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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
Recommended