電子回路論第6回

Electric Circuits for Physicists #6

東京大学理学部・理学系研究科

物性研究所

勝本信吾

Shingo Katsumoto

https://www.energymanagertoday.com/bell-labs-reinvented-energy-efficient-complex-0169126/

Bell labs is reinvented as an energy efficient complex

Outline

Supplements to feedback circuit and system control

Example of active element: Bipolar transistor

Disturbance and noise on feedback circuits

+

𝐷(𝑠)

𝑁(𝑠) +

+

- 𝐺𝐶(𝑠)

+ 𝐺(𝑠)

𝐻(𝑠) +

𝑌(𝑠) 𝑅(𝑠)

Circuit treatment of fluctuations: • Prepare external power sources

• Express them as transfer functions

+ +

-

a

b

c

a+b - c

PID control

+

+ 𝐺(𝑠)

𝑌(𝑠)

+

𝐾𝐼

𝑠

𝐾𝑃

𝐾𝐷𝑠

+

-

𝑅(𝑠)

P: proportional, I: integral, D: derivative Compensator

(controller)

PID controllers

4.4 Example of active element: Transistors

John Bardeen, William Shockley,

Walter Brattain 1948 Bell Labs.

Bipolar junction transistor

n n p

Field effect transistor p

n

Two types of transistors

Three types of semiconductors

𝐸F

Intrinsic

𝐸G band gap

conduction band

valence band

𝐸F

p-type

holes: p

𝐸F

n-type

electrons: n E

diffusion

vacuum

for holes

vacuum

for electrons

pn junction

- - -

+ + +

acceptor

doping donor

doping

Majority

carrier

Minority

carrier

p-type

n-type

hole

hole

electron

electron

pn junction thermodynamics

Consider electrons

+

+

+ +

+

donors

e-

e-

e- e-

e-

Vacuum

for electrons

diffusion

- - - - -

+ + + + +

voltage (polarization) → energy cost

𝐹 = 𝑈 − 𝑇𝑆

Voltage (internal energy cost) Diffusion (entropy)

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

Helmholtz free energy:

Two kinds of “current” in semiconductors

𝐹 = 𝑈 − 𝑇𝑆

Drift current: Motive force (electric field)

gives rise to non-uniformity in the momentum

space.

U drift current, -TS diffusion current 𝑓(𝑘)

𝑘

Diffusion current: There are some origins

(particle sink/source) to cause non-

uniformity in the real space (spatial density),

which causes flow of charges.

f: distribution function in phase space

D: diffusion constant

4.4.1 I-V characteristics of pn junctions

- - -

+ +

+ -

Forward bias

overcomes 𝑉𝑏𝑖

: go

- - -

+ +

+ -

Reverse bias

enhances 𝑉𝑏𝑖

: no go

Minority carrier

injection

Shockley theory

J

V

0

p n

circuit symbol

Injection of minority carriers

minority carrier

current Barrier overflow

Fate of injected minority

carriers:

Radiative recombination

ℎ𝜈

light emitting

diode

0

V

J

Nick Holonyak Jr.

Solar cells (injection of minority carriers with light)

V

J

0

0

dark

illuminated

𝑒𝑣𝑛Δ𝑛𝑝

Gerald Pearson, Daryl Chapin

and Calvin Fuller at Bell labs. 1954

external injection

ℎ𝜈 electron-hole

pair creation

Minority carriers which diffuse to the

junction region are swept out to the

other side.

4.3.2 Bipolar transistors

PNP type NPN type

Similar characteristics PNP and NPN: complementary

Base-Collector characteristics

n n p

E B C

−𝐽C

𝐽E

𝑉BC

−𝐽 C

𝑉BC

𝐽E 𝐽C increases with 𝑉BC.

Light illumination in solar cells

Emitter current

What is common? Minority carrier injection

How a bipolar transistor amplifies signal?

Base Emitter Collector n n p

A

VEC JC

𝜙

How a bipolar transistor amplifies signal?

Base Emitter Collector n n p

A JB

A

VEC JC

e+

e+ e+

e+

e+ e+

Diffusion

e-

e- e-

e- e- e- e-

e-

e-

Collector-Emitter characteristics

n

n

p

E

B

C

𝐽𝐶

Current amplification: Linearize with quantity selection

𝐽𝐶 = ℎ𝐹𝐸 𝐽𝐵

Emitter-common current gain

Linear approximation of bipolar transistor

Collector current (mA)

Norm

aliz

ed v

alu

e at

Jc

=2

mA

Hybrid matrix

h-parameters

Linear approximation for non-linear devices:

Take derivative to have local linear matrices.

(lower case:

local linear approximation)

Concept of bias circuits for non-linear devices

+𝑉

For small amplitude (high-

frequency) circuits

All the capacitors can be viewed as

short circuits.

For bias (dc) circuits

All the capacitors can be viewed as

break line.

Bias circuit determines

the action point.

Signal lines can

be separated

with reactive

elements.

Concept of equivalent circuit

n

n

p

E

B

C

A A

ℎfe𝑗b

𝑗b

ℎie

C B

E

But resistors for bias circuit still work in

signal circuit as long as they are not short

circuited by capacitors.

Concept of equivalent circuit: Feedback?

Ξ(𝑠)

ℎ(𝑠)

𝑗𝑏 ℎ𝑓𝑒𝑗𝑏

RE 𝑗𝑏 ℎ𝑓𝑒𝑗𝑏

In the case of current-current linear circuit, a

way for feedback is to have a common part

the current loops.

Concept of equivalent circuit: Feedback?

Ξ(𝑠)

ℎ(𝑠)

In the case of current-current linear circuit, a

way for feedback is to have a common part

the current loops.

Current amplification: Emitter follower

a input

output

Vcc +15V

2SC2458

10m

10m

10m

10k

680W

10k

𝑣𝑜

𝑣𝑖=

𝑗𝑏(1 + ℎ𝑓𝑒)(𝑅𝐸 ∥ 𝑅𝑜)

𝑗𝑏[ℎ𝑖𝑒 + 1 + ℎ𝑓𝑒 𝑅𝐸 ∥ 𝑅𝑜 ]

≈ 1 (ℎ𝑓𝑒 ≫ 1)

𝑣𝑜 does not depend on load

resistance

⇒ Very low output resistance

Complementary transistors

n

n

p

E

B

C

A A

p

p

n

A A

a

input output

Vcc +5V

2SC3668

10m

10m

10m

10k

10k

a

Vcc -5V

2SA1428

Symmetric characteristics:

Complementary

Symmetric: Small collector

current (idling current) for

zero input.

Example of transistor datasheet

Example of transistor datasheet (2SC1815(L))

Example of transistor datasheet (2SC1815(L))

ℎ𝑓𝑒 linear model availability

in the range of 𝐽𝐶 . Cut-off frequency as a function of 𝐽𝐶

Simulation example