九州大学大学院システム情報科学府

電気電子工学専攻

令和２年度入学試験問題【令和元年８月２６日（月）、２７日（火）】情報学専攻九州大学大学院システム情報科学府 情報知能工学専攻
電気電子工学専攻修士課程 入学試験問題 （令和元年８月２６日）

令和２年度

数 学 (Mathematics)（7 枚中の 1）

解答上の注意 (Instructions):

1. 問題用紙は，『始め』の合図があるまで開いてはならない．
Do not open this cover sheet until the start of examination is announced.
2. 問題用紙は表紙を含め 7 枚，解答用紙は 3 枚つづり (1 分野につき 1 枚) である．
You are given 7 problem sheets including this cover sheet, and 3 answer sheets (1 sheet foreach ﬁeld).
3. 以下の 6 分野から 3 分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．
Select 3 ﬁelds out of the following 6 ﬁelds and answer the questions. You must use a separate
answer sheet for each of the ﬁelds you selected.

1
2
3
4
5
6

分野
線形代数
微分方程式
ベクトル解析
複素関数論
確率・統計
記号論理学

ﬁeld
Linear algebra
Diﬀerential equation
Vector analysis
Complex function theory
Probability and statistics
Symbolic logic

page
2
3
4
5
6
7

4. 解答用紙の全部に，専攻名，コース名（情報学専攻を除く），選択分野番号（○で囲む），受験番号および氏名を記入すること．

Fill in the designated blanks at the top of each answer sheet with the department name,course name (except the department of informatics), the selected ﬁeld number (mark with acircle), your examinee number and your name.
5. 解答は解答用紙に記入すること．スペースが足りない場合は裏面を用いても良いが，その場合は，裏面に解答があることを明記すること．

Write your answers on the answer sheets. You may use the backs of the answer sheets when
you run out of space. If you do so, indicate it clearly on the sheet.
6. 解答は，日本語，英語のいずれかで記入すること．
Your answers must be written in Japanese or English.

情報学専攻九州大学大学院システム情報科学府 情報知能工学専攻
電気電子工学専攻修士課程 入学試験問題 （令和元年８月２６日）

令和２年度

数 学 (Mathematics)（7 枚中の 2）
6 分野のうちから 3 分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．
Select 3 ﬁelds out of the 6 ﬁelds and answer the questions. Use a separate answer sheet
for each ﬁeld.

1. 【線形代数 (Linear algebra) 分野】
数列 a0 ，a1 ，a2 ，…は，a0 = 3，a1 = 1，a2 = 3 およびan = an−1 + an−2 + 2an−3（n = 3，4，5，…）で定義される．

(1) a3 ，a4 ，a5 を求めよ．
(2) 各 n = 0，1，2，…について次が成立つような行列 T を答えよ．




an+1
an




an+2  = T an+1 
an+3
an+2
(3) T のすべての固有値とそれぞれに対応する固有ベクトルを求めよ．
 
3
 
(4) ベクトル 1 を，前問で求めた固有ベクトルの線形結合として表せ．
3(5) an を求めよ．
Consider the sequence a0 , a1 , a2 , . . . deﬁned by a0 = 3, a1 = 1, a2 = 3 andan = an−1 + an−2 + 2an−3 ,n = 3, 4, 5, . . . .

(1) Find a3 , a4 and a5 .
(2) Give the matrix T such that





an
an+1




an+2  = T an+1 
an+2
an+3

for all n = 0, 1, 2, . . . .
(3) Find all eigenvalues of T and their corresponding eigenvectors.
 
3
 
(4) Express the vector 1 as a linear combination of the eigenvectors obtained in the3previous question.
(5) Find an .

情報学専攻九州大学大学院システム情報科学府 情報知能工学専攻
電気電子工学専攻修士課程 入学試験問題 （令和元年８月２６日）

令和２年度

数 学 (Mathematics)（7 枚中の 3）
6 分野のうちから 3 分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．
Select 3 ﬁelds out of the 6 ﬁelds and answer the questions. Use a separate answer sheet
for each ﬁeld.

2. 【微分方程式 (Diﬀerential equation) 分野】

関数 y(x) の微分方程式(x4 − 1)dy
= y 2 + 2x3 y − 3x2dxについて以下の問いに答えよ．

(1) 与えられた微分方程式は，yp (x) = ax3 の形の特殊解を持つ．yp (x) を求めよ．ただし aは定数とする．

(2) 特殊解 yp (x) と関数 u(x) を用いて y = yp +1とおき，一般解を求めよ．
uConsider the diﬀerential equation(x4 − 1)dy
= y 2 + 2x3 y − 3x2dx

for a function y(x). Answer the following questions.
(1) Find one of the particular solutions yp (x) of the form yp = ax3 , where a is a constant.
1(2) Obtain the general solution with the replacement y = yp + , where u is a function of x.
u情報学専攻九州大学大学院システム情報科学府 情報知能工学専攻
電気電子工学専攻修士課程 入学試験問題 （令和元年８月２６日）

令和２年度

数 学 (Mathematics)（7 枚中の 4）
6 分野のうちから 3 分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．
Select 3 ﬁelds out of the 6 ﬁelds and answer the questions. Use a separate answer sheet
for each ﬁeld.

3. 【ベクトル解析 (Vector analysis) 分野】

直交座標系において，x，y ，z 軸方向の単位ベクトルをそれぞれ i，j ，k とする．次の各問
に答えよ．

(
)2 (
)2
1
1
(1) 面 z = x + y と面 z = x − √+ y−√について，次の問いに答えよ．
2
2
(√ √
)
2 2 1
(a) 点 P,,が，いずれの面にも含まれることを示せ．
4 4 4
2

2

(b) 点 P において，それぞれの面の法線のなす角を求めよ．
(2) ベクトル場 A を A = x2 i − y 2 j + z 2 k とする．S を x2 + y 2 = 9，z = 0，z = 4 で囲まれた円筒の表面とするとき，面積分∫∫
A · n dS
S

を求めよ (ただし，n は S の外向き単位法線ベクトル).

The unit vectors on x, y and z axes of Cartesian coordinates are denoted by i, j and k,
respectively. Answer the following questions.
(
)2 (
)2
1
1
(1) Consider the surfaces z = x + y and z = x − √+ y−√. Answer the22
following questions.
(√ √
)
2 2 1
(a) Show the point P,,
is included in both surfaces.
4 4 4
2

2

(b) Evaluate the angle between the normals to the surfaces at the point P.
(2) Let the vector ﬁeld A = x2 i − y 2 j + z 2 k and S be the surface of the cylinder boundedby x2 + y 2 = 9, z = 0 and z = 4 . Evaluate the following surface integral∫∫
A · n dS ,S

where n is the outward-pointing unit normal vector of S.

情報学専攻九州大学大学院システム情報科学府 情報知能工学専攻
電気電子工学専攻修士課程 入学試験問題 （令和元年８月２６日）

令和２年度

数 学 (Mathematics)（7 枚中の 5）
6 分野のうちから 3 分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．
Select 3 ﬁelds out of the 6 ﬁelds and answer the questions. Use a separate answer sheet
for each ﬁeld.

4. 【複素関数論 (Complex function theory) 分野】
図に示す z 平面における x = 1, y = 1, y = 1 − x で囲まれた三角領域 S を考える．以下の変換で S が写像される w 平面の領域 S ′ を図示すると共に、S ′ を囲む境界の方程式を示せ．た√だし，z = x + iy ，w = u + iv は複素数，x, y, u, v は実数，i = −1 である．

√
3i)
√
πi
(2) w = 2e 6 z + (1 − 3i)(1) w = z + (1 −(3) w = z 2

y
1

S

0

1

x

Consider the triangle region S bounded by x = 1, y = 1 and y = 1 − x in the z plane asshown in the ﬁgure. Illustrate the region S ′ in the w plane into which S is mapped under
the following transformations and ﬁnd the equations of the curves which enclose the region,
where z = x + iy and w = u + iv are complex numbers, x, y, u, and v are real numbers and√i = −1.
√3i)√
πi(2) w = 2e 6 z + (1 − 3i)(1) w = z + (1 −(3) w = z 2

情報学専攻九州大学大学院システム情報科学府 情報知能工学専攻
電気電子工学専攻修士課程 入学試験問題 （令和元年８月２６日）

令和２年度

数 学 (Mathematics)（7 枚中の 6）
6 分野のうちから 3 分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．
Select 3 ﬁelds out of the 6 ﬁelds and answer the questions. Use a separate answer sheet
for each ﬁeld.

5. 【確率・統計 (Probability and statistics) 分野】
2

連続確率変数 X は区間 [0, 1) 上の一様分布に従うものとする．Y = − log (1−X)とする．ただ4
し log は自然対数とする．以下の各問に答えよ．

(1) 実数 t に対し，Y ≤ t の確率を求めよ．
(2) Y の期待値と分散をそれぞれ求めよ．

Let X be a continuous random variable uniformly distributed over the interval [0, 1). Let
2

Y = − log (1−X)
where log denotes the natural logarithm. Answer the following questions.
4(1) Find the probability of Y ≤ t for a real t.
(2) Find the expectation and the variance of Y , respectively.

情報学専攻九州大学大学院システム情報科学府 情報知能工学専攻
電気電子工学専攻修士課程 入学試験問題 （令和元年８月２６日）

令和２年度

数 学 (Mathematics)（7 枚中の 7）
6 分野のうちから 3 分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．
Select 3 ﬁelds out of the 6 ﬁelds and answer the questions. Use a separate answer sheet
for each ﬁeld.

6. 【記号論理学 (Symbolic logic) 分野】(1) 以下の命題論理式をそれぞれ ϕ1 , ϕ2 , ϕ3 で表す．
(p → q) ∧ (p ∧ q → r),¬(¬(r → s) ∨ (¬p ∧ r ∧ s)),p↔r

ϕ3 は ϕ1 ∧ ϕ2 の論理的帰結であることを resolution 法により示せ．
(2) 以下の述語論理式をそれぞれ ψ1 , ψ2 , ψ3 で表す．
∀x∀y ((¬P (x, y) → P (y, x)) ∧ (Q(x, y) → ¬Q(y, x))),∀x∀y (P (x, y) → Q(x, y)),∀x (∃y Q(x, y) ∧ ∀y (Q(x, y) → ¬P (y, x)))以下の式は充足可能か否か，理由とともに答えよ．

(a) ψ1 ∧ ψ2(b) ψ1 ∧ ψ3(1) Let ϕ1 , ϕ2 , and ϕ3 denote the following propositional formulas, respectively:(p → q) ∧ (p ∧ q → r),¬(¬(r → s) ∨ (¬p ∧ r ∧ s)),p ↔ r.

Show by the resolution method that ϕ3 is a logical consequence of ϕ1 ∧ ϕ2 .
(2) Let ψ1 , ψ2 , and ψ3 denote the following predicate logic formulas, respectively:∀x∀y ((¬P (x, y) → P (y, x)) ∧ (Q(x, y) → ¬Q(y, x))),∀x∀y (P (x, y) → Q(x, y)),∀x (∃y Q(x, y) ∧ ∀y (Q(x, y) → ¬P (y, x))).

Determine whether each of the following formulas is satisﬁable. Justify your answer.
(a) ψ1 ∧ ψ2(b) ψ1 ∧ ψ3

令和２年度 九州大学大学院システム情報科学府 電気電子工学専攻修士課程 入学試験問題 （2019 年 8 月 26 日）

専門Ⅰ(Special subjects I)

解答上の注意 (Instructions):
1.

問題用紙は，『始め』の合図があるまで開いてはならない．
Do not open this cover sheet until the start of examination is announced.

2.

問題用紙は表紙を含め 9 枚，解答用紙は 3 枚である．
You are given 9 problem sheets including this cover sheet, and 3 answer sheets.

3.

以下の 3 分野から 1 分野を選び解答すること．
Select 1 out of the following 3 fields and answer the questions.

4.

分野

field

1

電気回路

Circuit theory

2 ～

2

電子回路

Electronic circuits

4 ～

3

制御工学

Control engineering

6 ～

page

解答は解答用紙に記入すること．大問一つあたり一枚の解答用紙を用いよ．スペースが足りない場合は裏面を用いても良いが，その場合は，裏面に解答があることを明記すること．
Your answers should be written on the answer sheets. Use one sheet for eachquestion. You may continue to write your answer on the back of the answer sheets if
you need more space. In such a case, indicate this clearly.

5.

解答用紙の全部に，志望するコース名，選択分野名，受験番号，氏名および問題番号を記入すること．
Fill in the designated blanks at the top of each answer sheet with the course name,your selected field name, your examinee number, your name and the question
number.

6.

解答は，日本語，英語のいずれかで記入すること．
Your answers must be written in Japanese or English.

1

令和 2 年度 大学院システム情報科学府入学試験問題 電気回路（令和元年 8 月 26 日）
4 問中 3 問を選び，解答用紙欄に解答した問題番号を記入すること．
I1【問 1】 図 1 の回路について，以下の問いに答えよ．ただし，( )πI1= , R = 1 Ω である．
電流 I1 と電流 I2 の位相差 arg
I2
4

R
I2

(1) X1 と X2 の間の関係式を示せ．
√
2 2
|I1 |
(2)
=
であるとき， X1 , X2 の値を求めよ．
|I2 |
3

jX1

E

R
-jX2

図1

1
【問 2】 図 2 の回路について，以下の問いに答えよ．

(1) 端子対 1-1’ 間を，開放電圧 V0 ，内部インピーダンス
Z0 の等価回路と考えるとき，V0 と Z0 を求めよ．
(2) 端子対 1-1’ 間に可変インピーダンス Z = R + jX を接続したとする．R における消費電力 P が最大となる Z を求め，その時の P の値を求めよ．

j2 

-j

1

Z
j

10 V

1'
図2

𝑀

【問 3】 図 3 の回路について，以下の問いに答えよ．
ただし，電源電圧 E の角周波数を ω とする．

(1) 閉路電流 I1 , I2 , I3 を変数に用いて閉路方程式を
𝐸
立てよ．

𝐼2

𝐼1

𝐼3
𝑅

𝑅
𝐿1

𝐿2

(2) 電源 E から右を見たインピーダンス Z を求めよ．
図3【問 4】 図 4 の回路について，以下の問いに答えよ．ただし，E = 1 V，R1 = R2 = 1 Ω，C = 1 F，L = 1 H とする．

(1) スイッチ S を開いたまま回路が定常状態に達した後，
時刻 t = 0 において S を閉じるとする．このとき，
t > 0 における電流 i1 (t)，i2 (t) をそれぞれ求めよ．
(2) スイッチ S を閉じたまま回路が定常状態に達した後，
t = 0 において S を開くとする．このとき，t > 0 における電流 i1 (t) を求めよ．
(3) 上記 (2) において，t = 0 から定常状態に達するまでに抵抗 R1 ，R2 で消費されるエネルギーの合計 W を
求めよ．
2

S
R1

i1(t)

R2

E
C

q(t)

図4

L

i2(t)

Circuit Theory (2020 ISEE Entrance Examination (August 26, 2019))
Choose three out of the four questions and write the chosen question number on each answer sheet.
【Q1】 Consider) circuit shown in Fig. 1, where the phase
( the
π
I1
= , and R = 1 Ω. Answer the following
diﬀerence arg
I2
4
questions.

(1) Find the equation representing the relation between X1
and X2 .
√
2 2
|I1 |
=
.
(2) Determine the values of X1 and X2 , when
|I2 |
3

I1
R
I2
jX1

E

R
-jX2

Fig. 1【Q2】 Consider the circuit shown in Fig. 2. Answer the following questions.

(1) When the one-port circuit with terminals 1-1′ is equivalent to the voltage source with the open voltage V0 and
the internal impedance Z0 , find V0 and Z0

1
j2 

-j

1

Z
j

10 V
(2) Consider the variable impedance Z = R+jX is connected
between the terminals 1-1′ . Find the value of Z when
the power consumption P by the resistance R becomesmaximum. And find the value of P under the condition.

【Q3】 Consider the circuit shown in Fig. 3, where thesource E has the angular frequency ω. Answer the following questions.

1'Fig. 2

𝑀
𝐼2

𝐼1

𝐼3
𝑅

𝑅

(1) Write the mesh equations using the mesh currents 𝐸
I1 , I2 and I3 as variables.

𝐿1

𝐿2(2) Find the impedance Z seen from the source E.
Fig. 3【Q4】 Consider the circuit shown in Fig. 4 where E = 1 V,
R1 = R2 = 1 Ω, C = 1 F, L = 1 H. Answer the following
questions.

(1) Assume the switch S is opened and the circuit is insteady state. Then, the switch S is closed at the time
t = 0. Find the currents i1 (t) and i2 (t) for t > 0.
(2) Assume the switch S is closed and the circuit is in steadystate. Then, the switch S is opened at the time t = 0.
Find the current i1 (t) for t > 0.
(3) In the above-mentioned condition (2), find the total energy W consumed by the resistances R1 and R2 fromt = 0 to the steady state.
3

S
R1

i1(t)

R2

E
C

q(t)

Fig. 4

L

i2(t)

電子回路

試験問題

令和元年８月２６日（月）次の各問（１，２，３）に答えよ。解答はすべて解答用紙に記入すること。
１．次の問に答えよ。ただし，演算増幅器は理想的であるとする。
（１）図１に示す回路の伝達関数 G(s) = Vo (s) / Vi (s)を求めよ。
（２）G(j)について電圧利得および位相のボーデ図の概形を描け。
R1

R1

Vo

Vi
R2
C

図１
２．図２に示す差動増幅回路の出力電圧Voを求めよ。但し，すべての演算増幅器は理想的であるとする。

V1
R2

R3

R4
Vo

R1

R2

R3

R4

V2

図２
３．図３に示すLC発振器（コルピッツ発振器）について，次の問に答えよ。
（１）ループ利得 T を求めよ。
（２）発振が定常状態にある時の発振周波数と振幅条件を求めよ。

L

Ri

Vb

gm Vb

C2

増幅器

図３

4

C1

ELECTRONIC CIRCUITS
Aug.26, 2019
Answer the following questions (1, 2, 3). All answers should be written on the answer sheets.

1. Answer the following questions. The operational amplifiers are assumed to be ideal.
(1) Derive the transfer function G(s) = Vo (s) / Vi (s) for the circuit shown in Fig. 1.
(2) Sketch the Bode plots of the voltage gain and the phase for G(j).
R1

R1

Vo

Vi
R2
C

Fig. 1

2. Derive the output voltage Vo of the differential amplifier circuit shown in Fig. 2.
All operational amplifiers are assumed to be ideal.
V1
R2

R3

R4
Vo

R1

R2

R3

R4

V2

Fig. 2

3.

Fig. 3 shows an LC oscillator which is called the Colpitts oscillator.
(1) Derive the loop gain T.
(2) Obtain the oscillation frequency and the condition of the amplitude for steady-state oscillation.

L

Ri

Vb

gm Vb

C2

AmplifierFig. 35C1

令和 2 年度電気電子工学専攻修士課程入学試験問題
制御工学分野 （令和元年 8 月 26 日実施）
次の各問（[1]，[2]，[3]）に答えよ．文中，t は時刻，s はラプラス演算子，ω は角周波数，e は自然対数の底を表す．

[1] 図 1 のブロック線図で表されるシステムがある．u(t), v(t), w(t), x(t), y(t) はすべてスカ
ラー値をとる信号である．システム内の要素 G1 , G2 , G3 は以下のように特徴づけられる．
• G1 のインパルス応答は g1 (t) = e−t + 2e−2t である．
• G2 の入力 v(t) と出力 y(t) は方程式dy(t) = v(t) − y(t) を満たす．
dt• G3 に t < 0 のとき 0，t ≥ 0 のとき 1 の値をとる信号を入力した時の t ≥ 0 における
1
G3 からの出力は (1 − e−2t ) である．
2(1) 各要素 G1 , G2 , G3 の伝達関数を求めよ．
(2) 図 1 のシステムの x(t) から y(t) までの伝達関数を求めよ．

図1[2] 伝達関数が以下のように与えられる二つのシステムがある．
G1 (s) =K1, K1 > 0,
1+s

G2 (s) =K2, K2 > 0(1 + s)(1 + 10s)(1 + 100s)

G1 のゲイン-周波数特性と位相-周波数特性をそれぞれ g1 (ω) [dB]，ϕ1 (ω) [deg]，G2 のゲイン-周波数特性と位相-周波数特性をそれぞれ g2 (ω) [dB]，ϕ2 (ω) [deg] とする．
(1) lim ϕ1 (ω) と lim ϕ2 (ω) を求めよ．
ω→∞

ω→∞

(2) K1 = 1, K2 = 1 のときのゲイン-周波数特性 g1 (ω), g2 (ω) のそれぞれを表す折れ線近似のグラフを描け．
(3) G1 , G2 のそれぞれによって構成される図 2 の閉ループ系を考える．K1 および K2 の値を大きくすると，それぞれの閉ループ系は不安定になる可能性があるか？ 理由とともに答えよ．

図2

6

[3] 次のシステムの制御系設計を考える．
dx(t) = x(t) + u(t) + d(t)dty(t) = x(t)ここで x(t) は状態量，u(t) は入力，y(t) は出力，d(t) は外乱であり，すべてスカラー値をとる．

(1) 以後 d(t) は一定値である，すなわち dtd d(t) = 0 であると仮定する．d(t) を状態量と考え，状態ベクトルを [x(t) d(t)]T としたシステムの状態方程式を導け．このシステム
を拡大系と呼ぶこととする．
(2) 拡大系は出力を y(t) とすると可観測であることを示せ．
(3) 拡大系の状態ベクトルを推定する完全次元オブザーバの状態方程式を導け．完全次元
ˆ T
オブザーバーの極は s = −2 および s = −3 とせよ．[x(t) d(t)]T の推定値は [x̂(t) d(t)]と表すものとする．
(4) 制御則を
ˆ + r(t)u(t) = −k y(t) − d(t)とする．ここで k はフィードバックゲイン，r(t) は目標値である．閉ループシステム
が BIBO 安定となるような k のとりうる範囲を示せ．

(5) r(t) が一定値の場合，y(t) は一定値に収束することを示せ．ただし k は閉ループシス
テムが BIBO 安定になるように選ばれているものとする．

7

令和 2 年度電気電子工学専攻修士課程入学試験問題
Control engineering (26 August 2019)
Answer the following questions ([1], [2] and [3]). In the following, t denotes the time, s
the Laplace operator, ω the angular frequency, and e the base of natural logarithm.
[1] Consider the system depicted by the block diagram in Fig.1, where u(t), v(t), w(t), x(t)and y(t) are all scalar-valued signals. The elements G1 , G2 and G3 in the system arecharacterized as follows:• the impulse response of G1 is g1 (t) = e−t + 2e−2t ;• the input v(t) and the output y(t) of G2 satisfy the equationdy(t) = v(t) − y(t);dt• when the signal that takes the value 0 for t < 0 and 1 for t ≥ 0 is fed to G3 , the1
output from G3 for t ≥ 0 is (1 − e−2t ).
2(1) Find the transfer function of each of the elements G1 , G2 and G3 .
(2) Find the transfer function from x(t) to y(t) of the system shown in Fig.1.

Fig. 1[2] There are two systems whose transfer functions are given below:
G1 (s) =K1, K1 > 0,
1+s

G2 (s) =K2, K2 > 0.
(1 + s)(1 + 10s)(1 + 100s)

Let the gain-frequency characteristics and phase-frequency characteristics of G1 are denoted, respectively, by g1 (ω) [dB] and ϕ1 (ω) [deg]. Also the gain-frequency and phasefrequency characteristics of G2 are g2 (ω) [dB] and ϕ2 (ω) [deg], respectively.
(1) Find lim ϕ1 (ω) and lim ϕ2 (ω).
ω→∞

ω→∞

(2) Draw the graphs of gain-frequency characteristics g1 (ω) and g2 (ω) approximated bystraight lines when K1 = 1 and K2 = 1.
(3) Consider the two closed loop systems shown in Fig. 2 each of which consists of either
G1 or G2 . Can increasing K1 and K2 make each closed loop system unstable? Answer
with the reason.

Fig. 28

[3] Consider a controller design for a system represented by the following equations.
dx(t) = x(t) + u(t) + d(t),dty(t) = x(t),where x(t) is the state variable, u(t) is the input, y(t) is the output, and d(t) is thedisturbance, all of which are scalar-valued.
(1) Hereafter assume that d(t) is constant, i.e., dtd d(t) = 0. Regarding d(t) as anotherstate variable, show the state equation of the augmented system of which the statevector is [x(t) d(t)]T .
(2) Show that the augmented system is observable regarding y(t) as the output of thesystem．
(3) Show the state equation of the full order observer that estimates the state vector of
the augmented system, assigning the poles of the observer at s = −2 and s = −3.
ˆ T be the estimate of [x(t) d(t)]T .
Let [x̂(t) d(t)](4) Consider the control law
ˆ + r(t),u(t) = −k y(t) − d(t)where k is the feedback gain, and r(t) is the reference. Show the admissible regionof k by which the closed loop system is BIBO stable.
(5) Show that y(t) converges to a constant under the condition that r(t) is constant andthat the value of k is chosen in the admissible region.

9௧࿴㸰ᖺᗘ ஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ 㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟 㸦2019 ᖺ 8 ᭶ 26 ᪥㸧ᑓ㛛ϩ (Special subjects ϩ)ゎ⟅ୖࡢὀព (Instructions):1.

ၥ㢟⏝⣬ࡣࠊࠗጞࡵ࠘ࡢྜᅗࡀ࠶ࡿࡲ࡛㛤࠸࡚ࡣ࡞ࡽ࡞࠸㸬
Do not open this cover sheet until the start of examination is announced.

2.

ၥ㢟⏝⣬ࡣ⾲⣬ࢆྵࡵ 21 ᯛࠊゎ⟅⏝⣬ࡣ 3 ᯛ࡛࠶ࡿ㸬You are given 21 problem sheets including this cover sheet, and 3 answer sheets.

3.

௨ୗࡢ 3 ศ㔝࠿ࡽ 1 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬
Select 1 out of the following 3 fields and answer the questions.

4.

ศ㔝  

field

page

1

㟁☢ẼᏛ   Electromagnetism

2 㹼2༙ᑟయࢹࣂ࢖ࢫ

Semiconductor device

8 㹼3ィ⟬ᶵᕤᏛ

Computer engineering

14 㹼ゎ⟅ࡣゎ⟅⏝⣬࡟グධࡍࡿࡇ࡜㸬኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬ࢫ࣮࣌ࢫࡀ㊊ࡾ࡞࠸ሙྜࡣ⿬㠃ࢆ⏝࠸࡚ࡶⰋ࠸ࡀ㸪ࡑࡢሙྜࡣ㸪⿬㠃࡟ゎ⟅ࡀ࠶ࡿࡇ࡜ࢆ᫂グࡍࡿࡇ࡜㸬Your answers should be written on the answer sheets. Use one sheet for eachquestion. You may continue to write your answer on the back of the answer sheets if
you need more space. In such a case, indicate this clearly.

5.

ゎ⟅⏝⣬ࡢ඲㒊࡟㸪ᚿᮃࡍࡿࢥ࣮ࢫྡ㸪㑅ᢥศ㔝ྡ㸪ཷ㦂␒ྕ㸪Ặྡ࠾ࡼࡧၥ㢟␒ྕࢆグධࡍࡿࡇ࡜㸬
Fill in the designated blanks at the top of each answer sheet with the course name,your selected field name, your examinee number, your name and the question
number.

6.

ゎ⟅ࡣ㸪᪥ᮏㄒ㸪ⱥㄒࡢ࠸ࡎࢀ࠿࡛グධࡍࡿࡇ࡜㸬Your answers must be written in Japanese or English.

1௧࿴㸰ᖺᗘ኱Ꮫ㝔ධᏛヨ㦂ၥ㢟 㟁☢ẼᏛ      㸦௧࿴ඖᖺ㸶᭶㸰㸴᪥㸧ḟࡢྛၥ㸦ၥ㸯 ၥ㸰 ၥ㸱㸧࡟⟅࠼ࡼ㸬ၥ㸯 ᅗ  ࡟♧ࡍࡼ࠺࡟㸪ㄏ㟁⋡ H0 ࡢ┿✵୰࡟ 3 ಶࡢ༙ᚄaࡢᑟయ⌫ $㸪%㸪& ࡀ 1 ㎶ࡢ㛗ࡉ / a << L ࡢṇ୕ゅᙧࡢ㡬Ⅼ࡟㓄⨨ࡉࢀ࡚࠸ࡿ㸬ᑟయ⌫ $ ࡢ㟁Ⲵࡣ Q1㸪㟁఩ࡣ V0, ᑟయ⌫% ࡢ㟁Ⲵࡣ Q2㸪㟁఩ࡣ 0, ᑟయ⌫ & ࡢ㟁Ⲵࡣ Q3㸪㟁఩ࡣ 0 ࡛࠶ࡿ㸬(1) 㟁఩ಀᩘࢆ⏝࠸࡚ྛ⌫ࡢ㟁఩ࢆ⾲ࡏ(2) ྛ㟁఩ಀᩘࢆ Q1 Q2 V0 ࢆ⏝࠸࡚⾲ࡏ㸬(3) ḟ࡟㸪3 ಶࡢᑟయ⌫ࡢ㟁఩ࡀ඲࡚ V ࡟࡞ࡿࡼ࠺࡟ྛᑟయ⌫ࡢ㟁Ⲵࢆኚ໬ࡉࡏࡓ㸬ࡇࡢሙྜࡢྛ⌫ࡢ㟁Ⲵࢆ Q1 Q2V0V ࢆ⏝࠸࡚⾲ࡏ㸬(4) ḟ࡟㸪3 ಶࡢᑟయ⌫ࡢ㟁Ⲵࢆ඲࡚ Q1 ࡟ኚ໬ࡉࡏࡓ㸬ࡇࡢሙྜࡢྛ⌫ࡢ㟁఩ࢆ Q1 Q2 V0 ࢆ⏝࠸࡚⾲ࡏ

Q1, V0
ᑟయ⌫ A

ᑟయ⌫ B

ᑟయ⌫ C

Q2, 0

Q3, 0
ᅗ㸯

2

௧࿴㸰ᖺᗘ኱Ꮫ㝔ධᏛヨ㦂ၥ㢟 㟁☢ẼᏛ      㸦௧࿴ඖᖺ㸶᭶㸰㸴᪥㸧ၥ㸰 ௨ୗࡢၥ࠸࡟⟅࠼ࡼ㸬ࡓࡔࡋ㸪S = 3.14,┿✵ࡢㄏ㟁⋡H0= 8.85x10-12 F/m㸪⣲㟁Ⲵ e =1.60x10-19 C ࡜ࡍࡿ㸬(1) ┿✵୰࡟⨨࠿ࢀࡓ༙ᚄaࡢᑟయ⌫ࡀ㟁ⲴQ࡛ᖏ㟁ࡋ࡚࠸ࡿ㸬ࡇࡢᑟయࡢ㟁఩࡜㟼㟁ᐜ㔞ࢆồࡵࡼ㸬
(2) ┿✵୰࡟⨨࠿ࢀࡓ༙ᚄ 2.00 nm ࡢᑟయ⌫ࡢ㟼㟁ᐜ㔞ࢆồࡵࡼ㸬ࡇࡢᑟయ⌫࡟㟁Ꮚ 1 ಶࡢ㟁Ⲵe ࢆ୚࠼ࡓሙྜࡢ㟁఩ࡢኚ໬࡜⵳✚ࡉࢀࡿ㟼㟁࢚ࢿࣝࢠ࣮ࢆồࡵࡼ㸬(3) ┿✵୰࡟⨨࠿ࢀࡓ༙ᚄ 2.00 m ࡢᑟయ⌫ࡢ㟼㟁ᐜ㔞ࢆồࡵࡼ㸬ࡇࡢᑟయ⌫࡟㟁Ꮚ 1 ಶࡢ㟁Ⲵe ࢆ୚࠼ࡓሙྜࡢ㟁఩ࡢኚ໬࡜⵳✚ࡉࢀࡿ㟼㟁࢚ࢿࣝࢠ࣮ࢆồࡵࡼ㸬(4) ༙ᚄ 2.00 nm ࡢᑟయ⌫࡜༙ᚄ 2.00 m ࡢᑟయ⌫ࡢ࡝ࡕࡽࡀ㟁఩ࡢᇶ‽࡟㐺ࡋ࡚࠸ࡿ࠿㸬ࡲࡓ㸪ࡑࡢ⌮⏤ࢆ㏙࡭ࡼ

3௧࿴㸰ᖺᗘ኱Ꮫ㝔ධᏛヨ㦂ၥ㢟 㟁☢ẼᏛ      㸦௧࿴ඖᖺ㸶᭶㸰㸴᪥㸧ၥ㸱 ┿✵୰࡟⨨࠸ࡓ༙ᚄ a ཌࡉ d ࡢ෇⟄≧ࡢⷧ࠸㟁ᴟ࡟㟁ὶࢆὶࡋࡓ㝿࡟⏕ࡌࡿ☢⏺࡟ࡘ࠸࡚⪃࠼ࡿ㸬㟁ᴟࡢ㛗ࡉࡣ඘ศ㛗ࡃ➃ࡢຠᯝࡣ↓どฟ᮶ࡿࡶࡢ࡜ࡋ㸪෇⟄ࡢ୰ᚰ㍈ࢆ z ㍈࡟࡜ࡾ㸪ᅗ㸱࡟♧ࡍࡼ࠺࡟ z ㍈࡟┤஺ࡍࡿ xy ᖹ㠃ࢆ⪃࠼ࡿ㸬㟁ᴟࡢཌࡉࡣ༙ᚄ࡟ẚ࡭඘ศⷧࡃ㸪㟁ὶࡣ㸪⣬㠃ࡢ⾲㠃࠿ࡽ⿬㠃࡟ྥ࠿ࡗ࡚㟁ᴟෆࢆ୍ᵝ࡟ὶࢀ࡚࠸ࡿ࡜ࡍࡿ㸬㟁ὶᐦᗘࡢ኱ࡁ
ࡉࢆ J㸪ࡲࡓ㸪┿✵ࡢ㏱☢⋡ࢆ P0 ࡜ࡍࡿ㸬(1) ᅗ㸱(a) ୰ࡢⅬ P(0,113a )㸪Q( a , 0)㸪R(0, a )࡟⏕ࡌࡿ☢⏺࣋ࢡࢺࣝࡢᡂศࢆࡑࢀࡒࢀ
2
2
2

⟅࠼ࡼ㸬
(2) ᅗ㸱(b)࡟♧ࡍࡼ࠺࡟㸪෇⟄ࡀ y ㍈࡜஺ࢃࡿ఩⨨࡟ษࡾ㎸ࡳࢆධࢀ㸪ᚤᑠᖜ ΔW ࡢ⥺≧ࡢࢫࣜࢵࢺࢆ㟁ᴟࡢ඲㛗࡟ࢃࡓࡗ࡚㛗ᡭ᪉ྥ࡟ᙧᡂࡋࡓ㸬Ⅼ P㸪Q㸪R ࡟⏕ࡌࡿ☢⏺࣋ࢡࢺࣝࡢᡂศࢆࡑࢀࡒࢀ⟅࠼ࡼ㸬ࡓࡔࡋ㸪㟁ᴟࢆὶࢀࡿ㟁ὶᐦᗘࡢ኱ࡁࡉࡣ๓ၥ࡜ྠᵝ J ࡢࡲࡲ୍ᐃ࡛࠶ࡿ࡜ࡍࡿ㸬

  

y

y

3
a
2

R

3
a
2

1
a
2

P

1
a
2

z

Q
1
a
2

z

x

(a)

(b)
ᅗ㸱

4

R
W
P

Q
1
a
2

x

2020 Entrance Examination for ISEE (Electromagnetism)
Answer the following questions(August 26, 2019)([Question 1] [Question 2] [Question 3]).

[Question 1] As shown in Fig. 1, three spherical conductors A, B and C of radius a are placed atvertices of an equilateral triangle of a side length L a << L in vacuum with permittivity H0. The
electrical charges of the spherical conductors A, B and C are Q1 Q2 and Q3, respectively. The
electrostatic potentials of the spherical conductors A, B and C are V0, 0 and 0, respectively.
(1) Give the electrostatic potential of each sphere using coefficients of potential.
(2) Give each coefficient of potential using Q1 Q2 V0.
(3) Next, the charge of each conductor is changed so that all the potential of three sphericalconductors become V. Give the charge of each conductor using Q1 Q2V0V.
(4) Next, all the charges of three spherical conductors are changed to be Q1. Give the potential ofeach conductor using Q1 Q2 V0.

Q1, V0Sphere A

Sphere B

Sphere C

Q2, 0

Q3, 0
Fig. 15

2020 Entrance Examination for ISEE (Electromagnetism)(August 26, 2019)[Question 2] Answer the following questions, where S= 3.14 permitvity of vacuumH0=
8.85x10-12 F/m㸪elementary chargee = 1.60x10-19 C.
(1) A spherical conductor of radius a placed in vacuum is charged with a charge Q. Give electrostatic
potential and capacitance of this conductor.
(2) Give capacitance of a spherical conductor of radius 2.00 nm. Give the change of potential and thestored electrostatic energy when the charge of an electron -e is given to the conductor.
(3) Give capacitance of a spherical conductor of radius 2.00 m. Give the change of potential and thestored electrostatic energy when the charge of an electron -e is given to the conductor.
(4) Which is more appropriate for the potential reference, either a spherical conductor of radius 2.00
nm or 2.00 m? And give the reason.

6

2020 Entrance Examination for ISEE (Electromagnetism)(August 26, 2019)[Question 3] Consider the magnetic field produced when a current is applied to a thin cylindricalelectrode with a radius a and a thickness d placed in a vacuum. Assume that the length of thecylindrical electrode is sufficiently long and the edge effect is negligible, and consider an xy planeorthogonal to the z-axis, as shown in Fig. 3, with the central axis of the cylinder taken as the z-axis. It
is assumed that the thickness of the electrode is sufficiently thin compared to the radius, and thecurrent flows uniformly in the electrode from the front surface to the back surface of the paper. Let J
be the magnitude of the current density, and P0 be the magnetic permeability of the vacuum.

(1) Answer the components of the magnetic field vectors that occur at points P(0,and R(0,11a ), Q( a , 0),223a ) in Fig. 3(a).
2(2) As shown in Fig 3 (b), a cut was made at a position where the cylinder intersected the y-axis, anda linear slit having a small width ΔW was formed in the longitudinal direction over the entire
length of the electrode. Answer the components of the magnetic field vectors at points P, Q, andR, respectively. It is assumed that the magnitude of the current density flowing through theelectrode remains constant at J as in the previous question.

  

y

y

3
a
2

R

3
a
2

1
a
2

P

1
a
2

z

Q
1
a
2

z

x

(a)

(b)
Fig. 3

7

R
W
P

Q
1
a
2

x

 ᖺ  ᭶  ᪥ಟኈㄢ⛬ධᏛヨ㦂༙ᑟయࢹࣂ࢖ࢫ
ḟࡢྛၥ㸦㸯 㸰  㸱 㸧࡟⟅࠼ࡼ㸬㸯ୗグࡢタၥ࡟⟅࠼ࡼ㸬 㔠ᒓ㸪༙ᑟయ㸪⤯⦕యࡢࣂࣥࢻᅗࢆᶍᘧⓗ࡟♧ࡍ࡜࡜ࡶ࡟㸪ࣂࣥࢻᅗࢆ⏝࠸࡚༙ᑟయࡢᑟ㟁⋡ࡀ㔠ᒓ࡜⤯⦕యࡢ୰㛫ࡢ್ࢆ᭷ࡍࡿࡇ࡜ࢆㄝ᫂ࡏࡼ㸬 እᅉᛶ༙ᑟయ ୙⣧≀༙ᑟయ ࡢᑟ㟁⋡ࡢࢀ࡚ᑟ㟁⋡ࡀቑຍࡍࡿ
ࡇࡢࡼ࠺࡞

ᗘ౫Ꮡᛶࢆ

ᗘ㡿ᇦ࡜ῶᑡࡍࡿ

ᐃࡍࡿ࡜㸪 ᗘࡢୖ᪼࡟ࡘ

ᗘ㡿ᇦࡢ  ࡘࡢᗘ㡿ᇦࡀᏑᅾࡍࡿ㸬ᗘ౫ᏑᛶࡀⓎ⌧ࡍࡿ⌮⏤ࢆㄝ᫂ࡏࡼ㸬  ࡘࡢ༙ᑟయ A, B, C ࡀ࠶ࡿ㸬ࡇࢀࡽࡢ⮬⏤㟁Ꮚࡢ᭷ຠ㉁㔞 m*e㸪ṇᏍࡢ᭷ຠ㉁㔞 m*h㸪࢚ࢿࣝࢠ࣮ࢠࣕࢵࣉ ⚗ไᖏᖜ Eg ࡢ኱ᑠ㛵ಀࢆ௨ୗࡢ⾲࡟♧ࡍ㸬ࡲࡓ㸪༙ᑟయ A ࡜
B ࡣ┤᥋㑄⛣ᆺ㸪༙ᑟయ C ࡣ㛫᥋㑄⛣ᆺ࡛࠶ࡿ㸬༙ᑟయ A, B, C ࡢ࢚ࢿࣝࢠ࣮ࣂࣥ
ࢻᵓ㐀ࢆᶍᘧⓗ࡟ᥥࡅ㸬ࡇࡢሙྜ㸪⦪㍈ࢆ㟁Ꮚࡢ࢚ࢿࣝࢠ࣮E㸪ᶓ㍈ࢆἼᩘ k ࡛⾲♧ࡍࡿࡇ࡜㸬ᅗ࡟ࡣ࢚ࢿࣝࢠ࣮ࢠࣕࢵࣉ ⚗ไᖏᖜ Eg ࢆグධࡏࡼ㸬


༙ᑟయ A

༙ᑟయ B

༙ᑟయ C

m*e

ᑠ

኱

୰

m*h

ᑠ

୰

኱

Eg

኱

୰

ᑠ

㑄⛣ᆺ

┤᥋

┤᥋

㛫᥋


 ձ┤᥋㑄⛣ᆺ༙ᑟయ࡟ࡣ࡝ࡢࡼ࠺࡞ࡶࡢࡀ࠶ࡿ࠿㸬ᑐᛂࡍࡿ༙ᑟయࡢ໬Ꮫᘧࢆ㸯ࡘ♧ࡏ㸬ղ㛫᥋㑄⛣ᆺ༙ᑟయ࡟ࡣ࡝ࡢࡼ࠺࡞ࡶࡢࡀ࠶ࡿ࠿㸬ᑐᛂࡍࡿ༙ᑟయࡢ໬Ꮫᘧࢆ㸯ࡘ♧ࡏ㸬 Ⓨගࢹࣂ࢖ࢫᮦᩱ࡜ࡋ࡚ࡣ㸪┤᥋㑄⛣ᆺ༙ᑟయ࡜㛫᥋㑄⛣ᆺ༙ᑟయࡢ࠸ࡎࢀࡀ㐺ࡍࡿ࠿㸽⌮⏤࡜ඹ࡟⟅࠼ࡼ㸬


8

㸰
࢔ࢡࢭࣉࢱ࣮ᐦᗘ NA ࡢ p ᆺࢩࣜࢥࣥ(Si)࡜ࢻࢼ࣮ᐦᗘ ND ࡢ n ᆺ Si ࡛ᵓᡂࡉࢀࡿ pn᥋ྜࢲ࢖࣮࢜ࢻ࡟㛵ࡍࡿୗグࡢタၥ࡟⟅࠼ࡼ㸬ࡓࡔࡋ㸪⤯ᑐᗘࢆ T㸪┿ᛶ࢟ࣕࣜ࢔ᐦ

ᗘࢆ ni㸪㟁Ẽ⣲㔞ࢆ q ࡜ࡍࡿ㸬 ࡇࡢ pn ᥋ྜࢲ࢖࣮࢜ࢻࡢ㸪㡰᪉ྥࣂ࢖࢔ࢫ≧ែ(ࣂ࢖࢔ࢫ㟁ᅽ:VF)㸪࠾ࡼࡧ㏫᪉ྥࣂ࢖࢔ࢫ≧ែ(ࣂ࢖࢔ࢫ㟁ᅽ: −VR)࡟࠾ࡅࡿࣂࣥࢻᅗࢆᥥࡅ㸬ᅗ࡟ࡣ㸪౯㟁Ꮚᖏࡢ㡬 EV㸪
ఏᑟᖏࡢᗏ EC㸪ࣇ࢙࣑࣭࢚ࣝࢿࣝࢠ࣮ࡢ఩⨨ EF㸪pn ᥋ྜࡢᣑᩓ㟁఩ VD㸪࠾ࡼࡧ༳ຍࣂ࢖࢔ࢫࡢ኱ࡁࡉ(VF, −VR)ࢆグධࡍࡿࡇ࡜㸬ࡓࡔࡋ㸪VF ࠾ࡼࡧ VR ࡣṇࡢ್࡛࠶ࡿ㸬(2) ㏫᪉ྥࣂ࢖࢔ࢫ≧ែ࡟࠾ࡅࡿ㸪✵ஈᒙෆࡢ㡿ᇦ࡛ᡂ❧ࡍࡿ୍ḟඖࡢ࣏࢔ࢯࣥ᪉⛬ᘧࢆ♧ࡏ㸬ࡓࡔࡋ㸪p ᆺ㡿ᇦ࡜ n ᆺ㡿ᇦࡢቃ⏺ࢆ x = 0 ࡜ࡋ㸪p ᆺ㡿ᇦࡣ x < 0㸪n ᆺ㡿ᇦࡣ x > 0 ࡜ࡍࡿ㸬p ᆺഃࡢ✵ஈᒙ➃ࡢ఩⨨ࢆ x = −Wp㸪n ᆺഃࡢ✵ஈᒙ➃ࡢ఩⨨ࢆ
x = Wn ࡜ࡋ㸪Si ࡢㄏ㟁⋡ࢆ εs ࡜ࡍࡿ㸬(3) ୖグࡢ࣏࢔ࢯࣥ᪉⛬ᘧࢆゎ࠸࡚㸪p ᆺ㡿ᇦࡢ✵ஈᒙࡢᖜ Wp㸪n ᆺ㡿ᇦࡢ✵ஈᒙࡢᖜWn㸪࢔ࢡࢭࣉࢱ࣮ᐦᗘ NA㸪࠾ࡼࡧࢻࢼ࣮ᐦᗘ ND ࡢ㛵ಀᘧࢆᑟࡅ㸬(4) ㏫᪉ྥࣂ࢖࢔ࢫ(−VR)ࢆ༳ຍࡋࡓ p+n ∦ഃ㝵ẁ᥋ྜ(NA» ND)ࡢ༢఩㠃✚ᙜࡓࡾࡢ✵ஈᒙᐜ㔞 C ࢆ୚࠼ࡿᘧࢆ♧ࡏ㸬(5) p+n ∦ഃ㝵ẁ᥋ྜࡢ༢఩㠃✚ᙜࡓࡾࡢ✵ஈᒙᐜ㔞 C 䜢㏫᪉ྥ䝞䜲䜰䝇㟁ᅽ䛾㛵ᩘ䛸䛧䛶ᐃ䛧䛯ᐇ㦂䝕䞊䝍䜢ゎᯒ䛩䜛䛸䠈䝗䝘䞊ᐦᗘ ND 䛸ᣑᩓ㟁఩ VD 䛜ồ䜑䜙䜜䜛䠊ND 䛸 VD䜢ồ䜑䜛᪉ἲ䜢ㄝ᫂䛫䜘䠊9㸱࢚࣑ࢵࢱ㡿ᇦ(E)㸪࣮࣋ࢫ㡿ᇦ(B)㸪ࢥࣞࢡࢱ㡿ᇦ(C)࠿ࡽ࡞ࡿࣂ࢖࣏࣮ࣛ᥋ྜࢺࣛࣥࢪ
ࢫࢱ࡟㛵ࡍࡿୗグࡢタၥ࡟⟅࠼ࡼ㸬(1) pnp ࣂ࢖࣏࣮ࣛ᥋ྜࢺࣛࣥࢪࢫࢱࡢ᩿㠃ᵓ㐀ࢆᶍᘧⓗ࡟ᥥࡁ㸪ࡑࡢ୰࡟㸪E㸪B㸪C ࢆ᫂♧ࡏࡼ㸬ࡉࡽ࡟㸪࣮࣋ࢫ᥋ᆅᅇ㊰࡜ࡋ࡚⏝࠸ࡿ᫬ࡢ࣮࣋ࢫ࣭࢚࣑ࢵࢱ㛫࡟༳ຍࡍࡿ࢚࣑ࢵࢱ㟁ᅽ VE㸪࣮࣋ࢫ࣭ࢥࣞࢡࢱ㛫࡟༳ຍࡍࡿࢥࣞࢡࢱ㟁ᅽ VC ࢆ㸪ᴟᛶ࡟ὀពࡋ㸪┤ὶ㟁ᅽ※ࡢᅇ㊰グྕࢆ⏝࠸࡚ᅗ୰࡟グධࡏࡼ㸬(2) pnp ࣂ࢖࣏࣮ࣛ᥋ྜࢺࣛࣥࢪࢫࢱࡢάᛶ≧ែ࡟࠾ࡅࡿࣂࣥࢻᅗࢆᥥࡅ㸬ࣂࣥࢻᅗ࡟ࡣ㸪౯㟁Ꮚᖏࡢ㡬 EV㸪ఏᑟᖏࡢᗏ EC㸪ࣇ࢙࣑࣭࢚ࣝࢿࣝࢠ࣮ࡢ఩⨨ EF㸪pn ᥋ྜࡢᣑᩓ
㟁఩ VD㸪࠾ࡼࡧ༳ຍࣂ࢖࢔ࢫࡢ኱ࡁࡉ(࢚࣑ࢵࢱ㟁ᅽ VE ࠾ࡼࡧࢥࣞࢡࢱ㟁ᅽ VC)ࢆグ㍕ࡍࡿࡇ࡜㸬ࡓࡔࡋ㸪㟁Ẽ⣲㔞ࢆ q ࡜ࡍࡿ㸬(3) pnp ࣂ࢖࣏࣮ࣛ᥋ྜࢺࣛࣥࢪࢫࢱࡢ࣮࣋ࢫ࣭࢚࣑ࢵࢱ㛫ࡢ pn ᥋ྜ࡟࢚࣑ࢵࢱ㟁ᅽ VEࢆ༳ຍࡍࡿ㸬ࡇࡢ᫬㸪࣮࣋ࢫഃࡢ✵ஈᒙ➃࡟ὀධࡉࢀࡿṇᏍࡢᐦᗘ(⇕ᖹ⾮ṇᏍᐦᗘ࠿ࡽࡢቑຍ㔞) Δpn ࢆ୚࠼ࡿᘧ㸪࠾ࡼࡧ࢚࣑ࢵࢱഃࡢ✵ஈᒙ➃࡟ὀධࡉࢀࡿ⮬⏤㟁Ꮚࡢᐦᗘ(⇕ᖹ⾮⮬⏤㟁Ꮚᐦᗘ࠿ࡽࡢቑຍ㔞) Δnp ࢆ୚࠼ࡿᘧࢆᑟࡅ㸬ࡇࢀࡽࡢᘧ࡟⏝࠸࡚Ⰻ࠸≀⌮㔞ࡣ㸪࣮࣋ࢫ㡿ᇦࡢ୙⣧≀ᐦᗘ NB㸪࢚࣑ࢵࢱ㡿ᇦࡢ୙⣧≀ᐦᗘ NE㸪┿ᛶ࢟ࣕࣜ࢔ᐦᗘ ni㸪࣎ࣝࢶ࣐ࣥᐃᩘ k㸪⤯ᑐᗘ T㸪㟁Ẽ⣲㔞 q㸪࢚࣑ࢵࢱ㟁ᅽ VEࡢࡳ࡜ࡍࡿ㸬
(4) ࢚࣑ࢵࢱ㟁ὶ IE ࡜ࢥࣞࢡࢱ㟁ὶ IC ࡢ㛵ಀࢆ࣮࣋ࢫ᥋ᆅ㟁ὶቑᖜ⋡ α ࢆ⏝࠸࡚♧ࡏ㸬(5) ࣮࣋ࢫ᥋ᆅ㟁ὶቑᖜ⋡ α ࢆ኱ࡁࡃࡍࡿ࡟ࡣ㸪࢚࣑ࢵࢱࡢ୙⣧≀ᐦᗘ(NE)ࢆ࣮࣋ࢫࡢ
୙⣧≀ᐦᗘ(NB)ࡼࡾᑠࡉࡃࡍࡿᚲせࡀ࠶ࡿ࠿㸽ࡶࡋࡃࡣ㸪኱ࡁࡃࡍࡿᚲせࡀ࠶ࡿ࠿㸽ୖグࡢタၥ(3)࡛ᑟ࠸ࡓᘧࢆཧ⪃࡜ࡋ࡚㸪⌮⏤࡜࡜ࡶ࡟⟅࠼ࡼ㸬(6) ࣮࣋ࢫ᥋ᆅ㟁ὶቑᖜ⋡ α ࢆ኱ࡁࡃࡍࡿ࡟ࡣ㸪࣮࣋ࢫ㡿ᇦࡢᖜࢆ኱ࡁࡃࡍࡿᚲせࡀ࠶ࡿ࠿㸽ࡶࡋࡃࡣ㸪ᑠࡉࡃࡍࡿᚲせࡀ࠶ࡿ࠿㸽⌮⏤࡜࡜ࡶ࡟⟅࠼ࡼ㸬(7) ࢚࣑ࢵࢱ᥋ᆅᅇ㊰ࡢ㟁ὶቑᖜ⋡ β ࢆ α ࢆ⏝࠸࡚⾲ࡏ㸬(8) ࡼࡾ㧗࠸࿘Ἴᩘ࡛ືసࡍࡿቑᖜᅇ㊰ࢆᵓᡂࡍࡿ࡟ࡣ㸪pnp ࣂ࢖࣏࣮ࣛ᥋ྜࢺࣛࣥࢪࢫ
ࢱ࡜ npn ࣂ࢖࣏࣮ࣛ᥋ྜࢺࣛࣥࢪࢫࢱࡢ࠸ࡎࢀࡀ㐺ࡍࡿ࠿㸽⌮⏤࡜ඹ࡟⟅࠼ࡼ㸬

10

Entrance Examination, Aug. 26. 2019

6HPLFRQGXFWRUGHYLFHV
Answer the following questions (㸯 㸰  㸱 ).㸯
Answer the following questions.
(1) Sketch energy band diagrams for a metal, a semiconductor, and an insulator; and describe thereason why the electrical conductivity of a semiconductor shows an intermediate value
between those of a metal and an insulator by using the diagrams.
(2) When the temperature dependence of the electrical conductivity of an extrinsic semiconductor(impurity semiconductor) is measured, two temperatures regions, where the conductivity
increases in one region and decreases with increasing temperature in the other region, areobserved. Explain the reason why such temperature dependence is generated.
(3) There are three semiconductors A, B, and C, whose free electron effective mass (m*e), holeeffective mass (m*h), and energy gap (Eg) are qualitatively summarized in the table below. In
addition, semiconductors A and B have a direct-energy-gap, and semiconductor C has anindirect-energy-gap. Sketch energy band diagrams for semiconductors A, B, and C. Use thevertical axis of energy E and horizontal axis of wave number k. The energy gap Eg must be
given in the drawings.
A

B

C

m*e

small

large

mediumm*hsmall

medium

large

Eg

large

mediumsmall

band structure

directdirectindirect(4) ձ Give an example of chemical formulae of semiconductor materials with a direct-energygap
ղ Give an example of chemical formulae of semiconductor materials with an indirectenergy-gap.
(5) Which is suitable for light-emitting device application, a direct-energy-gap semiconductor oran indirect-energy-gap semiconductor? Answer with the reason.


11

㸰
Answer the following questions about a pn junction diode consisting of a p-type silicon (Si)with an acceptor concentration NA and n-type Si with a donor concentration ND. The absolutetemperature, the intrinsic carrier concentration, and the elementary charge are represented as T, ni,
and q, respectively.
(1) Sketch the band diagram for the pn junction diode under the forward bias condition (applied
voltage: VF) and that under the reverse bias condition (applied voltage: −VR). The energy
positions of the top of the valence band EV, the bottom of the conduction band EC, the positionof the Fermi level EF, the diffusion potential VD of the pn junction, and the applied voltage (VF,−VR) must be given in the drawings, where VF and VR are positive values.
(2) Show one-dimensional Poisson’s equation for the depletion region under the reverse bias
condition. The boundary between p- and n-type regions is x = 0, where p-type region is x < 0,and n-type region is x > 0. The position of the edge of the depletion layer in the p-type regionis x = −Wp, the position of the edge of the depletion layer in the n-type region is x = Wn, andthe dielectric constant of Si is represented as εs.
(3) Solve the Poisson’s equation obtained above, and derive an equation showing the relationshipamong the width of the depletion layer in the p-type region Wp, the width of the depletion layer
in the n-type region Wn, the acceptor concentration NA, and the donor concentration ND.
(4) Show an equation for the capacitance C of the depletion layer per unit area for p+n one-sided
step junction (NA» ND) applied with a reverse bias (−VR).
(5) The values of the donor concentration ND and the diffusion potential VD can be obtained byanalyzing experimental data of the depletion-layer capacitance C per unit area as a function of
the reverse bias. Explain the procedure to obtain the values of ND and VD.

12

㸱
Answer the following questions about a bipolar junction transistor (BJT) consisting of theemitter (E), base (B), and collector regions (C).
(1) Draw a cross section of a pnp BJT schematically. The E, B, and C should be given in thedrawing. In addition, the polarity of the bias voltages VE and VC between base and emitter andbetween base and collector, respectively, in the common base circuit should be shown in thedrawing by using the circuit symbols of DC (direct current) voltage sources.
(2) Sketch a band diagram of a pnp BJT under the normal active condition. The energy positionsof the top of the valence band EV, the bottom of the conduction band EC, the Fermi level EF,
the diffusion potential VD of the pn junction, and the applied biases (emitter voltage VE andcollector voltage VC) should be given in the drawing. The elementary charge is represented asq.
(3) An emitter voltage of VE is applied to the pn junction between base and emitter of a pnp BJT.
Derive an equation for the concentration of holes Δpn injected into the edge of the depletion
region in the base region (the increase in hole concentrations from that under a thermal
equilibrium condition) and that for the concentration of free electrons Δnp injected into theedge of the depletion region in the emitter region (the increase in free electron concentrations
from that under a thermal equilibrium condition). Physical quantities which can be used in theequations are limited to the impurity concentration in the base region NB, the impurity
concentration in the emitter region NE, the intrinsic carrier concentration ni, Boltzmann
constant k, the absolute temperature T, the elementary charge q, and the emitter voltage VE.
(4) Show an equation for the relation between the emitter current IE and the collector current IC
using the the common base current gain D.
(5) In order to increase the common base current gain D, should the impurity concentration in theemitter region NE be smaller or larger than the impurity concentration in the base region NB?
Answer with the reason on the basis of the equations derived in question (3) above.
(6) In order to increase the common base current gain D, should the width of the base region belarge or small? Answer with the reason.
(7) Represent the current gain in a common emitter circuit E using the common base current gain
D.
(8) Which is suitable for application to circuits, which operate at higher frequency, a pnp BJT oran npn BJT? Answer with the reason.

13令和２年度ฏ੒̏̍೥౓ɹ‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊

2019.08.26‫( ֶ޻ػࢉܭ‬Computer Engineering)ʢ8 ຕதͷ 1ʣ࣍ͷ֤໰͍ʢʲ໰ 1ʳʙʲ໰ 3ʳʣʹ౴͑Αɽେ໰Ұͭ͋ͨΓҰຕͷղ౴༻ࢴΛ༻͍Αɽ
ʲ໰ 1ʳ࿦ཧؔ਺ H(a, b, c, d) ΛਤͰࣔ͞ΕΔΑ͏ʹؔ਺ F (a, b, c, d)ɼG(a, b, c, d) ͓Αͼ NANDήʔτΛ࢖࣮ͬͯ‫͢ݱ‬Δ͜ͱΛߟ͑Δɽؔ਺ H ͓Αͼؔ਺ G ͷਅཧ஋ද͕ҎԼͷΑ͏ʹ༩͑ΒΕΔ࣌ɼؔ਺ F ͷ࠷؆ੵ࿨‫ܗ‬Λࣔͤɽ

a

b

c

d

H

G

0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1

0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1

0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1

0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1

0
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1

1
1
0
1
0
1
0
0
0
1
1
1
1
1
0
0

a
b
c
d

F
H
G

14

令和２年度ฏ੒̏̍೥౓ɹ‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊

2019.08.26‫( ֶ޻ػࢉܭ‬Computer Engineering)ʢ8 ຕதͷ 2ʣ
ʲ໰ 2ʳ 16 ϏοτͷΞυϨεʢadr[15:0]ʣΛೖྗͱ͢Δ 2 ΢ΣΠɾηοτΞιγΞςΟϒΩϟο
γϡͷઃ‫͑ߟ͍ͯͭʹܭ‬ΔɽόΠτΞυϨογϯάํࣜͰ͋Γ 1 ‫ޠ‬͸ 4 όΠτͱ͢Δɽ
֤ΩϟογϡΞΫηεʹ͓͍ͯɼadr[15:8]ɼadr[7:4] ͳΒͼʹ adr[3:0] ͸ɼͦΕͧΕɼλάϑΟʔϧυɼΠϯσοΫεϑΟʔϧυɼΦϑηοτϑΟʔϧυͱͯ͠ࢀর͞ΕΔɽҎԼͷ໰͍ʹ౴͑Α(1) ΩϟογϡϒϩοΫαΠζΛ౴͑Αɽ(2) ΩϟογϡαΠζΛ౴͑Αɽ

15令和２年度ฏ੒̏̍೥౓ɹ‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊

2019.08.26‫( ֶ޻ػࢉܭ‬Computer Engineering)ʢ8 ຕதͷ 3ʣ
ʲ໰ 3ʳ࣍ͷ֤໰͍ʹ౴͑Αɽ=======================
ҎԼͷؔ܎ Item(id, name, price), User(uid, name, mail), Transaction(item,

uid, star), Follow(fid, uid1, uid2, time) ʹର͠ɼ࣍ͷ֤໰͍ʹ౴͑Αɽ
Item

User

id | name | price------------------001 | abc |150
002 | def |120
003 | ghi |230
004 | jkl |180005 | mno |350
006 | pqr |140
007 | stu |150

uid | name | mail--------------------u01 | A| s@a.com
u02 | B| t@b.com
u03 | C| u@c.com
u04 | D| v@d.com
u05 | E| w@e.com
u06 | F| x@f.com
u07 | G| y@g.com

Transaction

Follow

item| uid | star---------------001 | u01 | 3
001 | u03 | 4
001 | u06 | 5
002 | u03 | 4002 | u05 | 5002 | u06 | 5
003 | u02 | 2

fid| uid1| uid2| time---------------------------f0001 | u01 | u03 | 20190801
f0002 | u05 | u06 | 20190810
f0003 | u04 | u03 | 20190810
f0004 | u01 | u02 | 20190811
f0005 | u02 | u01 | 20190813
f0006 | u06 | u05 | 20190817
f0007 | u04 | u02 | 20190820(a) ؔ܎ Follow ͸ uid1 ͷϢʔβ͕ uid2 ͷϢʔβΛ࣌ࠁ time ʹϑΥϩʔͨ͠ͱ͍͏ࣄ
࣮ΛूΊͨू߹Ͱ͋Δɽ͜ͷؔ܎ͷީิΩʔΛશͯ౴͑Αɽͨͩ͠ɼ͋ΔϢʔβ͕
͋ΔϢʔβΛϑΥϩʔͰ͖Δͷ͸ߴʑҰ౓͚ͩͰ͋ΔͱԾఆ͢Δɽ(b) ؔ܎ Transaction ͸ uid ͷϢʔβ͕ id ͷ঎඼Λߪೖ͠ɼ1ʙ5 ͷ஋ star Ͱ঎඼ΛධՁͨ͠ͱ͍͏ࣄ࣮ΛूΊͨू߹Ͱ͋Δɽid ͕ 002 Ͱ͋Δ঎඼ΛങͬͨϢʔβͷ uidͷू߹Λ‫ٻ‬ΊΔؔ܎୅਺ͷԋࢉΛ౴͑Αɽ(c) ؔ܎ User ͸Ϣʔβ ID ͕ uid Λ࣋ͭϢʔβ name ͷϝʔϧΞυϨε͕ mail Ͱ͋Δ͜ͱΛද͢ू߹Ͱ͋Δɽid ͕ 002 Ͱ͋Δ঎඼ΛങͬͨϢʔβͷϝʔϧΞυϨεͷू߹Λ‫ٻ‬ΊΔؔ܎୅਺ͷԋࢉΛ౴͑Αɽ(d) id ͕ 002 Ͱ͋Δ঎඼ΛങͬͨϢʔβΛϑΥϩʔ͍ͯ͠ΔϢʔβͷू߹Λ‫ٻ‬ΊΔؔ܎୅਺ͷԋࢉΛ౴͑Αɽ(e) ؔ܎ Item ͸঎඼ͷࣝผࢠ idɼͦͷ໊લ nameɼͦͷՁ֨ price ΛूΊͨू߹Ͱ͋Γɼ͜ͷؔ܎͔ΒҎԼͷΑ͏ͳ݁ՌΛಘΔଐੑ price ͱෆ౳߸Λ༻͍ͨؔ܎୅਺ͷԋࢉΛ౴͑Αɽ
id | price16令和２年度ฏ੒̏̍೥౓ɹ‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊

2019.08.26‫( ֶ޻ػࢉܭ‬Computer Engineering)ʢ8 ຕதͷ 4ʣ-----------003 |
230
005 |
350
(f) ؔ܎ R ͱ Follow ͷࣗવ݁߹ (R[uid = uid1]Follow)[fid, uid1, uid2] ͰɼҎԼͷΑ͏ͳ݁Ռ͕ಘΒΕͨɽ͜ͷ࣌ͷؔ܎ R Λɼؔ܎ Transaction ͱଐੑ star Λ༻͍ͯ౴͑Αɽfid
| uid1| uid2-----------------f0002 | u05 | u06
f0006 | u06 | u05(g) ؔ܎ Follow ͱ R ͷ঎ Follow[uid1, uid2] ÷ R ͰɼҎԼͷΑ͏ͳ݁Ռ͕ಘΒΕͨɽ͜ͷ࣌ͷؔ܎ R Λ౴͑Αɽ
uid1---u01u04

17

2019.08.26
令和２年度ฏ੒̏̍೥౓ɹ‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊‫( ֶ޻ػࢉܭ‬Computer Engineering)ʢ8 ຕதͷ 5ʣ
Answer the following questions (ʲQ1ʳʲQ2ʳ).
【Q1】~【Q3】ʲ
Q1ʳ Let H(a, b, c, d) be a logic function. Consider that H is composed of another subfunctions【Q1】

F (a, b, c, d), G(a, b, c, d) and a NAND gate as shown in the ﬁgure below. When thetruthtables of H and G are given below, show the minimum sum of products form of F .
a

b

c

d

H

G

0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1

0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1

0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1

0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1

0
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1

1
1
0
1
0
1
0
0
0
1
1
1
1
1
0
0

a
b
c
d

F
H
G

18

令和２年度ฏ੒̏̍೥౓ɹ‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊

2019.08.26‫( ֶ޻ػࢉܭ‬Computer Engineering)ʢ8 ຕதͷ 6ʣ
ʲ【Q2】Q2ʳ Consider designing a two-way set-associative cache memory that accepts a 16-bit address(adr[15:0]). Suppose a byte-addressing scheme and the word size is 4 bytes. On eachcache access, adr[15:8], adr[7:4], and adr[3:0] are referenced as a tag ﬁeld, an index ﬁeld,
and an oﬀset ﬁeld, respectively. Answer the following questions.
(1) Answer the cache block size.
(2) Answer the cache size.

19令和２年度ฏ੒̏̍೥౓ɹ‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊

2019.08.26‫( ֶ޻ػࢉܭ‬Computer Engineering)ʢ8 ຕதͷ 7ʣ
ʲ【Q3】Q2ʳ Answer the following questions. For the following relations Item(id, name, price),

User(uid, name, mail),
Follow(fid, uid1, uid2, time), and Transaction(item, uid, star), answer thefollowing questions.
Item

User

id | name | price------------------001 | abc |150
002 | def |120
003 | ghi |230
004 | jkl |180005 | mno |350
006 | pqr |140
007 | stu |150

uid | name | mail--------------------u01 | A| s@a.com
u02 | B| t@b.com
u03 | C| u@c.com
u04 | D| v@d.com
u05 | E| w@e.com
u06 | F| x@f.com
u07 | G| y@g.com

Transaction

Follow

item| uid | star---------------001 | u01 | 3
001 | u03 | 4
001 | u06 | 5
002 | u03 | 4002 | u05 | 5002 | u06 | 5
003 | u02 | 2

fid| uid1| uid2| time---------------------------f0001 | u01 | u03 | 20190801
f0002 | u05 | u06 | 20190810
f0003 | u04 | u03 | 20190810
f0004 | u01 | u02 | 20190811
f0005 | u02 | u01 | 20190813
f0006 | u06 | u05 | 20190817
f0007 | u04 | u02 | 20190820(a) Relation Follow is a set of facts that a user of uid1 follows another user of uid2 attime of time. Answer all candicate keys, where we assume that a user can follow
one user at most once.
(b) Relation Transaction is a set of facts that a user of uid purchases an item of idand puts a score from 1 to 5 as star. Answer operations of the relational algebra
to obtain the set of uids who purchase items whose ids are 002.
(c) Relation User is a set of facts that a user whose ID is uid and name is name has amail address mail. Answer operations of the relational algebra to obtain the set ofmails of users who purchase items whose ids are 002.
(d) Answer operations of the relational algebra to obtain the set of users following theusers who purchased items whose ids are 002.
(e) Relation Item is a set of items whose IDs are id, their names names, and theirprices. Answer operations of the relational algebra to obtain the following result
using attribute price and a sign of inequality.
id | price20令和２年度ฏ੒̏̍೥౓ɹ‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊

2019.08.26‫( ֶ޻ػࢉܭ‬Computer Engineering)ʢ8 ຕதͷ 8ʣ-----------003 |
230
005 |
350
(f) When the following result is obtained using the natural join
(R[uid = uid1]Follow)[fid, uid1, uid2] of two relations R and Follow, answer relation R using relation Transaction and attribute star.
fid| uid1| uid2-----------------f0002 | u05 | u06
f0006 | u06 | u05(g) When the following result is obtained using the division Follow[uid1, uid2] ÷ R of
two relations Follow and R, answer relation R.
uid1---u01
u04

21