2015.11.26 Electric Circuit for Physicists 電子回路論 第７回

東京大学理学部・理学系研究科 物性研究所 勝本信吾 Shingo Katsumoto Exercise 3-3 P C - R3 - V + out + R1 In the circuit shown in the left, at point P, a waveform in the lower panel was R 2 observed. Here V+ and V- are power source voltages for + and – respectively.

V+ Draw a rough sketch of the

waveform for Vout. t

V- Review

Feedback

푈(푠) 푊(푠) 푊 푠 = Ξ 푠 푈(푠) − Ξ(푠) 푊 푠 = Ξ 푠 [푈 푠 − ℎ 푠 푊 푠 ] Ξ 푠 푊 푠 = 푈 푠 ℎ(푠) 1 + Ξ 푠 ℎ 푠

≝ 퐺 푠 푈(푠) 1 + Ξ 푠 ℎ 푠 > 1: Negative , < 1: Positive feedback ⊥ ∨ ≳ ∩ ∱ ≇ ∨ ≳∩ ∱ ≪ ≪ ⋀ ∡ ⊼ ≨ ∨ ≳ ∩ Condition for 1 + Ξ 푠 ℎ 푠 > 1: Negative feedback, < 1: Positive feedback

Im[Ξ 푠 ℎ 푠 ] Ξ 푠 ℎ(푠) 1 + Ξ 푠 ℎ(푠) ≝ 퐷(푠)

(-1,0) Re[Ξ 푠 ℎ 푠 ]

Negative feedback Oscillation point Positive feedback

If Ξ 푠 ℎ 푠 = −1 has solutions, the circuit may be unstable.

How can we judge? Criteria (Routh-Hurwitz, Nyqust, Liapunov, …) Zeros and poles of 퐷(푠)

Assumption 1: Ξ 푠 , Ξ 푠 ℎ 푠 are stable → Poles are on the left half plane of s.

Assumption 2: Ξ 푖휔 , Ξ 푖휔 ℎ 푖휔 → 0 for |휔| → ∞

푄(푠) 푞(푠) Ξ 푠 = , ℎ 푠 = ∶ 푃 푠 , 푄 푠 , 푝 푠 , 푞 푠 polynomials 푃(푠) 푝 푠

deg 푃 > deg 푄 , deg (푝) ≥ deg (푞) 푃 푠 푝(푠) 퐷 푠 = 1 + Ξ 푠 ℎ 푠 = 푃 푠 푝 푠 + 푄 푠 푞(푠)

(푠 − 훽1) ⋯ (푠 − 훽푛) 퐷 푠 = 퐷0 The same order (푠 − 훼1) ⋯ (푠 − 훼푛) Zeros and poles of 퐷(푠)

(푠 − 훽1) ⋯ (푠 − 훽푛) 퐷 푠 = 퐷0 (푠 − 훼1) ⋯ (푠 − 훼푛)

훽푖 : Zeros of 퐷(푠) → Poles of 퐺(푠)

∃훽푖 ∈ right half plane of 푠 → The circuit is unstable. 푛 푛

arg 퐷 = arg 푠 − 훽푖 − arg (푠 − 훼푖) 푖=1 푖=1 푠 = 푖휔 (on imaginary axis) Left half plane Right half plane +∞ 휔: −∞ → +∞ +∞ −휋 Number of zeros on the 휋 right half plane: 푚 훽푖 훽푖 ∆ arg 퐷 = 푛 − 푚 휋 − 푚휋 −∞ 푖휔 푖휔 −푛휋 = −2푚휋 −∞ Nyquist Plot and Criterion Ξ 푠 ℎ 푠 Ξ 푠 ℎ 푠

Harry Nyquist (1889–1976)

(−1,0)

∆ arg 퐷 = 0 ∆ arg 퐷 = −4휋 Stable Unstable Frequency Dependent Characteristics of OP-Amps

100 Cut-off frequency

휔푇 = 2휋푓푇 10 Phase rotates by 휋/2

1 A (Gain) A

0.1 10-3 10-2 10-1 100 101 102 103 푓 푓 휔 = 2휋푓 푇1 푇2 106 Multiple cut-off frequency: 4 10 Phase rotates more than 휋

2 10 If gain is larger than 1 at A (Gain) A phase shift p : 0 10 Dangerous!

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 푠 ℎ 푠

p phase shift

휔 = 0 Phase compensation Phase compensation Oscillator with an op-amp

gain phase

Seed of oscillation 4.3 Example of active element: Transistors Three types of semiconductors doping Intrinsic p-type n-type conduction band electrons band gap 퐸F 퐸F 퐸G

퐸F valence band holes

vacuum - for electrons -- diffusion vacuum for holes + + + pn junction pn junction thermodynamics

Consider electrons donors e- - + Vacuum + e- - + for electrons + e- - + + - + + e- - diffusion + e- + voltage (polarization) → energy cost 퐹 = 푈 − 푇푆

Voltage (internal energy cost) Diffusion (entropy)

Minimization of 퐹 → Built-in (diffusion) voltage 푉푏푖 4.3.1 I-V characteristics of pn junctions

------+ + - ++ ++ Reverse bias Forward bias Minority overcomes 푉 : go enhances 푉푏푖 : no go 푏푖 carrier injection

Rectification

푒푉 퐽 = 퐽0 exp − 1 푘B푇

Shockley theory Solar cell (injection of minority carriers)

16 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 AT&T Bell Laboratories Static characteristics

Bipolar transistor structures and symbols Why it can amplify?

cf. Solar cell: optical injection of minority carriers

Injection of minority carriers

Diffusive conduction

Mostly absorbed into the collector Why it can amplify? Current amplification : Linearize with quantity selection

퐽퐶 = ℎ퐹퐸 퐽퐵

Emitter-common current gain Linear approximation of bipolar transistor Hybrid matrix ≖ ∱ ∽ ≈ ∱ ∱ ≈ ∱∲ ≊ ∱ ∺ ≊ ∲ ≈ ∲ ∱ ≈ ∲∲ ≖ ∲ 푗2 ≶ ∱ ∽ ⋃≨ ∱∡∱ ≨ ∱⋃∲ ≪ ∱ ∽∡ ⋃≨ ≩ ∡≨ ≲ ≪ ∱ ≪ ∲ ≨ ∲∱ ≨ ∲∲ ≶ ∲ ≨ ≦ ≨ ≯ ≶ ∲ 푗1 ⋃h-parameters∡ ⋃ ∡⋃ ∡ ⋃ ∡⋃ ∡ (lower case: 푉1 local linear approximation)

푉2 Bias circuit for transistor For bias (dc) circuits Common emitter All the capacitors can be viewed as break line.

+푉 Small amplitude linear circuit for transistor

For small amplitude (high- frequency) circuits

All the capacitors can be viewed as Common emitter amplifier short circuits. Small amplitude linear circuit for transistor

Kirchhoff ≶ ≩ ∽ ≨ ≩≥ ≪≢ ∫ ≒ ≅ ∨ ≪≢ ∫ ≨ ≦ ≥ ≪≢ ∩ ≶ ≯ ∽ ≨ ≦ ≥ ≪≢ ≒ ≃

≁ ∽ ≶ ≯ ∽ ≨ ≦ ≥ ≒ ≃ ≶ ≩ ≨ ≩≥ ∫ ≒ ≅ ∨ ∱ ∫ ≨ ≦ ≥ ∩ ≒ ≃ ≨ ≦ ≥ ∱ Negative feedback ⊼ ≒ ≅ ⋀ Common emitter (grounded emitter) amplifier circuit 4.4 Field effect transistor (FET) (field effect transistor, FET) Junction FET (JFET) MESFET, MOSFET Static characteristics of FET

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

岡村迪夫 「OPアンプ回路の設計」 CQ出版社 松澤昭 「基礎電子回路工学」 電気学会

A. Agarwal, J. H. Lang “Foundations of Analog and Digital Electronic Circuits” (Elsevier, 2005).

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