝑠 [𝑈 𝑠 − ℎ 𝑠 𝑊 𝑠 ]

ℎ(𝑠)

Ξ 𝑠
𝑊 𝑠 =
𝑈 𝑠
1+Ξ 𝑠 ℎ 𝑠
≝ 𝐺 𝑠 𝑈(𝑠)1+Ξ 𝑠 ℎ 𝑠

> 1: Negative feedback, < 1: Positive feedbackCondition for negative feedback
1+Ξ 𝑠 ℎ 𝑠

> 1: Negative feedback, < 1: Positive feedback

Im[Ξ 𝑠 ℎ 𝑠 ]Ξ 𝑠 ℎ(𝑠)1 + Ξ 𝑠 ℎ(𝑠) ≝ 𝐷(𝑠)(-1,0)

Re[Ξ 𝑠 ℎ 𝑠 ]Negative feedback

Oscillation point
Positive feedback
If Ξ 𝑠 ℎ 𝑠 = −1 has solutions, the circuit may be unstable.

How can we judge?

Criteria (Routh-Hurwitz, Nyqust,
Liapunov, …)

Zeros and poles of 𝐷(𝑠)Assumption 1: Ξ 𝑠 , Ξ 𝑠 ℎ 𝑠 are stable→ Poles are on the left half plane of s.

Assumption 2: Ξ 𝑖𝜔 , Ξ 𝑖𝜔 ℎ 𝑖𝜔 → 0 for |𝜔| → ∞
Ξ 𝑠

𝑄(𝑠)
= 𝑃(𝑠), ℎ 𝑠𝑞(𝑠)= 𝑝 𝑠 ∶ 𝑃 𝑠 ,𝑄 𝑠 ,𝑝 𝑠 ,𝑞 𝑠

polynomials

deg 𝑃 > deg 𝑄 , deg(𝑝) ≥ deg(𝑞)𝑃 𝑠 𝑝(𝑠)𝐷 𝑠 =1+Ξ 𝑠 ℎ 𝑠 =𝑃 𝑠 𝑝 𝑠 + 𝑄 𝑠 𝑞(𝑠)(𝑠 − 𝛽1 ) ⋯ (𝑠 − 𝛽𝑛 )𝐷 𝑠 = 𝐷0(𝑠 − 𝛼1 ) ⋯ (𝑠 − 𝛼𝑛 )The same orderZeros and poles of 𝐷(𝑠)(𝑠 − 𝛽1 ) ⋯ (𝑠 − 𝛽𝑛 )𝐷 𝑠 = 𝐷0(𝑠 − 𝛼1 ) ⋯ (𝑠 − 𝛼𝑛 )𝛽𝑖 : Zeros of 𝐷(𝑠) → Poles of 𝐺(𝑠)∃𝛽𝑖 ∈ right half plane of 𝑠 → The circuit is unstable.
𝑛arg 𝐷 =

𝑛arg 𝑠 − 𝛽𝑖 −𝑖=1Left half planearg(𝑠 − 𝛼𝑖 )𝑖=1

Right half plane

+∞

+∞

−𝜋
𝜋

𝛽𝑖

𝛽𝑖
−∞

𝑖𝜔

𝑖𝜔

−∞

𝑠 = 𝑖𝜔 (on imaginary axis)
𝜔: −∞ → +∞Number of zeros on theright half plane: 𝑚∆ arg 𝐷 = 𝑛 − 𝑚 𝜋 − 𝑚𝜋−𝑛𝜋 = −2𝑚𝜋Nyquist Plot and Criterion
Ξ 𝑠 ℎ 𝑠Ξ 𝑠 ℎ 𝑠Harry Nyquist(1889–1976)(−1,0)∆ arg 𝐷 = 0
Stable∆ arg 𝐷 = −4𝜋
UnstableFeedback in terminal-pair circuits with resistors
+
in

+

+

-

-

+
-

out

in

+

+

-

-

out

(i) input: parallel, output: parallel(ii) input: series, output: parallel

out

out

in

+

+

-

-

in
(iii) input: parallel, output: series

+

+

-

-

(iv) input: series, output: series

Operational amplifier (OP amp.)
•
•
•
•

+

Differential amplifier
Input impedance ~∞Open loop gain 𝐴𝑜 ≫ 1
Output resistance ≈ 0Case (iv)

𝑅𝑓
𝑣𝑖𝑛

𝑅𝑖𝑛
𝐽𝑖𝑛 = 0+Virtual short circuit
𝐽
𝑣𝑜𝑢𝑡
Inverting amplifier

Opamp packages

Various applications of OP amps

Integration circuit

Logarithmic amplifier

Logarithmic Amplifier

Instrumentation amplifier

Instrumentation amplifierOP-amp data sheet

1st

2nd differentialOP-amp data sheet
Parameters

OP-amp data sheet
Common mode rejection ratio (CMRR)

OP-amp data sheet

Unity gain frequency+
Voltage follower

Frequency Dependent Characteristics of OP-Amps
Cut-off frequency
𝜔 𝑇 = 2𝜋𝑓𝑇

A (Gain)100

10Phase rotates by 𝜋/2
1

0.1
10-3

10-2

10-1

100

101

102

103

𝜔 = 2𝜋𝑓

𝑓𝑇1

𝑓𝑇2

A (Gain)

106

Multiple cut-off frequency:
Phase rotates more than 𝜋

104

102

If gain is larger than 1 atphase shift p :Dangerous!

100

10-3

10-2

10-1

100

101

102

103

Phase compensation
Why dangerous?
p phase shift: negative feedback → positive feedback
In Nyquist plot

Ξ 𝑠 ℎ 𝑠p phase shift
𝜔=0Phase compensation

Phase compensation

Inverting amplifier and cut-off frequency

A =200 fT =30kHz

A =10 fT =300kHz

A =50 fT =90kHz

A =2 fT =2MHz

Oscillation of OPamp

gain

Seeds of oscillation

Phase電子回路論第6回
Electric Circuits for Physicists

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoOutline
4.3 Feedback control
4.3.1 Disturbance and noise
4.3.2 PID control

4.4 PN junction transistors
4.4.1 Diodes
4.4.2 Bipolar junction transistors
4.5 Field effect transistors

Comment: Use of OP-amp at saturation voltages
𝑉𝑜𝑢𝑡
-

𝑉+

𝑉𝑜𝑢𝑡

+

𝑉𝑖𝑛

0

Compare 𝑉𝑖𝑛 with 0𝑉−Comparator
𝐵𝐴

rail to rail
characteristics

gradient 𝐴0+

𝑉𝑜𝑢𝑡 ∶ 𝐴 > 𝐵?

𝑉𝑖𝑛

Hurwitz criterion
Adolf Hurwitz
1859 - 1919

Pole equation:(Otherwise the system is unstable.)

Hurwitz matrix

Hurwitz criterion
Hurwitz determinants

Hurwitz criterion

𝐻1 , 𝐻𝑛 > 0 is trivialfrom the assumption.

Another expression:
Divide the denominator to odd and even parts 𝑂(𝑠) and 𝐸(𝑠).
If the zeros of 𝑂(𝑠) and 𝐸(𝑠) are aligned on the imaginary axis
alternatively, the system is stable.

Disturbance and noise on feedback control
Circuit treatment of fluctuations: • Prepare external power sources• Express them as transfer functions

𝐷(𝑠)𝑅(𝑠) +
-

𝑌(𝑠)

+
𝐺(𝑠)

𝐺𝐶 (𝑠)++

𝐻(𝑠)𝑁(𝑠) +

PID control
Compensator(controller)

P: proportional, I: integral, D: derivative𝐾𝐼𝑠
𝑅(𝑠) +

𝑌(𝑠)

+

𝐾𝑃

-

𝐺(𝑠)
+
+

𝐾𝐷 𝑠

PID controllers

4.4 Example of active element: Transistors
Three types of semiconductors
p-typeIntrinsic

dopingn-typeelectrons

conduction band
𝐸F

𝐸F

band gap

𝐸G

𝐸F

valence band

holes
--

vacuumfor electrons

diffusion
vacuumfor holes+++
pn junction

pn junction thermodynamics
Consider electrons
Vacuumfor electrons
diffusion

donorse+ e+ e++ e- + e-

-

+
+
+
+
+
voltage (polarization) → energy cost

𝐹 = 𝑈 − 𝑇𝑆
Voltage (internal energy cost)

Diffusion (entropy)

Minimization of 𝐹 → Built-in (diffusion) voltage 𝑉𝑏𝑖

4.4.1 I-V characteristics of pn junctions

-

--

+
++
Reverse bias
enhances 𝑉𝑏𝑖 : no go

+

--

-

++
Forward bias
Minority
overcomes 𝑉𝑏𝑖 : go carrier injection

Rectification𝑒𝑉
𝐽 = 𝐽0 exp−1
𝑘B 𝑇
Shockley theory

Injection of minority carriersJ

minority carrier
current0V0Fate of injected minority carriers:
Radiative recombination

Barrier overflow
light emittingℎ𝜈
diode
Nick Holonyak Jr.

Solar cell (injection of minority carriers with illumination)J

𝑒𝑣𝑛 Δ𝑛𝑝

dark

0External injection
illuminated0
V
Gerald Pearson,
Daryl Chapin
and Calvin Fuller
at Bell labs. 1954

4.3.2 Discovery and invention of bi-polar transistors
The first point contact transistor(Dec. 1947
The paper published in June 1948.)

John Bardeen, William Shockley,
Walter Brattain 1948 Bell Labs.

Bipolar junction transistorn p nField effect transistorpnBipolar transistor structures and symbols

PNP type

NPN type

Similar characteristics PNP and NPN: complementary

B

C

n

p

n

Base-Collector characteristics

−𝐽C

E

𝐽E
−𝐽C
𝑉BC

𝐽E

𝑉BC

How a bipolar transistor amplifies?

𝜙

Emitter

BaseCollector

n

p

n

VEC

A

JC

How a bipolar transistor amplifies?

JB

A
e+
e+
e+

e+

e-

ee- e-ee-e+e+Diffusione-ee-Emitter

BaseCollector

n

p

n

VEC

A

JC

Base-Collector characteristics
n

n
p

e+
e+
e+

e+
e+

e- eeee-

e-

𝐽𝐸

𝐽𝐶

𝑉BC

Collector-Emitter characteristics
n
E

p
B

n
C

𝐽𝐶

Current amplification : Linearize with quantity selection

𝐽𝐶 = ℎ𝐹𝐸 𝐽𝐵
Emitter-common current gainLinear approximation of bipolar transistor

⋃ ∡ ⋃
∡⋃ ∡
≖∱ ∽ ≈∱∱ ≈∱∲ ≊∱ ∺≊∲ ≈∲∱ ≈∲∲ ≖∲⋃ ∡ ⋃∡⋃ ∡ ⋃ ∡⋃ ∡≶∱ ∽ ≨∱∱ ≨∱∲ ≪∱ ∽ ≨≩ ≨≲ ≪∱≪∲≨∲∱ ≨∲∲ ≶∲ ≨≦ ≨≯ ≶∲𝑗2

𝑗1

h-parameters(lower case:local linear approximation)

𝑉1
𝑉2

Normalized value at Jc =2 mA

Hybrid matrix

Collector current (mA)Concept of bias circuits for non-linear devices
Common emitter amplifier

For bias (dc) circuits
All the capacitors can be viewed as
break line.
+𝑉

For small amplitude (highfrequency) circuits
All the capacitors can be viewed as
short circuits.

Concept of equivalent circuit
C
B

A

B

n

𝑗b
C

p

ℎie

A

n
E

E

ℎfe 𝑗b

Concept of equivalent circuit: Where is feedback?
B

𝑗b
C
ℎie

ℎfe 𝑗b

E

Ξ(𝑠)
ℎ(𝑠)

𝑗𝑏

ℎ𝑓𝑒 𝑗𝑏𝑗𝑏

REℎ𝑓𝑒 𝑗𝑏≶≁ ∽ ≶≯≩≨≒≦≥∽ ≨≩≥ ∫ ≒≅ ∨∱≃∫ ≨≦ ≥∩≒⊼ ≒≃≅ ≨≦ ≥ ⋀ ∱Current amplification: Emitter follower
𝑗𝑏 (1 + ℎ𝑓𝑒 )(𝑅𝐸 ∥ 𝑅𝑜 )𝑣𝑜=𝑣𝑖𝑗𝑏 [ℎ𝑖𝑒 + 1 + ℎ𝑓𝑒 𝑅𝐸 ∥ 𝑅𝑜 ]≈1(ℎ𝑓𝑒 ≫ 1)

𝑣𝑜 does not depend on load
resistance⇒ Very low output resistanceVcc +15V
10m

10k

10m

input

a
10k

2SC245810m
680W

outputComplementary transistors

B

A

n

Vcc +5V
10m

10k

p

A

n

a

10m

2SC3668

output

E

input

a

C

10m
2SA1428

10kp

Vcc -5VnApA
Symmetric characteristics: Complementary
Symmetric: Small collector current(idling current) for zero input.

Example of transistor datasheet

Example of transistor datasheet

Example of transistor datasheetℎ𝑓𝑒 linear model availability
in the range of 𝐽𝐶 .

Cut-off frequency as a function of 𝐽𝐶

Common emitter (grounded emitter) amplifier circuit4.4 Field effect transistor (FET)
Junction FET (JFET)

Circuit symbols
D

G

D
G

S
n-channelSp-channel

Pinch-offMES-FETMOS-FETenhancement

Simplified
CMOS inverter
circuitdepletion

Low leakage
current

Single gate input
both on/off switch

inversion

Static characteristics of FET⊵ ≀ ≊≄ ⊶≊≇ ∧ ∰ ∻≧≭ ⊴ ≀≖≇≓∻ transconductance
≖
∽
≣
≯
≮
≳
≴
∺
≄
⊵
⊶
≊≄ ∽ ≦ ∨≖≇∻ ≖≄∩≀≖≲≤ ⊴ ≀ ≊≄≄ ≖ ∽≣≯≮≳≴∺ Drain resistance≇≓≶≤≪≤ ∽ ≧ ≭ ≶ ≧ ≳ ∫ ≲ ≤Locally linear approximation

References
Feedback
 土谷武士，江上正「現代制御工学」(産業図書, 2000)
 J. J. Distefano, et al. “Schaum’s outline of theory and problems of
feedback and control systems” 2nd ed. (McGraw-Hill, 1990)

OP amp. circuit design 岡村迪夫 「OPアンプ回路の設計」 CQ出版社
 J. K. Roberge, K. H. Lundberg, “Operational Amplifiers: Theory
and Practice” (MIT, 2007).
http://web.mit.edu/klund/www/books/opamps181.pdf

BJT, FET circuits
 松澤昭 「基礎電子回路工学」 (電気学会, 2009).

 S. M. Sze, K. K. Ng, “Physics of Semiconductor Devices”(Wiley, 2007).

Exercise C-1
C

P

-

R3

+

-

Vout

+

R1

Draw a rough sketch of the
waveform for Vout.

R2

V+
t

V-

In the circuit shown in theleft, at point P, a waveform in
the lower panel wasobserved. Here V+ and V- arepower source voltages for +and – respectively.

“Rough sketch” should
contain the levels and thetiming of folding points.
Write a short comment why
Vout should be in such a form.

Exercise C-2
𝐴Consider a differential amplifier withthe open loop gain

+
-

1−𝛾 𝑅
𝛾𝑅

So the gain diverges with 𝑠 → 0 but here we ignore thisinstability. The input impedance is ∞, and the output
impedance is 0.
It is now placed in a circuit with a feedback shown in theleft.
Obtain the stability condition for 𝛾.
(hint) Apply the Hurwitz criterion for zeros of even andodd parts of the denominator.
Or just calculate 𝐻2 .

Exercise C-3
𝑗2

𝑗1
C

B

ℎie
ℎfe 𝑗bE
𝑣1

Let us view a bipolar transistor plus
an emitter resistance as a four
terminal circuit as shown in the left
figure.
Obtain the Y (admittance) matrix
defined below for this circuit.

𝑣2 Calculate each element in the Y𝑟𝑒matrix for
𝑟𝑒 =25Ω, ℎ𝑖𝑒 =500 W, ℎ𝑓𝑒 = 200電子回路論第7回
Electric Circuits for Physicists

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoOutline
4.5 Field Effect Transistors (FETs)
Ch.5 Distributed constant circuits
5.1 Transmission lines
5.1.1 Coaxial cables
5.1.2 Lecher lines
5.1.3 Micro-strip lines
5.2 Wave propagation through transmission lines
5.2.2 Connection and termination of transmission lines

Combination of an OP-amp and discrete transistors
10k
10m

Complementary
push-pull

a

𝑅𝑖
10m

a

𝑣𝑖𝑛 𝑅𝑖𝑛10k

𝐽𝑖𝑛 = 0++𝑅𝑓

𝑣𝑖𝑛 𝑅𝑖𝑛

𝑉+

𝑅𝑓

𝑣𝑜𝑢𝑡
𝑅𝑖

𝐽

𝑣𝑜𝑢𝑡
Inversion
amplifier𝑉−

Voltage, current booster

Depletion layer width with reverse bias voltage
Poisson equation
𝑉𝑏𝑖 + 𝑉

++
p
n
+
+
𝑤𝑛
−𝑤𝑝

Effective capacitance and reverse bias voltage

Doping profiler
−𝑉𝑏𝑖

V

KB505

Varicap diode

Frequency modulation
Phase lock loop

4.4 Field effect transistor (FET)
Junction FET (JFET)

Circuit symbols
D

G

D
G

S
n-channelSp-channel

Pinch-offStatic characteristics of FET

Ohmic area

Space charge limited area

Space-charge limitation of source-drain currentLGp+wd (y)-NDe
S

Vch(y)

2wt

Dn
conductivity
p+

G

Vg

y

pinch off (internal) voltage:

Only valid for wd < wt/2.

electric fieldchannel width

Static characteristics of FETtransconductanceLow bias current:
small power consumption

Drain resistance

Locally linear approximation

Amplification factor (voltage gain)

Biasing circuits for FETs
Fixed bias circuit

𝑅1

𝑅𝐷

𝑉𝐷𝐷
𝑉𝐷𝐷
𝑅𝐷

𝐽𝐷

𝐽𝐷

𝑉𝐺𝑆
𝑅2
𝑉𝐷𝐷

𝑉𝐷𝑆

Self-biasing circuit
Generally
more stable

𝑉𝐷𝐷
𝑅𝐷

𝑉𝐷𝐷
𝑅𝐷 + 𝑅𝑆

𝐽𝐷

𝐽𝐷
𝑅2

𝑉𝐺𝑆

𝑉𝐺𝑆

𝑅𝑆
𝑉𝐺𝑆

𝑉𝐷𝐷

𝑉𝐷𝑆

Equivalent signal circuits for FETS

GD

Drain grounded

Source grounded

𝑟𝑑
1+𝜇

𝑟𝑑

𝑣𝑖

+
𝜇𝑣𝑖

+

Gate grounded

𝑣𝑖

-

𝜇𝑣𝑖
1+𝜇

𝑣𝑖

+
𝜇𝑣𝑖

𝑟𝑑

MES-FETMOS-FETenhancement

Simplified
CMOS inverter
circuitdepletion

Low leakage
current

Single gate input
both on/off switch

inversionMOSFET switching characteristics
From datasheet CSD87381P power MOSFET (Texas instr.).

More than 7 orders change in 𝐽𝐷 within 3 V change of 𝑉𝐺𝑆 .

Ch.5 Distributed constant circuits

Submarine cable map

Distributed constant circuit concept
1. In what case we need to consider distributed constant circuits?

Characteristic sizes of devices ≳ wavelength of electromagnetic signal

2. A typical scheme to make the shift for distributed circuit
Lumped constantcircuit

1. Connection of unit circuits
2. Taking the infinitesimal limit

Distributed constantcircuit

3. Distributed constant circuits : transmission lines
Coaxial cables, Lecher lines, micro-strip lines, waveguides, optical fibers

5.1.1 Coaxial cableThin coaxial cable AWG50 (f25mm)

Transmission line as a series of infinitesimal terminal-pairs
Transmission line → divide into four terminal circuits⋯
Each unit should have delay. Ignore energy dissipation.
Oliver Heaviside
1850- 1925
C

C

C

Then take the infinitesimal limit

𝑑𝑥
𝐽

L

L

L

L

L

L

C

C

C

⋯

Width → 0, Number → ∞𝐽 + 𝑑𝐽

𝑍𝑑𝑥
𝑉 + 𝑑𝑉𝑉
𝑌𝑑𝑥

Telegraphic equation

Characteristic impedance(dimension: 𝐿−1 )-: Progressive, +: Retrograde
Characteristic impedancePure reactance 𝑌 = 𝑖𝜔𝐶, 𝑍 = 𝑖𝜔𝐿

For L and C model(dimension: velocity)

Coaxial cable setup

r

a

b
e, m

Maxwell theory𝑏

From Maxwell equations

𝑎
𝑦
𝑧
𝑥

However in TEM (transverse electric and magnetic) mode:i.e., the RHSs are zero.

For the fields along x and y to survive,

Propagation velocity
In such a case, from Maxwell equations:→ Potentials are conceivable for 𝑯 and 𝑬.

Maxwell theory

Cauchy-Riemann theorem

Characteristic impedance:

If we can express 𝑉 and 𝐽 in the form of distributed constantcircuit model (𝐿 and 𝐶 model), the equivalence is certified.
Capacitance part

Maxwell theory
Inductance partCore current 𝐽, shield current −𝐽

Flux per length:
Self inductance per length:

cf. Characteristic impedance of vacuumCoaxial cable 2

Coaxial connectors

Coaxial connectors

Coaxial connectors 2
SMA-typejack
K-typeplug
V-typeLEMO cables and connectors
http://www.lemo.com/
High-energy physics experiment,etc.

Lecher line

Micro strip line

⊽
⊷
⊵
⊶
⊸
⊾
⊡∱
∲
⊡
≚
≗
∱
⊲
∫
∱
⊼
≥
≗
⊲
∱
≥
⊼
≆
∰
≲
≲
≚ ∨≗∻ ≨∻ ⊲≲ ∩ ∽ ∲≰⊲≲ ∲≨ ∫ ⊼ ≬≯≧ ∴ ∫ ∲⊼⊲≲ ≬≯≧ ∲ ∲≨ ∫ ∰∺∹∴ ∲⊼⊲∲≲ ≬≯≧ ∱∶Wide (W/h>3.3) strip∸∼ ∲ ≳⊵ ⊶∲ ∳∹∽⊵⊶≚ ∨≗∻ ≨∻ ⊲≲ ∩ ∽ ⊼≰∲≚∨⊲≆≲∰∫ ∱∩ ∺≬≯≧ ∴ ∴≗≨ ∫ ∴≗≨ ∫ ∲ ∵ ⊡ ∱∲ ⊲⊲≲≲ ⊡∫ ∱∱ ≬≯≧ ⊼∲ ∫ ⊲∱≲ ≬≯≧ ⊼∴ ∻Narrow (W/h<3.3) strip

Connection and termination

𝑍0

𝑍1

-l

0

x

Connection and termination
At 𝑥 = 0:

𝐽 = 𝐽+ + 𝐽− (definition right positive)
progressive retrograde

𝑉 = 𝑉+ + 𝑉− = 𝑍0 𝐽+ − 𝐽−

Reflection coefficient:
𝑍1 = 𝑍0 : no reflection, i.e., impedance matching
𝑍1 = +∞ (open circuit end) : 𝑟 = 1, i.e., free end
𝑍1 = 0 (short circuit end) : 𝑟 = −1, i.e., fixed end

Connection and termination
Finite reflection → Standing wave
Voltage-Standing Wave Ratio (VSWR):

At x = - l

Reflection coefficient:SWR measurement

SWR Meters:

Desktop types

Cross-meter

Handy typeConnection and termination

𝑍0

𝑍1

-l

0

x

Connection and termination
Transmission line connection.
Characteristic impedance 𝑍0 , 𝑍0 ′At the connection point, only the local relation between
V and J affects the reflection coefficient.
The local impedance from the left hand side is 𝑍0 ′.

電子回路論第 8 回Electric Circuits for Physicists

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoOutline
5.1 Transmission lines
TEM mode Lecher line
Micro-strip line
TE, TM mode
Waveguide
Optical fiber
5.2 Propagation in transmission lines
Termination and connection
Smith chart
Scattering matrix
Impedance matching

Comment: bias + signal superposition
For bias (dc) circuits
All the capacitors can be viewed as break line.
For small amplitude (high-frequency) circuits
All the capacitors can be viewed as shortcircuits.

Self-bias
𝑅1

Source-grounded
𝑟𝑑

𝑅𝐷

-

𝐽𝐷

𝑣𝑖
𝑅2

𝜇𝑣𝑖
+

Coaxial cable 2

Coaxial connectors
coaxial connector (schematic)
coaxial cablemale contact female contact
Highest available frequencies for coaxial connectors

type

Plug

Jack

outer diam.

highest freq.

Coaxial connectors
N-type connectors

jack with flange

BNC-type connectors

plug
isolated jack(not for high freq.)plug [disassembled (b) ]jack with flange

plugCoaxial connectors 2
SMA-typejack
V-typeK-type

jack

plug

plug

jack

plug

LEMO cables and connectors
http://www.lemo.com/
High-energy physics experiment,etc.

Transmission lines with TEM mode
Transmission lines with two conductors are “families”.

Electromagnetic field confinement with parallel-plate capacitor
Roll up tocoaxial cableOpen to micro-strip line

TEM is the primary mode.

Shrink to dipole (Lecher line)

Lecher line

Micro strip line

Wide (W/h>3.3) strip

Narrow (W/h<3.3) strip

Waveguidecross section
metalhollow

Electromagnetic field is confined
into a simply-connected space.
TEM mode cannot exist.

Maxwell equations giveHelmholtz equation
𝐸𝑧 = 0: TE mode,
𝐻𝑧 = 0: TM mode

Optical fiber
step-typeoptical fiber
Difference in dielectric constantno dispersion

Termination of transmission linetransmission line
wave
source

𝑍1

𝑍0

load

x

-l

0

Termination of a transmission line with length l andcharacteristic impedance 𝑍0 at 𝑥 = 0 with a resistor 𝑍1 .
𝐽 = 𝐽+ + 𝐽−
At 𝑥 = 0:(definition right positive)

progressive retrograde
𝑉 = 𝑉+ + 𝑉− = 𝑍0 𝐽+ − 𝐽−Comment: Sign of Ohm’s law in transmission lines
reflects direction of waves (depends on the definitions).

Termination of transmission line

synthesized impedance:
reflection coefficient:

𝑍1 = 𝑍0 : no reflection, i.e., impedance matching
𝑍1 = +∞ (open circuit end) : 𝑟 = 1, i.e., free end

𝑍1 = 0 (short circuit end) : 𝑟 = −1, i.e., fixed end

Connection and termination
Finite reflection → Standing wave

1 − |𝑟|1 + |𝑟|Voltage-Standing Wave Ratio (VSWR):

SWR measurement

SWR Meters:

desktop types

cross-meter

directional coupler
handy type

Synthesized impedance
total impedancefrom this side

𝑍0

𝑍1
x

At x = - l-l

Reflection coefficient:0Connection and termination
Transmission line connection.
Characteristic impedance 𝑍0 , 𝑍0 ′At the connection point, only the local relation between
V and J affects the reflection coefficient.
The local impedance from the left hand side is 𝑍0 ′.

𝑍0

𝑍′0

𝑍0

𝑍′0

5.2.3 Smith chart, Immittance chart
End impedance 𝑍1 : Normalized end impedance 𝑍𝑛 ≡ 𝑍1 /𝑍0

real:
imaginary:
x: constant →constant resistancecircley: constant →constant reactancecircle5.2.3 Smith chart, Immittance chartr : reflection coefficient

𝑉− 𝑍 − 𝑍0𝑟==
𝑉+ 𝑍 + 𝑍0

Im Z

x-constanty-constant

Z0
Z

Re Z

Im r

Im r

y=2

y=4

Re r
Re r

y=0

x =4

y = -4y = -2

5.2.3 Smith chart, Immittance chartx=2

𝑟 = 𝑢 + 𝑖𝑤

x=4

5.2.3 Smith chart, Immittance chart
y =1

y =2

y =0.5y =4
y =0.2y =0

y = -0.2

y =-4
y = -0.5
y =-1

𝑟 = 𝑢 + 𝑖𝑤

y =-2

5.2.3 Smith chart, Immittance chart
y =2

y =1
y =0.5y =4
y =0.2y =0

y = -0.2

x=2

x=4

y =-4
y = -0.5
y =-1

𝑟 = 𝑢 + 𝑖𝑤

y =-2Smith chart5.2.3 Smith chart, Immittance chart

Admittance chart𝑟 = 𝑢 + 𝑖𝑤

Smith chart5.2.3 Smith chart, Immittance chart𝑟 = 𝑢 + 𝑖𝑤

Immittance chart5.3 Scattering (S) matrix (S parameters)
How to treat multipoint (crossing point) systematically?
Transmission lines: wave propagating modes → Channels
Take |𝑎𝑖 |2 , |𝑏𝑖 |2 to be
powers (energy flow).

outputS-matrix

input

𝑎
𝑏1 1
𝑎2
S

𝑏2

important properties:
Reciprocity

𝑎3

𝑏3
𝑎4

𝑏4

Unitarity(In case, no dissipation, no amplification)

5.3 S matrix (S parameters)
Propagation with no dissipation
incident power wave
reflected (transmitted) power wave

Simplest example: series impedance 𝑍𝑆
input direction

input direction

Take voltage as the “flow”
quantity.
(assume common
characteristic impedance)

𝐽1
𝑉1

𝐽2

𝑍𝑆
𝑉2

5.3 S-matrix (S-parameters)

𝑍0

𝐽1
𝑍𝑆

𝑍0
𝐸1

𝐽2

𝑉1

Terminate 2 with 𝑍0 → 𝑎2 = 0

𝑍0
𝑉2

𝑎2 = 0𝑍05.3 S-matrix (S-parameters)
𝐽1

𝑎1 = 0

𝐽2

𝑍0

𝑍𝑆
𝑍0

𝑉1

𝑉2

Terminate 1 with 𝑍0 → 𝑎1 = 0 (should be symmetric)

Generally

𝐸2

Cascade connection of S-matrices

𝑟𝐿,𝑅 , 𝑡𝐿,𝑅 : complex reflection, transmission coefficients satisfying

Conduction channels in quantum transport
Electron (quantum mechanical) waves also havepropagating modes in solids.
→ Conduction channel
Landauer eq.:
the conductance of a single perfect quantum channel isconductanceAB ring S-matrix model

magnetic flux

Rolf Landauer

S-parameter representation of high-frequency devices

S11

S12

S21

S22

S-parameter representation of high-frequency devices
Ex) NE76084 MES FET 0.5～18GHz
The datasheet tells that we need
impedance matching circuitswith transmission lines.
𝑆11

If we know Z-parameters:
From Ho-Thevenin theorem

𝑍𝑖𝑗 : BJT (FET) Z-parameters, again Ho-Thevenin says

S-parameter representation of high-frequency devices

Impedance matching with S-parameters
Generally the unitarity does not hold for amplification.

Matching condition:
Solutionwithmaximum available power gainstability factor

Practical impedance matching with Simth chartImpedance matching designer
http://home.sandiego.edu/~ekim/e194rfs01/jwmatcher/matcher2.html

http://leleivre.com/rf_lcmatch.html

Exercise D-1
𝑗2

𝑗1
𝑟f
𝑣1

𝑟i

-

𝑔m 𝑣1 +𝑟o𝑣2

Obtain the Y matrix for the above equivalent circuit (p-shapecircuit).

Exercise D-2

l =1km の伝送線路がある．終端側を短絡したところ，電源側から測定したインピダンスは0.6i Wであった．一方，終端側を開放して電源側からアドミタンスを測定すると4x10-6i Sであった．
この伝送線路の特性インピダンスを求めよ．

Consider a transmission line with the length l = 1km. First we
short-circuited the end and measured the impedance from thesignal source and obtained 0.6i W. Next we opened the end andmeasured the admittance from the signal source and obtained
4x10-6i S.
What is the characteristic impedance of the transmission line?

Exercise D-2

0.6iW

𝑍0

Short

𝑍0

Open

4x10-6iS
-1km

0

x

Exercise D-3
𝐽1

𝐽2

𝑉2

𝑉1

Remember F-matrix (cascade matrix) defined above.
Write down the F-matrix form of the transmission line
shown below.
𝐽1

𝑉1

𝐽2

l
𝑍0

𝜅 = 𝑌𝑍

𝑉2

電子回路論第９回
Electric Circuits for Physicists

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoOutline
5.3 S-parameter representation of devices
Impedance matching with Z, S parameters
Impedance matching with immittance chart
5.4 Non-TEM mode transmission lines
5.5 Non-linear elements and Toda lattice
Ch.6 Noises and Signals
6.1 Fluctuations
6.2 Fluctuation-dissipation theorem

Review: Scattering (S) matrix (S parameters)
Transmission lines: wave propagating modes → Channels
Take |𝑎𝑖 |2 , |𝑏𝑖 |2 to be
powers (energy flow).

outputS-matrix

input

𝑎
𝑏1 1
𝑎2

𝑏2

S

𝑎3

𝑏3
𝑎4

In the case of two-terminal pair circuit
𝑎2
𝑎1
𝑏2
𝑏1
S
𝑏4

GND

GND

S-parameter representation of high-frequency devices

𝑎1 , 𝑏1
GND

S11

S12

S21

S22

𝑎2 , 𝑏2
GND

𝑆11 = 𝑟𝑙 , 𝑆22 = 𝑟𝑟

Review: Smith Chart

P.H. Smith
1905-1987

r : reflection coefficient𝑟=

𝑉− 𝑍 − 𝑍0 𝑧 − 1==
𝑉+ 𝑍 + 𝑍0 𝑧 + 1

Z0
Z

Im r

Im Z

x-constant

y=2

y=4

y-constant

Re r

y=0

Re Z
x=4
y = -4y = -2

Comment: Mirror effectAn amplifier may change the effective impedance of passive elements.
equivalent circuit
𝑗1

𝐶

𝑗1

𝐶𝑒𝑓𝑓

−𝐴

𝑣1

𝑣2 = −𝐴𝑣1

𝑣1

𝑗1 = 𝑠𝐶 𝑣1 − 𝑣2 = 𝑠𝐶(1 + 𝐴)𝑣1
𝑗1𝑠=
(1 + 𝐴)𝐶𝑣1: mirror effect+-𝑣2 = −𝐴𝑣1

S-parameter representation of high-frequency devices
Ex) NE76084 MES FET 0.5～18GHz

The datasheet tells that we need
impedance matching circuitswith transmission lines with
𝑍0 = 50 Ω.

𝑆11Insert input and output matching circuits to kill reflections.
50W

𝑜
𝑍𝑖𝑗

𝑖
𝑍𝑖𝑗

input
matching
circuit

BJT,
FET

outputmatching
circuit50W

Impedance matching circuits
The circuit is summarized at the boundaries as
𝑍𝑆
𝑉𝑆

BJT,
FET

𝑍𝐿

If we know Z-parameters of the input/output matching circuits,
from Ho-Thevenin’s theorem

Z-matrix, Ho-Thevenin’s theorem
𝐽1
Z-matrix

𝑉1

Ho-Thevenin

1.
2.

𝐽2
𝑍

𝑉2

3

Measure the open terminal voltage 𝑉0 .
Turn off all the power sources (voltage sources: short, current
sources: open). Measure the open circuit impedance 𝑍𝑖 .

Then

𝑍𝑆

Impedance matching circuits
The circuit is summarized at the boundaries as
𝑍𝑆
𝑉𝑆

𝑍𝑙

BJT,
FET

𝑍𝑟

𝑍𝐿

If we know Z-parameters of the input/output matching circuits,
from Ho-Thevenin’s theorem

𝑍𝑖𝑗 : BJT (FET) Z-parameters, again Ho-Thevenin says

matching condition: 𝑍𝑙 = 𝑍𝑆∗ , 𝑍𝑟 = 𝑍𝐿∗Impedance matching with S-parameters
In S-parameter treatment, we use complex reflection coefficientsto express load, source etc.
𝑟𝐿
𝑎
𝑎
1

2

𝑏1

𝑏2

𝑟𝑆

𝑎1
𝑏1
Matching condition:
Solution
with

𝑎2
𝑏2

Practical impedance matching with Smith chart
Series and parallel connection of passive elements and traces on charts

Smith chart

Admittance chartAn example of impedance matchingfrequency 100 MHz ≈ 628 Mrad/s
immittance chart
Im r

𝑍𝑠

𝐿𝑃

50 W

Re r

targetequalize:

start

Re[Z]=0.4

𝐶𝑆

Re[Y]=1

similarly𝐿𝑃 ≈ 65 nH

Impedance matching designer
http://home.sandiego.edu/~ekim/e194rfs01/jwmatcher/matcher2.html
http://leleivre.com/rf_lcmatch.html

Useful freeware: Smith v4.0
http://fritz.dellsperger.net/smith.html

Impedance matching with Smith V4 (1)Impedance matching with Smith V4 (2)5.4 Non-TEM mode transmission line𝑑𝑥
𝐽

𝐽 + 𝑑𝐽

𝑍𝑑𝑥
𝑉 + 𝑑𝑉𝑉
𝑌𝑑𝑥

Characteristic impedance
LC model: 𝑍 = 𝑖𝜔𝐿, 𝑌 = 𝑖𝜔𝐶
The inductance represents magnetic fields circulating the core and thecapacitance electric fields directing from the core to the shield.

: real, dispersionless (no w-k relation)
Non-linear w-term in Z or Y → dispersion (longitudinal components)

5.4 Non-TEM mode gives mass in transmission line
K

L

K

C L

K

C L

C

C: capacitance per unit length
L: inductance per inverse unit length
K: inductance per unit length

Constant finite mass: 𝐸 = ℏ𝜔 ∝ 𝑘 2(Schrodinger eq.: Parabolic partial differential equation)
Coupling between linear dispersions: mass mechanism cf. Higgs

5.4 Non-TEM mode gives mass in transmission line1
= 𝜔0 unchanged with 𝑑𝑥 → 0𝐿𝐶𝜔𝜔 = 𝑐 ∗𝑘𝜔0

ℏ𝜔0 = 𝑚∗ 𝑐 ∗2

k

5.4 Giving mass to LC transmission line
No dispersion
Velocity:5.5 Non-linear LC transmission line and Toda lattice

Toda lattice is a typical non-linear system with exact (soliton)
solutions. It is defined as follows:
The springs in (a) have Toda-potential:

Equation of motion:

For relative shift
𝑟𝑛 = 𝑢𝑛+1 − 𝑢𝑛Force of a spring:Solitons in Toda lattice

Soliton solution:
sech2 (𝑥)x

N = 2 soliton solution:Non-linear capacitance: Vari-cap
Varicap BB505⊱≖ ∡ ≖

L-Varicap transmission line

Solitons in non-linear circuitToda lattice circuit, Soliton circuit

電子回路論第10回
Electric Circuits for Physicists

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoComment: Impedance match/mismatch
Propagation of a wave:

Impedance match: complete absorption(propagation without reflection)
Mismatch: wave reflection

Impedance match/mismatch is an important concept applicable toa broad area of physics.

 Antenna: should be matched to the vacuum.
EM wave propagation simulation: boundary is shuntedwith the characteristic impedance of vacuum.
 Optics: impedance mismatch → disagreement in refractive index

 Plasma: should be matched to electrodes for excitation.
 Phonon impedance mismatch at low temperatures: Kapitza resistance Sound insulated booth: should have sound impedance mismatch.

Ch6. Noises and Signals

Chapter 6 Noises and Signals

Outline

6.1 Fluctuation
6.1.1 Fluctuation-Dissipation theorem
6.1.2 Wiener-Khintchine theorem
6.1.3 Noises in the view of circuits
6.1.4 Nyquist theorem
6.1.5 Shot noise
6.1.6 1/f noise
6.1.7 Noise units
6.1.8 Other noises
6.2 Noises from amplifiers
6.2.1 Noise figure
6.2.2 Noise impedance matching

Noises
Electric circuits transform:

1) Information
2) Electromagnetic poweron some physical quantities like voltages, current, …
Noises: stochastic (uncontrollable, unpredictable by human) variationin other words, fluctuation in such a quantity.

Internal
noise

Intrinsic noise: Thermal noise (Johnson-Nyquist noise),
Shot noise
Noise related to a specific physical phenomenon
Avalanche, Popcorn, Barkhausen, etc.

1/f noise: Name for a group of noises with spectra 1/f.
External
noiseEMI, microphone noise, etc.

6.1 Fluctuation
Quantity 𝑥, fluctuation 𝛿𝑥 = 𝑥 − 𝑥

𝑔 𝑥 : distribution function of x
Fourier transform:𝑢 𝑞 : characteristic function of the distribution
From Taylor expansion, any moment can be obtained as

Moments to high orders → reconstruction of 𝑔(𝑥)

6.1 Fluctuation
In electric circuits we need to consider two kinds of averages:

substance 1

substance 2

substance 3

𝑥1

𝑥2

𝑥3

𝑥𝑗 : independent
𝑥4

𝑥
𝑥1
𝑥0

𝑥6

Ensemble average: 𝑥

Markovian𝑥8𝑥𝑥10
𝑥3 5 𝑥
7
𝑥12
𝑥2
𝑥9 𝑥

m-th order Markovian

11

𝑥13

t

Time averageof fluctuating
variable:𝑥

Random process to distributionxtx
𝑥1 (𝑡)𝑥2 (𝑡)𝑥3 (𝑡)t

The averaging interval
should be longer thanm in m-th order Markovian.

6.1.1 Fluctuation-Dissipation Theorem
久保亮五
Ryogo Kubo 1920-1995

Harry Nyquist
1889-1976

Nobert Wiener
1894-1964

Aleksandr Khinchin
1894-1959

Power Spectrum
Consider probability sets in the interval [0,T).
set index: j(Power)∵ cross product terms are averaged out
Random process:
Gaussian distribution in timeThen(non-Markovian)Power Spectrum
Power spectrum 𝑮(𝝎)Frequency band width 𝛿𝜔 : separation between two adjacent frequencies

6.1.1 Fluctuation-Dissipation Theorem
R

V(t)

L

C

Johnson-Nyquist noise

𝑉(𝑡) noise power spectrum → 𝐺𝑣 (𝜔)Johnson-Nyquist noise
Thermal noise

White noise
One representation of the fluctuation-dissipation theorem

6.1.2 Wiener-Khintchine Theorem
Autocorrelation function

Wiener-Khintchine theorem

6.1.2 Wiener-Khintchine Theorem
Example)𝜏0 = 10−7 s (10MHz)

𝑓 −2

Pole

6.1.3 Electric circuit treatment of noise𝑍(𝑖𝜔)+ NoiseNoiseless impedance𝑍(𝑖𝜔)Noise power source6.1.4 Nyquist TheoremlJ𝑍0 = 𝑅

R
R
V

Mode density on a transmission line with length lBidirectional → Freedom × 2
Bose distribution

6.1.4 Nyquist Theorem
Thermal energy density per freedom

Thermal energy density in band ∆𝜔, a half of which flows in one-direction
Energy flowing out from the end:

equals the energy supplied from the noise source.

→ Noise Temperature

Thermal noise6.1.5 Shot Noise
Single ElectronTime domain: 𝛿-function approximation

Uniform 2e in frequency domain: fluctuation at each frequency
Coherent only at 𝑡 = 𝑡0
Current fluctuation density for infinitesimal band 𝑑𝑓

6.1.5 Shot Noise6.1.5 Shot Noise
Double Electron

𝜙: coherent phase shift → averaged out

N-Electron

Quantum mechanical correlation → Modification from random6.1.5 Shot Noise
Example: pn junction
Current-Voltage characteristics:
Differential
resistance6.1.6 1/f noise

Noise power
[dBm]

1

f  1White noise

Corner frequency

f fc6.1.6 1/f noise

J. S. Bach, Brandenburg Concerto No.1
A. Vivaldi, Four Seasons, Spring

Kawai Naoko, Smile for me

S. Sato, Keshin (incarnation) II“Unit” of Noise
Noise: Power spectrum per frequency

Other noises: Barkhausen noisePopcorn noise
Popcorn noise, Burst noise
pn junction

Two-level system

Amplitude distributions of random-type noises

Amplitude distribution of popcorn noiseAvalanche noise

white noise
avalanche or Zener breakdown

Zener voltage standard diodeAvalanche Photo-Diode (APD)6.2 Noises from Amplifiers
Amplifier with noise
𝑒𝑅

𝑉𝑠𝑖𝑔

𝑅

𝑒𝑛

𝑗𝑛
Noiseless
AmplifierNoise source6.2 Noises from Amplifiers
Amplifiers: the elements have characteristic noises,
power sources work as noise sources

Noiseless amplifier + Noise source = Amplifier with noise
Power gain 𝐺𝑝

Signal to noise ratio: S/N ratio

Noise Figure:6.2.2 Noise impedance matching

𝑒𝑛
𝑍𝑆

𝑒𝑆

𝑍𝑖

𝑗𝑛

6.2.2 Noise impedance matching
Noise temperature andmatched source impedance
Output noise temperature:

Minimize 𝑇𝑛 :Noise matching conditionReferences
C. Kittel, "Elementary Statistical Physics", (Dover, 2004).
遠坂俊昭「計測のためのアナログ回路設計」(CQ出版社，1997).
Anton F. P. van Putten, "Electronic Measurement Systems",(IOP pub., 1996).

寺本英，広田良吾，武者利光，山口昌哉「無限・カオス・ゆらぎ」(培風館，1985).

Exercise E-1
Obtain the dispersion relation in the following
transmission line.

C

L

L

L

K

C

K

C

K

Exercise E-2
Show that the power spectrum 𝐺(𝑓) of voltage noise acrossthe impedanceis given as
Assume that thermal noise energy per unit time is 𝑘𝐵 𝑇∆𝑓.

𝑍(𝑓)
𝑍′
𝑉

(hint) From the above assumption we canskip the discussion on the mode energy intransmission line. Instead consider the casein the left figure, in which 𝑍′ is matched to𝑍 asExercise E-3
𝐶2

(a)

𝐶1

L

600W

600W

𝐶2
50W

𝐶1

𝐿1

𝐿2

50W

(b)

A preamplifier with FETs for an FM receiver has the output impedance of 600W.
The FM receiver has the input impedance of 50W and we need to make
impedance matching. The central frequency is 85MHz, the effective with of
amplification is 10MHz. Obtain 𝐶1 , 𝐶2 , 𝐿 in the matching circuit with 3 digits
significant figures.

(hint) Express L with a parallel of L1 and L2 as shown in (b). The left resonancecircuit should be tuned to 85MHz, width10MHz. Then the left and the rightcircuit should be impedance matched.

Exercise E-3
𝐶2

(a)

𝐶1

L

600W

600W

𝐶2
50W

𝐶1

𝐿1

𝐿2

50W

(b)

FM受信機のプリアンプをFETで作ったところ，出力インピーダンスが600Wになった．受信機の入力インピーダンスは50Wなので，インピーダ
ンスマッチを取る必要がある．中心周波数を85MHz，有効周波数幅を10MHz，として(a)のような回路でマッチを取ると，回路定数𝐶1 , 𝐶2 , 𝐿はどうなるか．有効数字3桁で答えよ．

(ヒント) (b)のようにインダクタンスを２つに分割し，左の共鳴回路で85MHz，10MHz幅に同調させる．この後，左右のインピーダンスが一致するように定数を求める．

電子回路論第11回
Electric Circuits for Physicists
東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoOutline
6.2 Noise from amplifiers
6.2.1 Noise figure
6.2.2 Noise impedance matching
6.3 Modulation and signal transfer
6.3.1 Modulation/demodulation
6.3.2 Amplitude modulation
6.3.3 Angle modulation
6.3.4 Demodulation of frequency
modulated signal
6.3.5 Modulation and noise
FM broadcast test

6.2 Noise from Amplifiers

𝑒𝑅
𝑉𝑠𝑠𝑠

𝑅

Amplifier with noise6.2 Noise from Amplifiers
Amplifier with noise
𝑒𝑅
𝑉𝑠𝑠𝑠

𝑅

𝑒𝑛
𝑗𝑛
Noise source

Noiseless
Amplifier6.2 Noise from Amplifiers
Amplifiers: the elements have characteristic noises,
power sources work as noise sources
Noiseless amplifier + Noise source = Amplifier with noise
Power gain 𝐺𝑝
Signal to noise ratio: S/N ratio

Noise Figure:6.2.2 Noise impedance matching
𝑒𝑛
𝑍𝑆
𝑒𝑆

𝑍𝑖

𝑗𝑛

Optimization of S/N ratio including the noise-source in the amplifier(a care should be taken to the effect of noise to the object)
Noises from the signal source, amplifiers: repel as much as possible
Signals from the source: absorb …
Noise temperature method: not almighty

6.2.2 Noise impedance matching
Nyquist theorem:
Noise temperature definition (𝐽 𝑓 , 𝑉(𝑓))Noise temperature andmatched source impedance
Output noise temperature:

Minimize 𝑇𝑛 :Noise matching condition

6.3 Signal transmission
Baseband communication
Electric communication
Carrier communication
Input

Modulator

𝑓(𝑡)

𝑠(𝑡)

Transmission

Demodulator(Detector)

Carrier𝑐(𝑡)Amplitude modulation
Modulation
Frequency (Phase) modulation

AnalogPulse

6.3.2 Amplitude modulation
𝑓(𝑡)
𝑐(𝑡)
𝑠(𝑡)

𝑡
𝑡

𝑡

m: Modulation index
0<𝑚≤1

6.3.2 Amplitude modulation𝑓(𝑡): Real

−𝜔𝑐

𝜔𝑐

Upper side band (USB), Lower side band (LSB)

6.3.2 Amplitude modulation (circuit example)
Collector modulation circuit

C-class amplification (non-linear) region

Modulation voltage

Current throughthe inductor

6.3.2 Amplitude modulation (circuit example2)
Modulation of oscillator circuit
Soft limiter circuit
Vref
𝑎=

𝑅3
Vin

𝑅1

−
+

𝑅2

𝑅4

𝑅2 ∥ 𝑅6
𝑅1

𝑉𝑜𝑜𝑜 = −𝑎𝑉𝑖𝑖 + 𝑏

VL+

Vout

𝑅6
−Vref

Vout

𝑅5

𝑅4
𝑅4
𝑉𝐿− = − 𝑉𝑟𝑟𝑟 − 1 +
𝑉𝑡𝑡
𝑅3
𝑅3

Vin
𝑎=

𝑅2
𝑅1

VL−
𝑎=

𝑅2 ∥ 𝑅4
𝑅1: controllable with Vref

6.3.2 Amplitude modulation (circuit example2)
𝑅𝑓

−

𝑅𝑓

+

𝑅𝑏
C

𝑉𝑏𝑏𝑏𝑏

𝑉𝑚𝑚𝑚

𝑅𝑓
𝑅𝑓

𝑅𝑓

−

+

C

𝑅𝑑

𝑅1

−
+

𝑅2

𝑅𝑎
𝑅𝑎

𝑉𝑜𝑜𝑜

𝑅𝑏

Delayed feedback → Oscillation

The amplitude is softly limited with the modulation voltage.

6.3.2 Amplitude modulation (circuit example2)

6.3.2 Amplitude modulation (circuit example2)

6.3.2 Amplitude modulation (demodulation)

6.3.3 Angle modulation

6.3.3 Angle modulation

Differential angular frequency

Frequency 𝜔 component: only phase shift 𝜋/2 :No difference in signal outlook.

6.3.3 Angle modulation (Frequency modulation)sin 𝜔𝑝 𝑡 : Periodic function with 𝑇 = 2𝜋/𝜔𝑝Fourier series expansion

First kind Bessel function

6.3.3 Angle modulation (Frequency modulation)

Actual band width:

6.3.3 Angle modulation (circuit example)
Voltage Controlled Oscillator (VCO)
VC

Modulation
input
SignaloutputVoltage controlled resonatorPhase Lock Loop (PLL)

Phase comparator

Loop filter

Voltage Controlled
Oscillator (VCO)

Standard
Signalf
N-Frequency
DividerNf6.3.3 Angle modulation (circuit example)

Phase comparator

Loop filter+
Standard
Signalf

Voltage Controlled
Oscillator (VCO)
N-Frequency
DividerModulation
Signal6.3.4 Angle modulation (Frequency demodulation)

Double tuned circuit6.3.4 Angle modulation (frequency demodulation)
PLL FM demodulation
Loop filter
Phase
𝑋(𝑠)
input

+
−

Phase comparator

𝑋𝑒 (𝑠)× 𝑘𝑒𝑘𝑜𝑠

Voltage control
signal𝑌(𝑠)ここに数式を入力します。
𝐻(𝑠)𝑘𝑜 � 𝑦(𝑡)𝑑𝑑𝑔 𝑡 : Frequency modulation signal (original)

6.3.5 Modulation and noiseAM
Received signal
Demodulated output
Averaged signal power

Received

Demodulated

6.3.5 Modulation and noise𝜔W : Noise bandwidth (assumption: white)
Noise power:

Received

Demodulated:Power transmission efficiency

6.3.5 Modulation and noise
FM, PM

In phase

SignalOut of phase

Noise

Phase noise

Time-dependent part in 𝑉𝑟 (𝑡) can be cut with a limiter circuit.

6.3.5 Modulation and noise (Diode limitter)

Output

Input

V(t)

V(t)

Input
t

Outputt

6.3.5 Modulation and noise

Noise: White, Power spectrum density 𝑛𝑎 /2, Band width 𝜔𝐵

𝑛𝑎 𝜔𝐵
Noise power: 𝑁𝑖 =2𝜋
Phase modulation

Averaged signal power:
Averaged noise power:

𝐴2𝑟
Signal power:2
𝑘𝑝2 𝑓 2𝜙𝑛 (𝑡): Uniform in [0, 2π] → ignored

6.3.5 Modulation and noise

Frequency modulation
Demodulated output 𝑑𝜃/𝑑𝑑Signal, power: 𝑘𝑓 𝑓(𝑡),6.4 Discrete signal
6.4.1 Sampling theorem
Sampled signalx(t)δ-functions with the period τ

Isao Someya
1915-2007

Claude Shannon
1916-2001

1928 H. Nyquist
1949 C. Shannon
染谷勲

t
x(t)

X(ω)

τ
ω

t

6.4.1 Sampling theorem

6.4.1 Sampling theorem“Cutting out” the frequency spectrum
2π/τ𝜔ℎ : Highest frequency
in 𝑋�𝜏 (𝜔)6.4.1 Sampling theorem: Reconstructing signal

𝑃𝜋 (𝜔)
𝜏

sinc(x)

1

1

ω
𝜋
−
𝜏

𝜋
𝜏

0.5

0

-10

0

x

10

電子回路論第12回
Electric Circuits for Physicists

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoOutline
6.4 Discrete signal
6.4.1 Sampling theorem
6.4.2 Pulse amplitude modulation (PAM)
6.4.3 Discrete Fourier transform
6.4.4 z-transform
6.4.5 Transfer function of discrete time signal
Ch.7 Digital signals and circuits
7.2 Logic gates
7.3 Implementation of logic gates
7.4 Circuit implementation and simplification oflogic operation

6.4 Discrete signal
6.4.1 Sampling theorem
Sampled signalx(t)d-functions with the period t

Isao Someya
1915-2007

Claude Shannon
1916-2001

1928 H. Nyquist
1949 C. Shannon
染谷勲

t
x(t)

X(w)

t
w

t

6.4.1 Sampling theorem

6.4.1 Sampling theorem“Cutting out” the frequency spectrum
2p/t𝜔ℎ : Highest frequency
in 𝑋𝜏 (𝜔)

6.4.1 Sampling theorem: Reconstructing signal

1

𝜏sinc(x)𝑃𝜋 (𝜔)
1

w
𝜋
−
𝜏

𝜋
𝜏

0.5

0

-10

0

x

10

6.4.2 Pulse amplitude modulation (PAM)
s(t)
f(t)
t

t
Carrier: 𝛿𝜏 (𝑡)Demodulation = Reconstruction of continuous signal
from sampled data.

tDemodulation of PAM and a trick in the sampling theorem
In the sampling theorem, though we only have discrete-time
data, we can reconstruct complete original signal.
↑
Assumption: we have data in infinite period [−∞, +∞].

However in actual situations we can never have such data.

Need to consider handling data in a finite period.

6.4.3 Discrete Fourier transform
𝑓(𝑡)

f (t)

t
0

zt

-z

0

z

2z

Assumption:
can be assumed withoutloosing generality

Fourier expansion:

6.4.3 Discrete Fourier transformDiscreteness:

Twiddle factor:

6.4.3 Discrete Fourier transform

Twiddle factor:

Discrete Fourier transform:(DFT)6.4.3 Discrete Fourier transform6.4.4 z-transformDiscrete Laplace transform: z-transform

one-sided z-transform6.4.4 z-transform6.4.4 z-transform6.4.5 Transfer function for discrete time signalℎ𝑛 : (impulse) response to 𝛿(𝑛𝜏), response to discrete signal 𝑓𝑛 = 𝑓 (𝑛𝜏): Transfer function

Crystal lattice and X-ray diffraction
Max von Laue
1879-1960

Laue pattern

Diamond lattice

[100]

[100]
[010]
[111]

Optical Frequency Comb

Frequency Comb

Theodor Hänsch, Max-Planck Institute, Science 2008

Measurement of the Doppler effect in cosmic expansion

Byzantine mosaic

Chapter 7

Digital signal
and circuits
Chartres Blue
(Stained glass)Ch.7 Digital signal and circuitss (t)Discrete time analog

d (t)

t 2t

t

5t

Value discretized → Digital signal
Signal unit : 0 xor 1 (bit)Boolean algebra : F xor T
Voltage level : L xor H

Multiple bit → binary operation → parallel signalt7.2 Logic gates
Digital signal=logic value  Logic operation : logic gates
De Morgan's laws: 𝑥 + 𝑦 = 𝑥 ∙ 𝑦, 𝑥 ∙ 𝑦 = 𝑥 + 𝑦

Combinational logic  Truth table
Sequential logic  Timing chart7.2.1 Combinational logic: Single input gates

Truth table

Circuit symbol

7.2.2 Combinational logic: Double input gates7.2.3 Sequential logic: Flip-Flop (FF)RS (reset-set) Flip-Flop (FF)Truth table

SymbolEquivalent circuit with discrete gates7.2.3 Sequential logic: Flip-Flop (FF)JK Flip-Flop
Truth table

SymbolEquivalent circuit with discrete gates7.2.3 Sequential logic: D-FF, T-FF
D-FF

Symbol

T-FF

SymbolTruth table

Truth table7.2.4 Sequential logic: Counters
Unsynchronizedcounter(ripple counter)

Timingchart

7.2.4 Sequential logic: Counters
Synchronized counter

Equivalentcircuit withdiscrete gates

Timing chart7.3 Implementation of logic gates
NAND gates

TTL (transistor-transistor logic)

CMOS (complimentary MOS)

7.3 Implementation of logic gates

LT Spice
simulation

7.3 Implementation of logic gates
Voltage levels diagram

TTL logic family evolution

Legacy: don’t use
in new designs

Widely used today

CMOS logic family evolution
obsolete

General trend:• Reduction of dynamic losses throughsuccessively decreasing supply voltages:
12V  5V  3.3V  2.5V  1.8V
CD4000
LVC/ALVC/AVC• Power reduction is one of the keys toprogressive growth of integration

Summary
TTL
Logic
FamilyCMOS

TPD

Trise/fall

VIH,min

VIL,max

VOH,min

VOL,max

Noise
MarginExercise F-1
Show that the following circuit works as a demodulator of
frequency modulation (FM) signal (quadrature demodulator).
input

multiplierLow-pass
filterPhase shifter∆𝜃Here the phase shifter gives the shift
proportional to the frequency difference
between input and the career frequency
𝜔0 . The shift at 𝜔0 is 𝜋/2 as shown in
the right (this can be achieved withresonant circuits). The low-pass filter
cuts components with frequencies ashigh as 𝜔0 .

𝜋
2output𝜋∆𝜃 = 𝑎 𝜔0 − 𝜔 +
2

𝜔0

𝜔

Exercise F-1(hint)

Assume the original signal 𝑓(𝑡) is much slower thanthe carrier 𝐴 cos 𝜔0 𝑡 . Then the input can beapproximated as

Then the phase shifter output is

Taking product and high-frequency filtering gives …
(use 𝑎𝑘𝑓 𝑓(𝑡) ≪ 1).

Exercise F-2
In the following phase lock loop (PLL) circuit, the initial (𝑡 = 0)
oscillation frequency of voltage-controlled oscillator (VCO) 𝜔
deviates from 𝑁𝜔0 by ∆𝜔. Obtain the relaxation time of 𝜔 to 𝑁𝜔0 .
Input
freq. 𝜔0phase 𝜃𝑖

𝑉𝑐 (𝜃𝑖 − 𝜃𝑜 /𝑁) gain 𝐺𝑝
Phasecomparator

freq. 𝜔/𝑁

Loop
filter

Coef. 𝐴 for freq. shift
OutputVCO
freq. 𝜔phase 𝜃𝑜𝑁- divider(hint) Here we can put 𝜃𝑖 = 0 hence input = 𝑉𝑖 sin 𝜔0 𝑡 withoutloosing generality. Similarly output = 𝑉𝑜 sin 𝑁𝜔0 𝑡 + 𝜃𝑜 (𝑡) .
Now 𝜔 = 𝑁𝜔0 + 𝑑𝜃𝑜 /𝑑𝑡 and it is easy to write 𝑑𝜃𝑜 /𝑑𝑡 with 𝐴,
𝐺𝑝 , 𝑉𝑐 , 𝜃𝑜 (𝑡) and a constant.

Exercise F-3
Solve the difference equation below with z-transform.

(hint) z-transform of 𝑛 is

as in the table (slide no.14).

Then z-transform of 𝑥(𝑛) : 𝑋(𝑧) is easily obtained. Inverse ztransform gives 𝑥 𝑛 .

Answer sheet submission deadline: 11th Jan. 2017.

電子回路論第13回
Electric Circuits for Physicists

東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoOutline
7.2 Sequential digital circuit
7.3 Implementation of logic gates

7.4 Circuit implementation and simplificationof logic operation
7.5 DA/AD converter circuits
7.6 Digital filters (digital signal processing)7.2.3 Sequential logic: Flip-Flop (FF)RS (reset-set) Flip-Flop (FF)Truth table

SymbolEquivalent circuit with discrete gates7.2.3 Sequential logic: Flip-Flop (FF)JK Flip-Flop

Truth table

SymbolEquivalent circuit with discrete gates7.2.3 Sequential logic: D-FF, T-FF
D-FF

Symbol

T-FF

SymbolTruth table

Truth table7.2.4 Sequential logic: Counters
Unsynchronizedcounter(ripple counter)

Timingchart

7.2.4 Sequential logic: Counters
Synchronized counter
Equivalentcircuit withdiscrete gates

Timing chart

Standard gate logic IC packaging
Full pitch

Half pitch surface mountMounting of logic ICs

Printed board with soldering

Surface mounting

Wire wrapping7.3 Implementation of logic gates
NAND gates

TTL (transistor-transistor logic)

CMOS (complimentary MOS)

7.3 Implementation of logic gates

LT Spice
simulation

7.3 Implementation of logic gates
Voltage levels diagram

TTL logic family evolution

Legacy: don’t use
in new designs

Widely used today

CMOS logic family evolution
obsolete

General trend:• Reduction of dynamic losses throughsuccessively decreasing supply voltages:
12V 5V 3.3V 2.5V 1.8V
CD4000
LVC/ALVC/AVC• Power reduction is one of the keys toprogressive growth of integration

Summary
TTL
Logic
FamilyCMOS

TPD

Trise/fall

VIH,min

VIL,max

VOH,min

VOL,max

Noise
Margin7.4 Circuit implementation and simplification of logic operation
Truth table  Simplification  Circuit diagram
Simplification

Visual method: Karnaugh mapping

Quine-McClusky algorithm

Product of all the logic variables: canonical expansion

principal disjunctive canonical expansion (主加法標準展開)

Ex)

Or in binary: Y = 0011+1111+0111+1101+1100+1011

Quein-McClusky algorithm
Classification
with the numberof 1

Num.of 1

smallest

compress1

compress2

2

0011

0_11

__11

1100

_011

__11

0111

110_

1011

_111

1101

1_11

1111

11_1

3

4
Y = __11+110_+11_1

First simplification
smallest

0011
__11
110_

1100

0111

1011

◎

◎

◎
◎

Final form

1111
◎

◎
○

11_1
Y = __11+110_

1101

○

Wolfram AlphaDesign of sequential logic circuit: State diagram
State (transition) diagram:Transition

state

input

outputFF output:

Karnaughmap
simplification
Recursion equation:stateEx) 2-bit counter with two T-FF

Design of sequential logic circuit: State diagram
Characteristic equation(recursion equation)

T-FF :

y
x

TFF1

TFF2

7.5 AD/DA converter circuit
7.5.1 Digital to Analog conversion
Resistor string type DA convertern bits converter→ 2𝑛 outputs!, 2𝑛 resistors!

Input

MOS switches7.5.1 Digital to Analog conversion
Resistor ladder type DA converter

MOS switch array
ON: 𝑉𝑆
OFF: ground

Input 0,0, ⋯ , 0,1,0, ⋯ , 0

𝑑𝑘 = 1, others = 0
𝑉𝑆

2R

2R

2R

R

2R
1

𝐽1

2R

R
k

𝐽1

n

7.5.1 Digital to Analog conversion
𝑅f
2R

𝐽𝑙

𝐽𝑙 /2

2R

2R

𝐽out
𝐽𝑙 /2

From the superposition theorem:-+

7.5.1 Digital to Analog converter
Pulse width modulation (PWM)

Digital signal →PWM signal with a countert
Low pass
Analog signal

D-class amplifier7.5.2 Analog-Digital converter
Successive comparison type AD converter

-

-

+

+

Flush type
input

𝑉𝑆
Sampleandholdcomparators

R

R

R

R

⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
Decoder
Output7.5.2 Analog-Digital converter
Integrating Analog-Digital Converter
clock

Sampleandhold

𝑉count+-
comparator

Counter𝑉𝑖𝑛outputreset

𝑉𝑜slope =

𝑉𝑆
𝐶𝑅

𝑉𝑖𝑛

C
start
𝑉𝑜

R
-

𝑉𝑆

t

+
𝑉countt

Filter Circuit
Transmission

Transmission

High pass filter
Low pass filterw

w

Transmission

Transmission
Band passfilterNotch filterww

An example for retrieving data from noise

Signal with noise

Detection ofcarrierResults
Frequency filter

Original signal
1.5

1.0

0.5

0

1

2

3

4

5

6

7.6 Digital filter (as a digital signal processing)
Digital filtering:

Block diagram representation of operations

constant multiplieradder

delay (shift resistor)

Block diagram example

Feedback and transfer function

電子回路論第14回 (最終回)Electric Circuits for Physicists
東京大学理学部・理学系研究科
物性研究所
勝本信吾
Shingo KatsumotoFilter Circuit
Transmission

Transmission

High pass filter
Low pass filterw

w

Transmission

Transmission
Band passfilterNotch filterww

An example for retrieving data from noise

Signal with noise

Detection ofcarrierResults
Frequency filter

Original signal
1.5

1.0

0.5

0

1

2

3

4

5

6

7.6 Digital filter (as a digital signal processing)
Digital filtering:

Block diagram representation of operations

constant multiplieradder

delay (shift resistor)

Block diagram example

Feedback and transfer function

FIR filter
Finite impulse response (FIR) filter

A Simple example of FIR filter
Moving average, differentiation:

A Simple example of FIR filter
Differentiation of moving average:

IIR Filter
Infinite impulse response (IIR) filter:

Stability condition:

Conversion of z-transform: 𝑧 > 1 the poles should be in 𝑧 < 1

Design of FIR filter: Window function
Ideal low pass filter

Nyquist𝐺 𝑒 𝑖𝜔𝜏

𝑛cut = 20

Cut the series at a finite number

Ripples in frequency characteristics
𝜔/𝜔𝑁Design of FIR filter: Window function
Sudden cutting of z-transform series → Ripples

Cut with a smooth function
Kaiser window
𝐼0 : 0th order 1st type modified Bessel function
𝐺 𝑒 𝑖𝜔𝜏

𝑛cut = 20
𝐺 𝑒 𝑖𝜔𝜏

𝜔/𝜔𝑁

𝜔/𝜔𝑁

Design of IIR filter
Transfer function: a rational function (有理式)
A way to design IIR filter: modification of analog filter transfer function

Remember: Butterworth filter

Heaviside function
Time discretization
with 𝜏 = 1:

Design of IIR filter
Impulse invariant method:

Bilinear z-transform (双一次z変換法) :4th Butterworth:Digital filter design web application
http://t-filter.engineerjs.com/

Adaptive filter
input

outputFilter with variable
coefficients
Coefficients
correction program++
reference

Least mean square method:Adaptive filter (adaptive line enhancer)
Delay

Filter with variable
coefficients

Coefficients
correction program++Digital filter implementation
Digital signal processing (DSP) board

PLD/FPGA with HDL
Example of programmable
logic device (PLD)
circuitExample of fieldprogrammablegate array (FPGA)
circuitFPGA ∈ PLD

Hardware description language, HDL

RTL: register transfer level
Cf. SPICE

Spice+HDL mixed circuit simulation
Easier logic circuit design

Polyphony: Python

Verilog HDL http://www.sinby.com/PolyPhony/index.html

C-to-Hardware compiler (CHC) http://anvil.co.jp/?page_id=296

Mixed circuit simulation

Overview
Electric Circuits: Treasury of Languages and Concepts
Concept of lumped electromagnetic field
Metal
Lumped constantcircuit

Semiconductor
Classical mechanics

designability
Linear systems

Quantum mechanics

Concept of distributed constant circuit

Non-linear systems
Z-transform
Noise
SignalSampling
Real electromagnetic
field

DigitalInformation theory電子回路論レポート問題: 1. DA変換回路(1) 次の図のような抵抗ラダー型DA変換回路を考える．講義で扱ったものと端の処理だけが違っている．この回路に2進数列 𝑑𝑘 (𝑘 = 1, . . , 𝑛)が入力されたとき，出力電圧𝑉out を求めよ． 𝑉out は，高入力インピダンスアンプで受けるものとする．
d1

d n- 1

d2

dn

Vs

2R

2R

2R

2R

2 R
R

R

R

𝑉out

(2) 手元に，{R0/2k} (k =0, …, n)の抵抗値列を持つ抵抗，抵抗値𝑅f の抵抗，OPアンプ，電圧VSの標準電源，n +1個のnチャネルMOSスイッチ，同じくn +1個のpチャネ
ルMOSスイッチがある．これらを使って，2進数列{𝑑𝑘 }が入力された時に

≮
≘
≦
≫
≒
≯
≵
≴
≳
≫
≖ ∽⊡≖≒≤∲∰≫∽∰
を出力するDA変換器回路を考えなさい．

電子回路レポート問題 ： 1. DA変換回路(3) (2) と同様，に手元に，{2kC0} (k =0, …, n)の抵抗値列を持つキャパシタ，電圧VSの標準電源，n +1個のnチャネルMOSスイッチ，同じくn +1個のpチャネルMOSスイッチがある．これらを使って，2進数列{𝑑𝑘 }が入力された時にという出力が得られるようなDA変換回路を考えなさい．ただし，出力は入力イン
ピダンスが高くバイアス電流が無視できるような増幅器で受けることとする．

Problems for the final report: 1. DA conversion circuits(1) Let us consider the following resistance ladder DA conversion circuit. The
right end is a bit different from the one we treated in the lecture. Calculate theoutput voltage 𝑉out for the input 𝑑𝑘 (𝑘 = 1, ⋯ , 𝑛).
d1

d n- 1

d2

dn
Vs

2R

2R

2R

2R

𝑉out

2 R
R

R

R

(2) We have resistors with values 𝑅0 /2𝑘 (𝑘 = 0, ⋯ , 𝑛), and 𝑅f , an OP amp., astandard voltage source of the voltage 𝑉S , 𝑛 + 1 n-channel MOS switches, 𝑛 + 1
p-channel MOS switches. With these components, design a DA conversion
circuit which has the output
for the binary input 𝑑𝑘 .

Problems for the final report: 1. DA conversion circuits(3) We have capacitors with values 2𝑘 𝐶0 (𝑘 = 0, ⋯ , 𝑛), a standard voltage
source of the voltage 𝑉S , 𝑛 + 1 n-channel MOS switches, 𝑛 + 1 p-channel MOS
switches. With these components, design a DA convertor circuit which has theoutputfor the binary input 𝑑𝑘 .

電子回路論レポート問題：2. 分布定数回路
L/2
C

L

L

L

C

C

⋯

上図のように，端のインダクタンスL/2を除いてLとCが無限に繰り返す回路がある．この回路の，周波数軸上での透過域と減衰域を求めよ．
また，左の端子から見たインピーダンスの周波数特性(周波数wに対するインピーダンス)を求めよ．

Problems for the final report: 2. Distributed constant circuit
L/2
C

L

L

L

C

C

⋯

Consider the above circuit with L, C infinite repetition to the right and theinductor L/2 at the left end. Obtain the transmission range and the attenuation
range in the frequency domain. And what is the total impedance from the left
end for the frequency w.

電子回路論レポート問題：3. OPアンプ回路
次のような回路部品がある．
高精度OPアンプ ・・・ 4個
定電圧ダイオード (逆方向に電圧を印加するとツェナートンネルにより一定電圧を発生する) 2.5V ・・・ 1個
これらを使って，低温で試料の電気抵抗を測定するための回路を構成しなさい．
試料は下の図のように，電流端子，電圧端子が別に出た構造をしている．
電圧
電流

電圧
電流

ただし，
(a) 抵抗，キャパシタ，インダクタの類の受動素子は適当に追加してよい．
(b) OPアンプの電源を供給するためのトラッキング電源は準備されているものとする．
(c) 測定抵抗の範囲は100W～10kWで，接触抵抗を含めて50kW以内である．
(d) OPアンプのオフセット電圧，バイアス電流は無視できるとする．従ってオフセット調整回路を入れる必要はない．

Problems for the final report: 3. OP amp. circuit
We have the following components:
4 high precision operational amplifiers,
1 high precision Zener tunnel diode with the constant voltage 2.5V (this diodeprovide precise 2.5V for the reverse bias).
With these components, design a circuit to measure the electric resistance of asample at low temperatures. The shape of the sample is shown below:
Voltage
Current

Voltage

Current(a) You can add any passive elements (resistors, capacitors, inductors).
(b) The power supply for the OP amps. is ready.
(c) The sample resistance range is from 100W to 10kW, lower than 50kW
including the contact resistance.
(d) The offset voltages, the bias currents of the OP amps. can be ignored. No
need for the offset cancellation circuit.

電子回路論レポート問題：4. ディジタル・フィルター
𝑦𝑛

𝑥𝑛

+

+

𝑐0
𝑧 −1

+

+
𝑎1

𝑏1

𝑧 −1
𝑎2
(1) 上のブロックダイアグラムから，出力𝑦𝑛 , 𝑦𝑛−1 , 𝑦𝑛−2 と入力𝑥𝑛 , 𝑥𝑛−1 , 𝑥𝑛−2 との関係
式を示せ．
(2) 係数𝑎1 ， 𝑎2 ， 𝑏1 ， 𝑐0 が右上のような関係を満たすとき，このフィルターはどのような周波数特性を示すか．ただし，𝑔0 < 𝑓0 < 𝑓s /2 (サンプリング周波数)を満たすとする．(20𝑔0 = 10𝑓0 = 𝑓𝑠 としてグラフを描いてみよ．)

Problems for the final report: 4. Digital filter
𝑦𝑛

𝑥𝑛

+

+

𝑐0
𝑧 −1

+

+
𝑎1

𝑏1

𝑧 −1
𝑎2

(1) From the above diagram, write down the relation between the output
𝑦𝑛 , 𝑦𝑛−1 , 𝑦𝑛−2 and the input 𝑥𝑛 , 𝑥𝑛−1 , 𝑥𝑛−2 .
(2) When the coefficients 𝑎1 , 𝑎2 , 𝑏1 , 𝑐0 satisfy the above relations, obtain thefrequency characteristics of this filter. Draw a rough sketch of the graphfor 20𝑔0 = 10𝑓0 = 𝑓𝑠 .

電子回路論レポート問題: 5 離散フーリエ変換
http://kats.issp.u-tokyo.ac.jp/kats/electroniccircuit/report/reportdata.txtに，4096点のデータが入っている．各行に x f(x) という形で納められており，セパレーターはTAB記号 (ASCII番号9番)である．

このデータは，ある特定の周波数で励起した出力を取ったもので，信号は特定の周波数成分で応答する．ただし，その振幅は完全に一定とは限らない．
(1) 離散フーリエ変換 (現実的には高速フーリエ変換，FFTをかけることになると思われる)を施し，パワースペクトルの主ピークから信号の周波数を特定せよ．

(2) 特定の窓 (幅 256点程度が適当)を使って部分的にフーリエ変換を施し，上で求めた周波数に最も近い周波数成分振幅を求める．窓の位置を x が小さい位置から100点刻み程度に動かし，各点での振幅を窓位置に対してプロットせよ．
解答のみ記せばよい．プログラム等を示す必要はない．

Problems for the final report: 5 Discrete Fourier Transform
http://kats.issp.u-tokyo.ac.jp/kats/electroniccircuit/report/reportdata.txt
contains data of 4096 points. Each line has a point in the form x f(x).
The separator between x and f(x) is TAB (ASCII No.9).

The data are signal responding to an excitation with a particular frequency.
Hence the signal has the same central frequency but the amplitude is notnecessarily constant.
(1) Apply discrete Fourier transformation (DFT, practically fast Fouriertransformation) to the signal and extract the central frequency from themain peak.

(2) Carry out DFT to a window with a shorter period (256 points isappropriate) and obtain the amplitude of the central frequency component.
Shift the position of the window with a step size about 100 points. Plot theamplitude as a function of the window position.
You do not need to show your program codes for the analysis.

電子回路レポート問題： 6 インピーダンス整合
FETを用いたアンプで信号ラインの特性インピーダンスとFETの入力インピーダ
ンスとの間で整合を取りゲインを最大化すると，雑音も増幅され，雑音指数(Noise figure, NF) が悪くなる．
そこで，50 Ω ラインの信号をFETから見たときのインピーダンスが800 Ω程度になるように変換する．入力は80 MHz～90 MHzの共鳴フィルターを構成する．

L

50 W

𝑁1

𝑁2

C

800 W

これを左図のようにコイルに中間タッ
プを出すことで行う．全体のコイル巻
き数𝑁2 に対してタップの巻き数位置は
𝑁1 とする．入力中心周波数は85 MHz，共鳴幅(半値幅)は10 MHzとする．
CとL及び𝑁1 の𝑁2 に対する比を求めよ．
(有効数字3桁)
ここでは，コイルのインダクタンスは巻き数の2乗に比例するとする．

Problems for the final report: 6 Impedance matching
In amplifiers with FETs, the noise matching condition, that optimizes the noise figure,usually deviates from the power matching one. The impedance conversion circuit
shown below thus converts the transmission line characteristic impedance 50 W into
800 W. The input to FET also constitutes a resonance filter for 80 MHz to 90 MHz.

L

50W

𝑁1

𝑁2

C

800W

The coil in the left has a tap at the winding
number 𝑁1 in the total winding number 𝑁2 .
The central frequency of input is 85 MHz,the resonance width (peak width at halfheight) is 10 MHz. Calculate C, L, and theration of 𝑁1 to 𝑁2 . (significant digits =3)
Assume the inductances are proportional tosquares of the winding numbers.