電子回路論第 6 回 Electric Circuits for Physicists

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1 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

2 Outline 4.3 Feedback control Disturbance and noise PID control 4.4 PN junction transistors Diodes Bipolar junction transistors 4.5 Field effect transistors

3 Comment: Use of OP-amp at saturation voltages V out - V out V V in gradient A 0 rail to rail characteristics Compare V in with 0 0 V in Comparator V B A - V out A > B?

4 Hurwitz criterion Adolf Hurwitz Pole equation: (Otherwise the system is unstable.) Hurwitz matrix

5 Hurwitz criterion Hurwitz determinants Hurwitz criterion H 1, H n > 0 is trivial from 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.

6 Disturbance and noise on feedback control Circuit treatment of fluctuations: Prepare external power sources Express them as transfer functions D(s) R(s) G C (s) - Y(s) G(s) H(s) N(s)

7 PID control Compensator (controller) P: proportional, I: integral, D: derivative K I s R(s) - K P G(s) Y(s) K D s

8 PID controllers

9 4.4 Example of active element: Transistors Three types of semiconductors E F Intrinsic conduction band band gap E G p-type doping E F n-type electrons valence band E F holes vacuum for electrons diffusion vacuum for holes pn junction

10 pn junction thermodynamics Consider electrons Vacuum for electrons diffusion e- e- e- e- e- donors F = U TS voltage (polarization) energy cost Voltage (internal energy cost) Diffusion (entropy) Minimization of F Built-in (diffusion) voltage V bi

11 4.4.1 I-V characteristics of pn junctions Reverse bias enhances V bi : no go Forward bias overcomes V bi : go Minority carrier injection Rectification J = J 0 exp ev k B T 1 Shockley theory

12 J Injection of minority carriers 0 0 V Fate of injected minority carriers: Radiative recombination minority carrier current hν Barrier overflow light emitting diode Nick Holonyak Jr.

13 Solar cell (injection of minority carriers with illumination) J 0 ev n Δn p dark External injection 0 illuminated V Gerald Pearson, Daryl Chapin and Calvin Fuller at Bell labs. 1954

14 4.3.2 Discovery and invention of bi-polar transistors The first point contact transistor (Dec The paper published in June 1948.) John Bardeen, William Shockley, Walter Brattain 1948 Bell Labs. Bipolar junction transistor n p n Field effect transistor p n

15 Bipolar transistor structures and symbols PNP type NPN type Similar characteristics PNP and NPN: complementary

16 E B C n p n Base-Collector characteristics J E J C J C V BC J E V BC

17 How a bipolar transistor amplifies? φ Emitter n Base p Collector n V EC A J C

18 How a bipolar transistor amplifies? A J B Diffusion e e e Emitter n e e e e- e- e- e- e- e- e- e- e- Base p Collector n V EC A J C

19 n p Base-Collector characteristics n e e e e e e- e- e- e- e- e- J E J C V BC

20 n p n E B C J C Collector-Emitter characteristics

21 Current amplification : Linearize with quantity selection J C = h FE J B Emitter-common current gain

22 Normalized value at J c =2 ma Linear approximation of bipolar transistor Hybrid matrix j 2 j 1 h-parameters V 1 (lower case: local linear approximation) V 2 Collector current (ma)

23 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.

24 Concept of equivalent circuit C n B B p j b C A n A h ie h fe j b E E

25 Concept of equivalent circuit: Where is feedback? B j b C h ie h fe j b E Ξ(s) h(s) j b h fe j b j b R E h fe j b

26 Current amplification: Emitter follower v o j b (1 h fe )(R E R o ) = v i j b [h ie 1 h fe R E R o ] 1 (h fe 1) v o does not depend on load resistance Very low output resistance 10k 10m Vcc 15V input 10m a 2SC k 10m output 680W

27 Complementary transistors A C B n p n E A input 10m 10k a a 10m 2SC m 2SA1428 Vcc 5V output A p n p A 10k Vcc -5V Symmetric characteristics: Complementary Symmetric: Small collector current (idling current) for zero input.

28 Example of transistor datasheet

29 Example of transistor datasheet

30 Example of transistor datasheet h fe linear model availability in the range of J C. Cut-off frequency as a function of J C

31 Common emitter (grounded emitter) amplifier circuit

32 4.4 Field effect transistor (FET) Junction FET (JFET) Circuit symbols D D G G S S n-channel p-channel Pinch-off

33 MES-FET

34 MOS-FET enhancement Simplified CMOS inverter circuit depletion Low leakage current Single gate input both on/off switch inversion

35 Static characteristics of FET transconductance Drain resistance Locally linear approximation

36 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 2 nd 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). 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).

37 Exercise C-1 - R 1 P R 3 C - V out In the circuit shown in the left, at point P, a waveform in the lower panel was observed. Here V and V- are power source voltages for and respectively. R 2 Draw a rough sketch of the waveform for V out. V t Rough sketch should contain the levels and the timing of folding points. Write a short comment why V out should be in such a form. V-

38 Exercise C-2 - A Consider a differential amplifier with the open loop gain 1 γ R γr So the gain diverges with s 0 but here we ignore this instability. The input impedance is, and the output impedance is 0. It is now placed in a circuit with a feedback shown in the left. Obtain the stability condition for γ. (hint) Apply the Hurwitz criterion for zeros of even and odd parts of the denominator. Or just calculate H 2.

39 Exercise C-3 j 1 j 2 B C h ie h fe j b E v 1 v 2 r e 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. Calculate each element in the Y matrix for r e =25Ω, h ie =500 W, h fe = 200

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