
۟य୉ָ୉ָӅεητϞ৚ๅՌָැ
ు‫ـ‬ుࢢ޽ָ઒ߊ
ྫ࿪̔೧ౕ೘ָࢾ‫ݩ‬໲ୌʴྫ࿪̓೧̖݆̗̐ೖʤՒʥɼ̑̎ೖʤਭʥʵ
৘ใཧ޻ֶઐ߈
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊ʢྩ࿨̑೥݄̔̎̕೔ʣ

‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎

਺ ֶ (Mathematics)
ʢ7 ຕதͷ 1ʣղ౴্ͷ஫ҙ (Instructions):

1. ໰୊༻ࢴ͸ɼʰ࢝Ίʱͷ߹ਤ͕͋Δ·Ͱ։͍ͯ͸ͳΒͳ͍ɽ
Do not open this cover sheet until the start of examination is announced.
2. ໰୊༻ࢴ͸දࢴΛ‫ؚ‬Ί 7 ຕɼղ౴༻ࢴ͸ 3 ຕͭͮΓ (1 ෼໺ʹ͖ͭ 1 ຕ) Ͱ͋ΔɽYou are given 7 problem sheets including this cover sheet, and 3 answer sheets (1 sheet foreach ﬁeld).
3. ઢ‫ܗ‬୅਺ɼղੳֶɾඍੵ෼ͷ 2 ෼໺ʹՃ͑ͯɼϕΫτϧղੳ͓Αͼ֬཰ɾ౷‫͔ܭ‬Β 1 ෼໺Λબ୒͠ɼ߹‫ ܭ‬3 ෼໺ʹ͍ͭͯղ౴͢Δ͜ͱɽબΜͩ෼໺ຖʹղ౴༻ࢴΛผʹ͢Δ͜ͱɽ

Answer three ﬁelds in total, including Linear algebra and Analysis and calculus, and either
Vector analysis or Probability and statistics. You must use a separate answer sheet for eachof the ﬁelds you selected.

1
2
3
4

෼໺
ઢ‫ܗ‬୅਺
ղੳֶɾඍੵ෼
ϕΫτϧղੳ
֬཰ɾ౷‫ܭ‬ﬁeld
Linear algebra
Analysis and calculus
Vector analysis
Probability and statistics

page
2
4
6
7

4. ղ౴༻ࢴͷશ෦ʹɼઐ߈໊ɼड‫ݧ‬൪߸͓Αͼࢯ໊Λ‫ه‬ೖ͢Δ͜ͱɽ3 ຕ໨ͷղ౴༻ࢴʹ͍ͭͯ͸ɼબ୒ͨ͠෼໺൪߸ʢ3 ·ͨ͸ 4ʣΛ˓ͰғΉ͜ͱɽ

Fill in the designated blanks at the top of each answer sheet with the department name, yourexaminee number and your name. Mark the selected ﬁeld number (3 or 4) with a circle on
the third answer sheet.
5. ղ౴͸ղ౴༻ࢴʹ‫ه‬ೖ͢Δ͜ͱɽεϖʔε͕଍Γͳ͍৔߹͸ཪ໘Λ༻͍ͯ΋ྑ͍͕ɼͦͷ৔߹͸ɼཪ໘ʹղ౴͕͋Δ͜ͱΛ໌‫͢ه‬Δ͜ͱɽ

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. ղ౴͸ɼ೔ຊ‫ޠ‬ɼӳ‫ޠ‬ͷ͍ͣΕ͔Ͱ‫ه‬ೖ͢Δ͜ͱɽ
Your answers must be written in Japanese or English.

৘ใཧ޻ֶઐ߈
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊ʢྩ࿨̑೥݄̔̎̕೔ʣ

‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎

਺ ֶ (Mathematics)
ʢ7 ຕதͷ 2ʣ෼໺ຖʹղ౴༻ࢴΛผʹ͢Δ͜ͱɽ
Use a separate answer sheet for each ﬁeld.

1. ʲઢ‫ܗ‬୅਺ (Linear algebra) ෼໺ʳ
m × n ࣮ߦྻ A ͱ m ࣍‫࣮ݩ‬ϕΫτϧ b ʹରͯ͠ɼS = {x ∈ Rn | Ax = b}ɼf (x) = Ax Λఆٛ͢Δɽ͜ͷͱ͖ɼҎԼͷ໰͍ʹ౴͑Αɽͨͩ͠ɼҎԼͷࣄ࣮͸ূ໌ͳ͠ʹ༻͍ͯΑ͍ɽ
ࣄ࣮. R ্ͷϕΫτϧۭؒ (ઢ‫)ۭؒܗ‬V ͷ෦෼ू߹ W ͕ V ͷ෦෼ۭؒͰ͋Δඞཁे෼৚݅͸ҎԼͷ৚͕݅ຬͨ͞ΕΔ͜ͱͰ͋Δɽ

C1. 0 ∈ W ɽ
C2. u, v ∈ W ͳΒ͹ u + v ∈ W ɽ
C3. u ∈ W, c ∈ R ͳΒ͹ cu ∈ W ɽ⎤
2 0 2 0⎥⎢
(1) A = ⎣0 8 24 24⎦ Ͱ͋Δͱ͖ɼf ͷ֩ Ker(f ) ͷ࣍‫جͱݩ‬ఈΛҰ૊‫ٻ‬ΊΑɽ
0 2 6 6⎡(2) Ұൠʹ Ker(f ) ͕ Rn ͷ෦෼ۭؒͰ͋Δ͜ͱΛࣔͤɽ(3) S ͕ Rn ͷ෦෼ۭؒͰ͋Δͱ͖ɼb = 0 Ͱ͋Δ͜ͱΛࣔͤɽ(4) S ͕ Rn ͷ෦෼ۭؒɼA ͕ਖ਼ํߦྻͰ͋Δͱ͢Δɽ͜ͷͱ͖ɼA ͕ਖ਼ଇߦྻͰ͋ΔͳΒ͹ɼS = {0} Ͱ͋Δ͜ͱΛࣔͤɽ৘ใཧ޻ֶઐ߈
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊ʢྩ࿨̑೥݄̔̎̕೔ʣ

‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎

਺ ֶ (Mathematics)
ʢ7 ຕதͷ 3ʣ෼໺ຖʹղ౴༻ࢴΛผʹ͢Δ͜ͱɽ
Use a separate answer sheet for each ﬁeld.

For an m×n real matrix A and an m-dimensional real vector b, deﬁne S = {x ∈ Rn | Ax = b}
and f (x) = Ax. Answer the following questions. Note that the following fact can be usedwithout proof.
Fact. A subset W of a vector space (linear space) V over R is a subspace of V if and only if
the following conditions hold.
C1. 0 ∈ W .
C2. If u, v ∈ W , then u + v ∈ W .
C3. If u ∈ W and c ∈ R, then cu ∈ W .
⎤⎡2 0 2 0⎥⎢
(1) When A = ⎣0 8 24 24⎦, ﬁnd the dimension of the kernel Ker(f ) of f and a basis of
0 2 6 6
the kernel.
(2) Show Ker(f ) is a subspace of Rn in general.
(3) Suppose that S is a subspace of Rn . Show b = 0 .
(4) Suppose that S is a subspace of Rn and A is a square matrix. Show S = {0} if A isinvertible.

৘ใཧ޻ֶઐ߈
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊ʢྩ࿨̑೥݄̔̎̕೔ʣ

‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎

਺ ֶ (Mathematics)
ʢ7 ຕதͷ 4ʣ෼໺ຖʹղ౴༻ࢴΛผʹ͢Δ͜ͱɽ
Use a separate answer sheet for each ﬁeld.

2. ʲղੳֶɾඍੵ෼ (Analysis and calculus) ෼໺ʳ(1) ੵ෼ ∞I=Λ‫ͤࢉܭ‬Αɽͨͩ͠ɼ

x5 exp(−x4 )dx0√∞exp(−x2 )dx = π Λূ໌ͳ͠ʹ༻͍ͯΑ͍ɽ−∞(2) ࣍ͷඍ෼ํఔࣜͷҰൠղΛ‫ٻ‬ΊΑɽdy+ y = x sinh xdx(3) ෳૉؔ਺ f (z) =1
z4 + 1Λߟ͑Δɽ࣍ͷ֤໰͍ʹ౴͑Αɽ(a) f (z) ͷ‫ۃ‬Λ͢΂ͯ‫ٻ‬ΊΑɽ(b) Լਤʹࣔ͢൒ԁ C ʹԊͬͨෳૉੵ෼

Cf (z)dz Λ‫ٻ‬ΊΑɽͨͩ͠ɼR > 1 ͱ͢Δ.

৘ใཧ޻ֶઐ߈
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊ʢྩ࿨̑೥݄̔̎̕೔ʣ

‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎

਺ ֶ (Mathematics)
ʢ7 ຕதͷ 5ʣ෼໺ຖʹղ౴༻ࢴΛผʹ͢Δ͜ͱɽ
Use a separate answer sheet for each ﬁeld.

(1) Calculate the integral ∞I=

where you can use

x5 exp(−x4 )dx,0√∞2exp(−x)dx=
π without proving it.
−∞(2) Find the general solution to the following diﬀerential equation.
dy+ y = x sinh x.
dx(3) Consider the complex function f (z) =1
z4 + 1. Answer the following questions.

(a) Find all the poles of f (z).
(b) Calculate the complex integral
ﬁgure below, and R > 1.

C

f (z)dz, where C is a semicircle as shown in the৘ใཧ޻ֶઐ߈
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊ʢྩ࿨̑೥݄̔̎̕೔ʣ

‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎

਺ ֶ (Mathematics)
ʢ7 ຕதͷ 6ʣ෼໺ຖʹղ౴༻ࢴΛผʹ͢Δ͜ͱɽ
Use a separate answer sheet for each ﬁeld.

3. ʲϕΫτϧղੳ (Vector analysis) ෼໺ʳ௚ަ࠲ඪ‫͍͓ͯʹܥ‬ɼxɼy ࣠ํ޲ͷ୯ҐϕΫτϧΛͦΕͧΕiɼj ͱ͢Δɽ࣍ͷ (1)ɼ(2) ʹࣔ͢ϕΫτϧ৔ F ͱ‫ܦ‬࿏ C ʹ͍ͭͯɼઢੵ෼

C

F · dr ΛͦΕͧΕ‫ٻ‬ΊΑɽ(1) ϕΫτϧ৔ F Λ F = (2xy 3 − y) i + (3x2 y 2 − x) j ͱ͠ɼ‫ܦ‬࿏ C Λ‫ۂ‬ઢ y 3 + x2 − 2xy = 1ͷ఺ (−13, −4) ͔Β఺ (5, −4) ·Ͱͷ෦෼ͱ͢Δɽ(2) ϕΫτϧ৔ F Λ F = (3e2x + 4y) i + (x + 7yey ) j ͱ͢Δɽ ‫ܦ‬࿏ C ͸ɼ఺ P(0, 0)ɼ఺Q(2, 2)ɼ఺ R(4, 6)ɼ఺ S(1, 3) Λ௖఺ͱ͢Δ࢛ล‫Ͱܗ‬ɼPɼQɼRɼSɼP ͷॱʹ޲͖͚ͮΒΕ͍ͯΔͱ͢Δɽ

The unit vectors on x and y axes of Cartesian coordinates are denoted by i and j, respectively.
Find each line integralCF · dr of the vector ﬁeld F along the path C deﬁned in (1) and (2).

(1) Let F = (2xy 3 − y) i + (3x2 y 2 − x) j, and C be the curve y 3 + x2 − 2xy = 1 from(−13, −4) to (5, −4).
(2) Let F = (3e2x + 4y) i + (x + 7yey ) j, and C be the quadrilateral with vertices at points
P(0, 0), Q(2, 2), R(4, 6) and S(1, 3), which is directed from P to Q, Q to R, R to S and
S to P.

৘ใཧ޻ֶઐ߈
ి‫ֶ޻ࢠిؾ‬ઐ߈म࢜՝ఔɹೖֶࢼ‫ݧ‬໰୊ʢྩ࿨̑೥݄̔̎̕೔ʣ

‫۝‬भେֶେֶӃγεςϜ৘ใՊֶ෎

਺ ֶ (Mathematics)
ʢ7 ຕதͷ 7ʣ෼໺ຖʹղ౴༻ࢴΛผʹ͢Δ͜ͱɽ
Use a separate answer sheet for each ﬁeld.

4. ʲ֬཰ɾ౷‫( ܭ‬Probability and statistics) ෼໺ʳശͷதʹ N1 ‫ݸ‬ͷന͍Ϙʔϧͱ N2 ‫ݸ‬ͷࠇ͍Ϙʔϧ͕͋Γɼͦͷ૯਺Λ N = N1 + N2 ͱ͢
Δɽ͜ͷശ͔Β 2 ͭͷϘʔϧΛϥϯμϜʹબͼɼ྆ํ͕ന͍ϘʔϧͰ͋Δ֬཰͸ 1/2 Ͱ͋Δͱ͢Δɽ(1) N2 ͕‫਺ح‬ͷͱ͖ N1 ͷ࠷খ஋Λ‫ٻ‬ΊΑɽ(2) N2 ͕‫਺ۮ‬ͷͱ͖ N1 ͷ࠷খ஋Λ‫ٻ‬ΊΑɽ(3) N Λ஋ͷখ͍͞ॱʹ 3 ͭ‫ٻ‬ΊΑɽ
A box contains N1 white and N2 black balls, and the total number of balls is N = N1 + N2 .
When two balls are randomly drawn from the box, the probability that both balls are whiteis 1/2.
(1) Find the minimum value of N1 when N2 is an odd number.
(2) Find the minimum value of N1 when N2 is an even number.
(3) Find the three smallest values of N .

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（実施日 令和 5年 8月 29日）

専 門 科 目 (Specializedsubjects)()

解答上の注意 (Instructions):

 問題用紙は，『始め』の合図があるまで開いてはならない．
Donotopen thiscoversheet untilthestartofexaminationis announced.
 問題用紙は表紙を含め 枚，解答用紙は 3枚つづり 2部 (1分野につき 1部)である．
Youaregivenproblemsheetsincludingthiscoversheet,and2setsof3answersheets(1set
foreachﬁeld).
 以下の 5分野から 2分野を選び解答すること．解答用紙は１分野につき１部，大問１つあた
り１枚を使用すること．１枚に大問２問以上の解答を書いてはならない．

Select2ﬁeldsoutofthefollowing5ﬁeldsandanswerthequestions.Youmustuseaseparate
setofanswersheetsforeachoftheﬁeldsyouselected.Onesheetinasetisforonequestion.
Youmaynotuseonesheetfortwoormorequestions

A
B
C
D
E

分野
電気回路
電子回路
制御工学
電磁気学
半導体デバイス

ﬁeld
Circuit theory
Electronic circuits
Control engineering
Electromagnetism
Semiconductor device

page






4. 解答用紙の全部に，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 selected ﬁeld, yourexaminee number, your name, and the question number.
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 so 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.

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(2/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

A. 【電気回路 (Circuit theory) 分野】
次の各【問 1】∼【問 3】に答えよ．解答用紙の問題番号欄に解答した問題番号を記入すること．
【問 1】 図 1 の回路について，以下の問いに答えよ．ただし，端子間には e(t) =sin 2t [V] の交流電圧が印加されているとする．

(1) 端子間のインピーダンスを求めよ．















(2) 電流 i(t) を求めよ．

(3) i(t)，i1 (t) のフェーザ表記をそれぞれ I ，I1 としたとき， II1 はいくらになるか答えよ．



図1

【問 2】 図 2 の回路において，ZG を変えることで負荷インピーダンス ZL で消費される電

力を最大化したい．以下の問いに答えよ．た‫܈‬૚だし，J は交流電流源であり，Z1 で表される
二端子対網は以下に示すインピーダンス行列（Z 行列）のように表されるとする．また，図

‫ܬ‬

ͳ

ܼீ
݆

内の破線で囲まれた部分に示された数値は全てインピーダンスを示すものである．
√

Z1 =

1+j
−1 + 7√−1 + 7
1−j

ܼ௅
െ݆
b

b

図2

(1) 破線で囲まれた部分を二端子対網と見なし，この部分に相当する:::::::::::::::::::
アドミタンス行列（Y行列）Y2 を求めよ．ただし，破線内の数値はインピーダンスであることに留意せよ．さらに 1-1’ 端子と 2-2’ 端子からなる二端子対網の Z 行列 Z0 を求めよ．

(2) 2-2’ 端子から左側を見た出力インピーダンス Zout を ZG を含む式として求めよ．
(3) ZL = 1 + j2 のとき，ZL での消費電力が最大になる ZG を求めよ．

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(3/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

S【問 3】 図 3 の回路について，以下の問いに答えよ．ただし，E = 5 [V]，R1 = 1 [Ω]，R2 = 2 [Ω]，R3 = 3 [Ω]，R4 = 4 [Ω]，C =2 [F] であり，スイッチ S を閉じる前の回路は定常状態にあるとする．

(1) 時刻 t = 0 でスイッチ S を閉じた直後の電荷量 q(0) を求めよ．

t=0
i(t)

R1

R2
R4
R3

‫ܧ‬

q(t)
C

(2) t > 0 における電流 i(t) を求めよ．
図3

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(4/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

Answer the following questions (【Q1】【Q2】【Q3】) and write the number of the selected
question on the answer sheet.
【Q1】 Consider the circuit shown in Fig. 1,
where the alternating voltage e(t) = sin 2t[V] is applied to the terminals. Answer thefollowing questions.









(1) Find the impedance at the terminals.







(2) Find the current i(t).

(3) The phaser expressions of i(t) andi1 (t) are I and I1 , respectively. Find
the value of II1 .
【Q2】 Consider the circuit shown in Fig. 2. The
objective is to maximize the power consumed at
the load impedance ZL by optimizing ZG . Find
the solution by answering to the following questions, where J is the alternative current source,
and the two-port pair network represented by Z1
has the impedance matrix (Z-matrix) parameters
as follows. Note that all values shown in thedashed line in the ﬁgure are impedances.
Z1 =

√
1+j
−1 + 7√−1 + 71−jFig. 1‫܈‬૚

‫ܬ‬

ͳ

ܼீ
݆

ܼ௅
െ݆
b

b

Fig. 2(1) Consider the area enclosed by the dashed line as a two-port pair network described byadmittancematrix (Y-matrix) Y2 , and ﬁnd Y2 , where note that all values in the dashed:::::::::::::::::::
line are impedances. Furthermore, ﬁnd the Z-matrix Z0 of the two-port pair network
between 1-1’ and 2-2’ ports.
(2) Find the output impedance Zout considering left side from 2-2’ port as an expression
involving ZG .
(3) Find ZG that maximizes power consumption at ZL when ZL = 1 + j2.

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(5/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

S【Q3】 Consider the circuit shown in Fig.
3, where E = 5 [V], R1 = 1 [Ω], R2 = 2 [Ω],R3 = 3 [Ω], R4 = 4 [Ω], C = 2 [F], and thecircuit is in steady state before the switch
S is closed.

(1) Find the charge q(0) just after theswitch S is closed at t = 0.

t=0
i(t)

R1

R2
R4
R3

‫ܧ‬

q(t)
C

(2) Find the current i(t) at t > 0.
Fig. 3஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(6 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer sheets foreach of the fields you selected and use an answer sheet for each question.

B. ࠙㟁Ꮚᅇ㊰ (Electronic circuits) ศ㔝ࠚḟࡢྛၥ࠸㸦࠙ၥ㸯ࠚ࠙ၥ㸰ࠚ㸧࡟⟅࠼ࡼ㸬ゎ⟅⏝⣬ࡢၥ㢟␒ྕḍ࡟ゎ⟅ࡋࡓၥ㢟␒ྕࢆグධࡍࡿࡇ࡜㸬࠙ၥ㸯ࠚᅗ㸯࡟♧ࡍᅇ㊰࡟࠾࠸࡚㸪ධຊ㟁ᅽ ࡢࣛࣉࣛࢫኚ᥮ࢆ ( )㸪ฟຊ㟁ᅽ ࡢࣛࣉࣛࢫኚ᥮ࢆ ( )㸪ධຊ㟁ὶ ࡢࣛࣉࣛࢫኚ᥮ࢆ ( )࡜ࡍࡿ࡜ࡁ㸪ḟࡢၥ࠸࡟⟅࠼ࡼ㸬ࡓࡔࡋ㸪₇⟬ቑᖜჾࡣ⌮᝿ⓗ࡛࠶ࡿ࡜ࡍࡿ㸬㸦㸯㸧㟁ᅽ฼ᚓࡢఏ㐩㛵ᩘ ( ) =( )/ ( )ࢆồࡵࡼ㸬㸦㸰㸧㟁ᅽ฼ᚓࡢ࿘Ἴᩘఏ㐩㛵ᩘ (ࢻ⥺ᅗࢆᢡࢀ⥺㏆ఝ࡛ᥥࡅ㸬㸦㸱㸧ධຊ࢖ࣥࣆ࣮ࢲࣥࢫ ( ) =)࡟ࡘ࠸࡚㸪኱ࡁࡉ | ()|࡜఩┦∠ (( )/ ( )ࢆồࡵࡼ㸬㸦㸲㸧ධຊ࢖ࣥࣆ࣮ࢲࣥࢫࡢ࿘Ἴᩘ≉ᛶ (ࢢࣛࣇࢆᢡࢀ⥺㏆ఝ࡛ᥥࡅ㸬)࡟ࡘ࠸࡚㸪኱ࡁࡉ| (

1
1

2



ᅗ㸯




)ࡢ࣮࣎

)|࡜఩┦∠ ()ࡢ

஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(7 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer sheets foreach of the fields you selected and use an answer sheet for each question.

࠙ၥ㸰ࠚᅗ㸰࡟♧ࡍLCṇᘻἼⓎ᣺ჾ࡟ࡘ࠸࡚㸪ḟࡢၥ࠸࡟⟅࠼ࡼ㸬ࡓࡔࡋ㸪
₇⟬ቑᖜჾࡣ⌮᝿ⓗ࡛࠶ࡿ࡜ࡍࡿ㸬㸦㸯㸧ቑᖜᅇ㊰㸦Aࡢ㒊ศ㸧ࡢ㟁ᅽ฼ᚓGA ࢆồࡵࡼ㸬
㸦㸰㸧RLCᅇ㊰㸦Bࡢ㒊ศ㸧ࡢ㟁ᅽ฼ᚓ㸦ῶ⾶⋡㸧GB ࢆồࡵࡼ㸬㸦㸱㸧LCṇᘻἼⓎ᣺ჾࡢ࣮ࣝࣉ฼ᚓ T ࢆồࡵࡼ㸬
㸦㸲㸧Ⓨ᣺ࡀᐃᖖ≧ែ࡟࠶ࡿ᫬ࡢⓎ᣺ゅ࿘Ἴᩘ࡜᣺ᖜ᮲௳ࢆồࡵࡼ㸬


A

B

1

2


ᅗ㸰




஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(8 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer sheets foreach of the fields you selected and use an answer sheet for each question.

Answer the following questions ࠙Q1ࠚ,࠙Q2ࠚ and write the number of the selected question on
the answer sheet.

࠙Q1ࠚIn the circuit shown in Fig. 1, the Laplace transform of the input voltage
is ( ), theLaplace transform of the output voltage is ( ), and the Laplace transform of the input current
is ( ). Answer the following questions. The operational amplifier is assumed to be ideal.
(1) Derive the transfer function of the voltage gain ( ) =( )/ ( ).

(2) Sketch the straight-line Bode plots of the magnitude | (frequency transfer function of the voltage gain ( ).

)| and the phase ∠ ((3) Derive the input impedance( )=) for the( )/ ( ).

(4) Sketch the straight-line graphs of the magnitude | (frequency characteristics of the input impedance (1
1

2

Fig. 1

)| and the phase ∠ ().

) for the஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(9 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer sheets foreach of the fields you selected and use an answer sheet for each question.

࠙Q2ࠚAnswer the following questions on the LC sinusoidal oscillator shown in Fig. 2. The
operational amplifier is assumed to be ideal.
(1) Derive the voltage gain GA of the amplifier circuit (part A).
(2) Derive the voltage gain (attenuation rate) GB of the RLC network (part B).
(3) Derive the loop gain T of the LC sinusoidal oscillator
(4) Obtain the angular oscillation frequency and the condition of the amplitude for steady-stateoscillation.

A

B

1

2

Fig. 2

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(10/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

C. 【制御工学 (Control engineering) 分野】
次の各【問 1】∼【問 3】に答えよ．解答用紙の問題番号欄に解答した問題番号を記入するこ
と.

【問 1】
図 1 に示すフィードバック制御系について以下の問に答えよ．ただし，R(s) は目標値 r(t) のラプラス変換，Y (s) は制御量 y(t) のラプラス変換であり，t は時刻，s はラプラス変換の変数を表す．また，Kp > 0, Ti > 0, k > 0, T > 0 である．解答中の数値はすべて整数または小数で表現し，伝達関数は s の多項式を用いた分数式として表現せよ．

(1) この制御系の開ループ伝達関数を求めよ．
(2) この制御系の閉ループ伝達関数を求めよ．
(3) この制御系の安定判別を行え．

(4) この制御系に目標値として r(t) =0, t < 0
1, t ≥ 0を与えたときの定常偏差を求めよ．

(5) k = 1, T = 1 のとき，閉ループ伝達関数の極が −4 ± j3 となるように Kp , Ti の値を定めよ．ここで，j は虚数単位である．

(6) 任意の k > 0, T > 0 について，Kp , Ti の値を適切に設定することにより，任意の 2 個の負の実数あるいは任意の 1 組の共役複素数（ただし実部は負）を閉ループ伝達関数の極に割り当てることができるか否かを，理由とともに答えよ．

図1

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(11/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

【問 2】あるシステムは分母が s の 2 次多項式であって分子が定数である伝達関数で表される．このシステムに入力として u1 (t) = 1 を与えると，十分に時間が経過した後の出力 y1 (t) は 2 となった．また，入力として u2 (t) = sin 2t を与え，十分に時間が経過した後の入力 u2 (t) と出力 y2 (t)は図 2 に示す通りとなった．ただし，破線が u2 (t) を表し，実線が y2 (t) を表す．また縦の破線
は二つの曲線の位置関係を表すために描かれている．このシステムの伝達関数を定めよ．ただし，t は時刻，s はラプラス変換の変数を表す．

図2

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(12/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

【問 3】
次の状態方程式で表されるシステム G を考える．
ẋ(t) = Ax(t) + Bu(t),G:y(t) = Cx(t),
⎡
⎡
⎤
⎤
b1
−1
1
0


⎢
⎢
⎥
⎥
A = ⎣ 0 −1
0 ⎦ , B = ⎣ b2 ⎦ , C = 1 0 1
0
0 −1
b3
ただし b1 , b2 , b3 はスカラーの実定数であり，t は時刻を表す．以下の問に答えよ．

(1) 行列 A がフルビッツ安定であることを示せ．
(2) 行列指数関数 exp(At) を求めよ．
(3) 初期状態が x(0) = [ 1 1 1 ]T であり，入力が u(t) = 0 (0 ≤ t ≤ 1) であるとする．このとき，y(1) を求めよ．

(4) b1 , b2 , b3 をどのように選んでも，対 (A, B) は可制御にならないことを示せ．

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(13/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

Answer the following questions (【Q1】【Q2】【Q3】) and write the number of the selected
question on the answer sheet.

【Q1】

Answer the following questions regarding the feedback control system depicted in Fig.1, whereR(s) is the Laplace transform of the reference r(t), Y (s) stands for the Laplace transformof the controlled variable y(t), t stands for time, s is the variable in the Laplace transform.
Also, Kp > 0, Ti > 0, k > 0, and T > 0. When providing your answers, express numericalvalues as integers or decimals, and transfer functions as polynomial fractions in s.
(1) Find the open-loop transfer function of the control system.
(2) Find the closed-loop transfer function of the control system.
(3) Determine the stability of the control system.

0, t < 0is applied, as the reference, to the(4) Find the steady state error when r(t) =1, t ≥ 0
control system.
(5) Determine the values of Kp and Ti such that the poles of the closed-loop transfer function
are at −4 ± j3 when k = 1 and T = 1. Here j stands for the imaginary unit.
(6) Answer with the reason whether or not we can assign any two negative real values orany pair of conjugate complex numbers with negative imaginary parts to the poles of
the closed-loop transfer function by appropriately choosing values of Kp and Ti for any
k > 0 and T > 0.

Fig.1

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(14/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

【Q2】

A system is represented by a transfer function where its denominator is a second degree
polynomial in s and its numerator is a constant. We feed u1 (t) = 1 as the input to thesystem, and we have 2 as the value of output y1 (t) after a suﬃciently long time has passed.
Also, when we apply the input u2 (t) = sin 2t to the system, we obtain the input u2 (t) andthe output y2 (t) as shown in Fig.2 after a suﬃciently long time has passed. Here the dashedline indicates u2 (t) and the solid line represents y2 (t). The vertical dashed lines are drawn toclarify the relative positions of the two curves. Determine the transfer function of the system.
Here, t stands for time, and s is the variable in the Laplace transform.

Fig.2

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（実施日 令和 5 年 8 月 29 日）

専 門 科 目 (Specialized subjects)(15/25)
5 分野から 2 分野を選び解答すること．選んだ分野毎に解答用紙 1 部を用いよ．また，
大問一つあたり一枚の解答用紙を用いよ．
Select 2 ﬁelds out of the 5 ﬁelds and answer the questions. Use a set of answer sheetsfor each of the ﬁelds you selected and use an answer sheet for each question.

【Q3】

Consider the system G given by the following state equation.
ẋ(t) = Ax(t) + Bu(t),G:y(t) = Cx(t),
⎤
⎡
⎡
⎤
b1
−1
1
0


⎥
⎢
⎢
⎥
A = ⎣ 0 −1
0 ⎦ , B = ⎣ b2 ⎦ , C = 1 0 1 .
0
0 −1
b3
Here, b1 , b2 , and b3 are scalar and real constants and t stands for time. Answer the following
questions.
(1) Prove that the matrix A is Hurwitz stable.
(2) Calculate the matrix exponential exp(At).
(3) Suppose the initial state is given by x(0) = [ 1 1 1 ]T and the input is given by u(t) =0 (0 ≤ t ≤ 1). Calculate y(1).
(4) Prove that the pair (A, B) cannot be controllable for any b1 , b2 , and b3 .

஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴  ᖺ  ᭶  ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(16 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for eachti

D. ʴుࣕ‫( ָـ‬Electromagnetism) ෾໼ʵ௨ୗࡢྛၥ࠸ ࠙ၥࠚ࠙ၥࠚ ࡟⟅࠼ࡼ㸬ゎ⟅⏝⣬ࡢၥ㢟␒ྕḍ࡟ゎ⟅ࡋࡓၥ㢟␒ྕࢆグධࡍࡿࡇ࡜
࠙ၥ ࠚ ᅗ 1 䛾䜘䛖䛻䠈༙ᚄ a䠈b䠈c 䛾 3 䛴䛾ྠᚰᑟయ⌫Ẇ䛛䜙ᡂ䜛ᑟయ 1 䛸䠈༙ᚄ a䠈c 䛾 2 䛴䛾ྠᚰᑟయ⌫Ẇ䛛䜙ᡂ䜛ᑟయ 2 䜢䠈┿✵୰䛻୰ᚰ㛫㊥㞳 d 䛷⨨䛟䠊䛭䜜䛮䜜᭱䜒ෆഃ䛻䛒䜛༙ᚄ a䛾ᑟయ⌫Ẇ䛻㟁Ⲵ Q1䠈Q2 䜢୚䛘䛯䠊䛣䛣䛷䠈⌫Ẇ䛾ཌ䜏䛿↓ど䛷䛝䠈݀ ‫ܽ ب‬, ܾ, ܿ䛷䛒䜚䠈┿✵䛾ㄏ㟁⋡䜢ߝ଴ 䛸䛩䜛䠊
䠄1䠅㻌 ᭱䜒እ䛻䛒䜛༙ᚄ c 䛾ᑟయ⌫Ẇ䛾እ㠃䛻⌧䜜䜛㟁Ⲵ䜢䠈ᑟయ 1 䛸ᑟయ 2 䛭䜜䛮䜜䛻䛴䛔䛶♧䛫䠊
䠄2䠅㻌 ᭱䜒እ䛻䛒䜛༙ᚄ c 䛾ᑟయ⌫Ẇ䛾㟁఩䜢䠈ᑟయ 1 䛸ᑟయ 2 䛻䛴䛔䛶ồ䜑䜘䠊䛯䛰䛧↓㝈㐲䛷䛾㟁఩䜢 0 䛸䛩䜛䠊䠄3䠅㻌 ᑟయ 1 䛸ᑟయ 2 䛾㟼㟁䜶䝛䝹䜼䞊䜢䛭䜜䛮䜜ồ䜑䜘䠊
䠄4䠅㻌 ᑟయ 1 䛸ᑟయ 2 䛾᭱䜒እ䛻䛒䜛༙ᚄ c 䛾ᑟయ⌫Ẇ㛫䜢ᑟ⥺䛷᥋⥆䛧䛯䠊඘ศ᫬㛫䛜⤒䛳䛯ᚋ䛾䛣䛾⣔䛾඲㟼㟁䜶䝛䝹䜼䞊䜢ồ䜑䜘䠊䛯䛰䛧䠈ᑟ⥺䛻⵳䛘䜙䜜䜛㟁Ⲵ䛿↓ど䛷䛝䜛䜒䛾䛸䛩䜛䠊
ᑟయ 

ᑟయ 

ᅗ 

஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴  ᖺ  ᭶  ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(17 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for eachti࠙ၥࠚ ᅗ㸰࡟♧ࡍࡼ࠺࡟㸪‫ ؤ ݖ‬0 ࡢ㡿ᇦ࡟࠾࠸࡚㸪 ‫ ݔ‬㍈ࡢṇࡢྥࡁ࡟୍ᵝ࠿ࡘ᫬㛫ኚ໬ࡢ࡞࠸☢᮰ᐦᗘ B ࡀ࠶ࡿ㸬‫ ݖ‬൏ 0 ࡢ㡿ᇦ࡛ࡣ☢᮰ᐦᗘࡣ 0 ࡛࠶ࡿ㸬᢬ᢠࡀ ܴ ࡛ 1 ㎶ࡢ㛗ࡉࡀ ܽࡢṇ᪉ᙧࢥ࢖ࣝ ABCD ࡀⅬ P ሺ0, 0, ‫ݖ‬ሻ ࢆ୰ᚰ࡜ࡋ࡚㸪 ‫ݕ‬-‫ ݖ‬ᖹ㠃ෆ࡟㸪㎶ AB ࡀ ‫ ݕ‬㍈࡟ᖹ⾜࡟࡞ࡿࡼ࠺࡟┿✵୰࡟⨨࠿ࢀ࡚࠸ࡿ㸬ㄏᑟ㟁ὶ࡟ࡼࡗ࡚Ⓨ⏕ࡍࡿ☢⏺ࡣ↓ど࡛ࡁࡿ࡜ࡍࡿ㸬(1) ࢥ࢖ࣝ࡟እຊࢆຍ࠼㸪 ‫ ݖ‬㍈ࡢ㈇ࡢྥࡁ࡟୍ᐃࡢ㏿ࡉ ‫ݒ‬௖ ࡛㐠ືࡉࡏࡓ㸬ࡇࡢ᫬㸪ࢥ࢖ࣝ࡟⏕ࡌࡿㄏᑟ㉳㟁ຊ㸪࡞ࡽࡧ࡟ㄏᑟ㟁ὶࡢ኱ࡁࡉࢆồࡵࡼ㸬(2) (1)࡟࠾࠸࡚㸪ࢥ࢖ࣝࡢ୰ᚰ P ࡢ ‫ ݖ‬ᗙᶆࡀ െܽ/2 ൏ ‫ ݖ‬൏ ܽ/2 ࡢ㛫࡟࠶ࡿ᫬㸪ࢥ࢖ࣝࢆ㏿ࡉ ‫ݒ‬௖࡛ື࠿ࡍࡓࡵ࡟ᚲせ࡞እຊࡢ኱ࡁࡉࢆồࡵࡼ㸬ࡇࡇ࡛ࡣ㸪ṇ᪉ᙧࢥ࢖ࣝࡢ㉁㔞ࡣ↓ど࡛ࡁࡿ࡜ࡍࡿ.
(3) (1)࡟࠾࠸࡚㸪ࢥ࢖ࣝࡢ୰ᚰ P ࡢᗙᶆࡀ ሺ0, 0, ܽ/2ሻ ࠿ࡽ ሺ0, 0, െܽ/2ሻ ࡲ࡛ືࡃ᫬㸪እຊ࡟ࡼࡾࢥ࢖ࣝ࡟࡞ࡉࢀࡿ௙஦㸪࡞ࡽࡧ࡟ࢥ࢖ࣝ࡟Ⓨ⏕ࡍࡿࢪ࣮ࣗࣝ⇕ࢆồࡵࡼ㸬ࡲࡓࡑࢀࡽ
ࡢᑟฟ㐣⛬ࡶ♧ࡏ㸬ࡇࡇ࡛ࡣ㸪ṇ᪉ᙧࢥ࢖ࣝࡢ㉁㔞ࡣ↓ど࡛ࡁࡿ࡜ࡍࡿ.
(4) ࢥ࢖ࣝࢆ㔜ຊࡢࡶ࡜࡛ⴠୗࡉࡏࡿ㸬㔜ຊࡣ ‫ ݖ‬㍈ࡢ㈇ࡢྥࡁ࡟ാࡁ㸪㔜ຊຍ㏿ᗘࡣ ݃ ࡛࠶ࡿ㸬ࢥ࢖ࣝࡢ୰ᚰ P ࡢ ‫ ݖ‬ᗙᶆࡀ െܽ/2 ൏ ‫ ݖ‬൏ ܽ/2 ࡢ㛫࡟࠶ࡿ᫬㸪᫬้ ‫ࡢࣝ࢖ࢥࡿࡅ࠾࡟ ݐ‬㏿ࡉ ‫ݒ‬ሺ‫ݐ‬ሻ ࢆồࡵࡼ㸬ࡇࡇ࡛ࡣ㸪ࢥ࢖ࣝࡢ㉁㔞ࡣ↓ど࡛ࡁ࡞࠸࡜ࡋ㸪ࡑࡢ኱ࡁࡉࡣ ݉ ࡜ࡍࡿ㸬ࡲࡓ ‫ ݐ‬ൌ 0 ࡟࠾࠸࡚㸪ࢥ࢖ࣝࡢ㏿ࡉࡣ ‫ݒ‬ሺ‫ݐ‬ሻ ൌ 0 ࡜ࡋ㸪ࡑࡢ᫬ࢥ࢖ࣝࡢ୰ᚰ P ࡢ ‫ ݖ‬ᗙᶆࡣ െܽ/2 ൏ ‫ ݖ‬൏ ܽ/2 ࡢ㛫࡟࠶ࡿ࡜ࡍࡿ㸬ᅗ㸰஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴  ᖺ  ᭶  ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(18 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for eachtiAnswer the following questions ([Question 1] [Question 2]) and write the number of the selected
question on the answer sheet.
[Question 1] As shown in Fig. 1, Conductor 1 consists of three concentric spherical shells with radii
a, b, and c, and Conductor 2 consists of two concentric spherical shells with radii a and c, where they areplaced with a center-to-center distance of d in vacuum. The innermost shells of radius a in both
Conductor 1 and Conductor 2 are charged with Q1 and Q2, respectively. All spherical shells havenegligibly small thickness, ݀ ‫ܽ ب‬, ܾ, ܿ, and the electrical permittivity in vacuum is ߝ଴ .
(1) Give the electric charges which appear on the outside surfaces of the outermost spherical shells ofradius c for each of Conductor 1 and Conductor 2.
(2) Give the electric potentials of the outermost spherical shells of radius c for each of Conductor 1 andConductor 2. The electric field at infinity is taken as zero.
(3) Give the values of the electrostatic energy of Conductor 1 and Conductor 2, respectively.
(4) Two outermost spheres of radius c were connected to each other with a conductive wire. Find theelectrostatic energy of the system. Assume that the electric charge stored in the wire is negligible.
Conductor 1

Conductor 2

Fig. 1஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴  ᖺ  ᭶  ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(19 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for eachti[Question 2]  As shown in Fig. 2, a uniform static magnetic field B is in the positive ‫ ݔ‬direction in theregion ‫ ؤ ݖ‬0. The magnetic field is zero in the region ‫ ݖ‬൏ 0. A square coil ABCD of resistance ܴ andside length ܽ is placed in vacuum in the ‫ݕ‬-‫ ݖ‬plane, and the side AB is parallel to the ‫ ݕ‬axis. The centerof the coil, P, is at the point ሺ0, 0, ‫ݖ‬ሻ. Assume that the magnetic field generated by the induced current
flowing in the coil is negligible.
(1) Let an external force be applied so that the coil moves at a constant speed ‫ݒ‬௖ in the negative ‫ݖ‬
direction. Show the magnitude of the induced electromotive force and the magnitude of the induced
current in the coil.
(2) In the case of (1), show the magnitude of the external force needed to make the coil move at the
speed ‫ݒ‬௖ when the ‫ ݖ‬coordinate of the point P, the center of the coil, is in the range െܽ/2 ൏ ‫ ݖ‬൏ܽ/2. Here, the mass of the coil is negligible.
(3) In the case of (1), show the work done by the external force on the coil and the Joule heat generated
in the coil when the coordinate of the point P, the center of the coil, is moved from ሺ0, 0, ܽ/2ሻ to
ሺ0, 0, െܽ/2ሻ. Also, show how to derive them. Here, the mass of the coil is negligible.
(4) The coil is released and falls under the influence of gravity. The gravity is in the negative ‫ ݖ‬direction,and the acceleration of gravity is ݃. Find the speed ‫ݒ‬ሺ‫ݐ‬ሻ of the coil at a time ‫ ݐ‬when the ‫ ݖ‬coordinateof the point P, the center of the coil, is in the range െܽ/2 ൏ ‫ ݖ‬൏ ܽ/2. Here, the mass of the coil, ݉,
is not negligible. Assume that at ‫ ݐ‬ൌ 0, ‫ݒ‬ሺ‫ݐ‬ሻ ൌ 0 and the ‫ ݖ‬coordinate of the point P is in the range
െܽ/2 ൏ ‫ ݖ‬൏ ܽ/2.

Fig. 2஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(20 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for each question.

E. ʴൔ಍ରυώ΢η (Semiconductor device) ෾໼ʵḟࡢྛၥ࠸㸦࠙ၥ 1ࠚ࠙ၥ 2ࠚ㸧࡟⟅࠼ࡼ㸬ゎ⟅⏝⣬ࡢၥ㢟␒ྕḍ࡟ゎ⟅ࡋࡓၥ㢟␒ྕࢆグධࡍࡿࡇ࡜.
࠙ၥ ࠚ
n ᆺ༙ᑟయ A, B, C ࡟㛵ࡍࡿୗグࡢタၥ࡟⟅࠼ࡼ㸬ࡓࡔࡋ㸪A, B, C ࡢᐊ ࡟࠾ࡅࡿ⮬⏤㟁Ꮚ⃰ᗘࡣ㸪ࡑࢀࡒࢀ㸪1.0×1020, 1.0×1022, 1.0×1022 m-3 ࡛࠶ࡿ㸬A ࡢࣂࣥࢻࢠࣕࢵࣉࡣ B ࡜➼ࡋࡃ㸪C ࡢࣂࣥࢻࢠࣕ
ࢵࣉࡣ A ΓΕ΍୉͘͏㸬ࡲࡓ㸪A, B, C ࡢఏᑟᖏ࠾ࡼࡧ౯㟁Ꮚᖏࡢ᭷ຠ≧ែᐦᗘࡣ➼ࡋ࠸㸬A ࠾ࡼࡧ C ࡟ࡘ࠸࡚㸪⮬⏤㟁Ꮚ⃰ᗘࡢ ᗘ౫Ꮡᛶࢆ♧ࡍࢢࣛࣇࢆᶍᘧⓗ࡟ᥥࡅ㸬ࢢࣛࣇ࡟ࡣ㸪ࣙ༟ుࢢ⃰ᗘࡢᑐᩘ⾲♧ࢆ⦪(y)㍈㸪⤯ᑐ ᗘ T ࡢ㏫ᩘࢆᶓ(x)㍈࡟ྲྀࡾ㸪A ͕Γ; C ͹⮬⏤㟁Ꮚ⃰ᗘࢆͨΗͩΗᐇ⥺͕Γ;ഃતͲᥥࡅ㸬ࡲࡓ㸪ᗘ౫ᏑᛶࡢᵝᏊ࡟ᛂࡌ㸪ప

㸪୰

㸪㧗

ࡢ

㡿ᇦࢆᅗ♧ࡏࡼ㸬୰ 㡿ᇦ࡟࠾࠸࡚㸪A ࠾ࡼࡧ B ࡢ㟁Ꮚ⛣ືᗘࡢᗘ౫Ꮡᛶࡣ㸪ࡑࢀࡒࢀ㸪 ᗘ TA ͕Γࡧ TB ࡛ࣆ࣮ࢡࢆࡶࡘᒣᆺࡢ᭤⥺࡜࡞ࡿ㸬⛣ືᗘࡢᑐᩘ⾲♧ࢆ⦪(y)㍈㸪⤯ᑐ ᗘ T ࡢᑐᩘ⾲♧ࢆᶓ(x)㍈࡟ྲྀࡾ㸪A ࠾ࡼࡧ B ࡢ⛣ືᗘࡢ


୰

㡿ᇦࡢ㧗

ᗘኚ໬ࢆᶍᘧⓗ࡟ẚ㍑ࡋ࡚♧ࡏ㸬ഃ(T>TB)࡟࠾ࡅࡿ A ࡢ᢬ᢠ⋡ࡣ㸪B ࡜ẚ࡭࡚኱ࡁ࠸࠿㸪ᑠࡉ࠸࠿㸪ࡲࡓࡣྠ➼࠿㸽 ⌮⏤࡜࡜ࡶ࡟⟅࠼ࡼ㸬༙ᑟయ A ࢆ㸪㛗ࡉ 10 cm㸪᩿㠃✚ 1.0 cm2 ࡢᲬ࡟ษࡾฟࡋ࡚ᐊ ࡟ಖࡕ㸪Წࡢ୧➃͹ؔ࡟ుҒࠫ
20 V ࢆ༳ຍࡋࡓ㸬Წࡢ୧➃㛫ࡢ᢬ᢠɾ͕Γ;ᲬࢆὶࢀࡿṇᏍ㟁ὶࢆồࡵࡼ㸬ࡓࡔࡋ㸪ᐊ࡟࠾ࡅࡿ㟁Ꮚ࠾ࡼࡧṇᏍࡢ⛣ືᗘࢆ 0.14 ࠾ࡼࡧ 0.05 m2/Vs㸪┿ᛶ࢟ࣕࣜ࢔⃰ᗘࢆ 1.5×1016 m-3㸪⣲㟁Ⲵࢆ 1.6×10−19 C ࡜ࡍࡿ㸬஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(21 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for each question.

࠙ၥ ࠚ
㔠ᒓ㓟໬⭷༙ᑟయᵓ㐀ࢆᣢࡘ㟁⏺ຠᯝࢺࣛࣥࢪࢫࢱ㸦MOSFET㸧࡟ࡘ࠸࡚⪃࠼ࡿ㸬ࢯ࣮ࢫ㟁ᴟ㸪ࢻࣞ࢖ࣥ㟁ᴟ㸪࠾ࡼࡧ༙ᑟయᇶᯈࡣ᥋ᆅࡋ㸪ࢤ࣮ࢺ㟁ᴟ࡟ࢤ࣮ࢺ㟁ᅽ㸦㸧ࢆ༳ຍࡍࡿࡶࡢ

࡜ࡍࡿ㸬ᅗ (a)ࡣࢤ࣮ࢺ࡟

(< 0)ࢆ༳ຍࡋ࡚ࣇࣛࢵࢺࣂࣥࢻ࡟ࡋࡓ≧ែ㸪ᅗ (b)ࡣ

ᅗ (c)ࡣࢤ࣮ࢺ࡟㜈್㟁ᅽ㸦㸧ࢆ༳ຍࡋࡓ≧ែࢆ⾲ࡋ࡚࠸ࡿ㸬ᖏୗ➃࡜౯㟁Ꮚᖏୖ➃ࢆ㸪㸪㸪

= 0ࡢᖹ⾮≧ែ㸪ࡣࡑࢀࡒࢀ༙ᑟయࡢఏᑟ

ࡣࡑࢀࡒࢀ㔠ᒓ࡜༙ᑟయࡢࣇ࢙࣑ࣝ‽఩ࢆ㸪 ࡣ༙ᑟయࡀ┿ᛶ༙ᑟయࡔ࡜௬ᐃࡋࡓ࡜ࡁࡢྛⅬ࡛ࡢࣇ࢙࣑ࣝ‽఩ ┿ᛶࣇ࢙࣑ࣝ‽఩ ࢆ㸪 ࡣ⣲㟁Ⲵࢆ⾲ࡍ㸬
(a)

(b)

࢐Կຜ
P‫ܗ‬ൔ಍ର

ۜ଒
ΰʖφ
EFM

q\B

(c)

࢐Կຜ
P‫ܗ‬ൔ಍ର

EC
Ei
EFS
EV

ۜ଒
ΰʖφ
EFM

࢐Կຜ

EC

q\B

Ei
EFS
EV

P‫ܗ‬ൔ಍ର
q\B

ۜ଒
q\B
ΰʖφ
EFM

0
ਦ1ɿ(a) VG = VFBɾ(b) VG = 0ɾ(c) VG = VTͶ͕͜ΖMOSߑଆ͹ώϱχਦɿ

EC
Ei
EFS
EV

y

D

MOSFET ࡢ㜈್㟁ᅽ ࡢ⾲ᘧࢆồࡵࡿࡓࡵ㸪௨ୗࡢၥ࠸࡟⟅࠼ࡼ㸬㸦㸧 ࢤ࣮ࢺ㟁ᴟ࡟(>)ࢆ༳ຍࡍࡿ࡜㸪㓟໬⭷༙ᑟయ⏺㠃࡟࢟ࣕࣜ࢔ࡀ⵳✚ࡋ࡚ MOSFET ࡣ ON≧ែ࡜࡞ࡿ㸬

(>)ࢆ༳ຍࡋࡓ᫬࡟㓟໬⭷༙ᑟయ⏺㠃࡟⵳✚ࡉࢀࡿ࢟ࣕࣜ࢔ᐦᗘࢆ㸪 㸪ࢆ⏝࠸࡚⾲ࡏ㸬ణࡋ㸪 ࡣ㓟໬⭷ࡢ⭷ཌࢆ㸪࡛ࡣ㸪༙ᑟయࡢ⾲㠃㟁఩ࡣࡣ㓟໬⭷ࡢㄏ㟁⋡ࢆ⾲ࡍ㸬ࡲࡓ࡟Ṥ࡝౫Ꮡࡋ࡞࠸ࡶࡢ࡜㏆ఝࡋ㸪)ࡣ㓟໬⭷࡟༳ຍࡉࢀࡿࡶࡢ࡜ࡍࡿ㸬㸪ࡢ࡟ᑐࡍࡿ㉸㐣ศ(㸪>−஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(22 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for each question.

㸦㸧 ᅗ (c)࡟♧ࡍࡼ࠺࡟㸪=ࡢ᫬࡟ࡘ࠸࡚⪃࠼ࡿ㸬ࡇࡢ᫬㸪༙ᑟయࡢෆⶶ㟁఩ࡢ኱ࡁࡉ㸦ࣂࣥࢻࡢ᭤ࡀࡾࡢ኱ࡁࡉࢆ ࡛๭ࡗࡓࡶࡢ㸧ࡣ2఩⨨࡟࠾ࡅࡿ ࡜
࡚㸪

ࢆ

㸪 㸪

࡟➼ࡋࡃ࡞ࡿ㸬ࡇࡇ࡛ࡣ⏺㠃࠿ࡽ༑ศ㞳ࢀࡓࡢᕪ࡛࠶ࡿ㸬༙ᑟయഃࡢ⏺㠃㟁⏺ࡢ኱ࡁࡉࢆ 㸪༙ᑟయࡢㄏ㟁⋡ࢆ ࡜ࡋ㸪 㸪㸪 ࢆ⏝࠸࡚⾲ࡏ㸬ణࡋ㸪㓟໬⭷ෆ㒊ࡢ㟁ሙࡢ኱ࡁࡉࡣ

=࡛୚࠼ࡽࢀࡿࡇ࡜ࢆ⏝࠸࡚ࡼ࠸㸬㸦㸧 ༙ᑟయࡢ✵ஈᒙෆ࡛࣏࢔ࢯࣥ᪉⛬ᘧࢆゎࡃ࡜㸪௨ୗࡢ㟼㟁࣏ࢸࣥࢩࣕࣝ ( )ࡀᚓࡽࢀࡿ㸬( )=2( − ) ࡇࡇ࡛㸪 ࡣ㓟໬⭷༙ᑟయ⏺㠃࠿ࡽࡢ㊥㞳㸪 ࡣ✵ஈᒙࡢཌࡳ㸪 ࡣ༙ᑟయࡢ୙⣧≀⃰ᗘࢆ⾲ࡋ㸪 =࡛ࡢ✵ஈᒙ➃࡟࠾࠸࡚ ( ) = 0࡜ࡋ࡚࠸ࡿ㸬௦ධࡋ࡚ ࡢ⾲ᘧࢆồࡵࡼ㸬ࢆࢆ⏝࠸࡚⾲ࡋ㸪  ࡢࡢᘧ࡟஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(23 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for each question.

Answer the following questions ( 䛆Q1䛇,䛆Q2䛇). and write the number of the selected question on
the answer sheet.
䛆Q1䛇
Answer the following questions concerning n-type semiconductors A, B, and C. Here, the semiconductors
A, B, and C have free electron concentrations of 1.0×1020, 1.0×1022, 1.0×1022 m-3, respectively, at room
temperature. The band gap of A is the same as that of B, and that of C is larger than that of A. In addition, theeffective density of states of the conduction bands and valence bands for A, B, and C are the same value.

(1) Draw schematically a graph showing the temperature dependence of free electron concentrations in A and C.
In the graph, take the logarithmic free electron concentration as the y-axis and the inverse of the absolutetemperature (T) as the x-axis, and draw the free electron concentrations in A and C with solid and broken lines,
respectively. In addition, indicate the low, middle, and high temperature regions, based on the temperaturedependence.
(2) The temperature dependences of the electron mobility in A and B show convex curves with peaks attemperatures TA and TB, respectively. Draw schematically the temperature dependence of the electron mobility
in A and B, taking the logarithmic mobility as the y-axis and the logarithmic absolute temperature T as the xaxis.
(3) At a higher temperature (T>TB) in the middle temperature region, is the resistivity of A larger, smaller, or almost
equal compared with that of B? Answer with the reason.
(4) The semiconductor A is cut into a rod with a length of 10 cm and a cross section of 1.0 cm2 at room temperature,
and a potential difference of 20 V is applied between both edges. Evaluate the resistance between both edgesof the rod and the hole current flowing through the rod. Here, the electron and hole mobilities at room
temperature are 0.14 and 0.05 m2/Vs, respectively, the intrinsic carrier concentration is 1.5×1016 m-3, and theelementary charge is 1.6×10−19 C.

஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(24 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for each question.

䛆Q2䛇
Consider a field-effect transistor with a metal-oxide-semiconductor structure (MOSFET). The source
is applied toelectrode, the drain electrode, and the semiconductor substrate are grounded, and a gate voltage
the gate electrode.
(< 0) applied on the gate, Fig. 1(b) shows the equilibrium
Fig. 1(a) shows a flat-band state withstate with = 0, and Fig. 1(c) shows the state with a threshold voltage ( ) applied to the gate. andrepresent the lower conduction band edge and the upper valence band edge of the semiconductor,
represent the Fermi levels of the metal and the semiconductor, represents the Fermi level atandeach point if the semiconductor is an intrinsic semiconductor (intrinsic Fermi level), and represents theelementary charge.
(a)

(b)

Oxide
P-typesemiconductor

Metal
gate
EFM

P-typesemiconductor

EC

E
q\B i
EFS
EV

(c)

Oxide

Metal
gate
EFM

q\B

P-typeOxide semiconductor

EC
Ei
EFS
EV

q\B
Metal
gate q\B

EC

Ei
EFS
EV

EFM

0
D
Fig. 1ɿBand diagrams of the MOS structure at (a) VG = VFB, (b) VG = 0, and (c) VG = VT.

Answer the following questions to obtain an expression for the MOSFET threshold voltagey.

(1) When (> ) is applied to the gate electrode, charge carriers accumulate at the oxidesemiconductor interface and the MOSFET enters the ON state. Express the carrier density that(> ) is applied, by using
accumulates at the oxide-semiconductor interface when
, , , , and. Here, is the oxide thickness, andis the permittivity of the oxide. For >, it is assumed that the surface potential of the semiconductor is almost independent of byover , i.e., − , is applied to the oxide.
approximation and the excess of=as shown in Fig. 1(c). In this case, the magnitude of the built-in(2) Consider the situation at
potential in the semiconductor (the magnitude of the band bending divided by ) equals 2 , whereis the difference between andat a sufficient distance from the interface. Express using, , , ,
, and , whereis the magnitude of the interface electric field on thesemiconductor side and is the permittivity of the semiconductor. It can be used that the electricfield inside the oxide layer is=.

஑ᕞ኱Ꮫ኱Ꮫ㝔ࢩࢫࢸ࣒᝟ሗ⛉Ꮫᗓ㟁Ẽ㟁ᏊᕤᏛᑓᨷ
ಟኈㄢ⛬ ධᏛヨ㦂ၥ㢟㸦ᐇ᪋᪥௧࿴ 5 ᖺ 8 ᭶ 29 ᪥㸧ᑓ㻌㛛㻌⛉㻌┠㻌㻔㻿㼜㼑㼏㼕㼍㼘㼕㼦㼑㼐㻌㼟㼡㼎㼖㼑㼏㼠㼟㻕(25 / 25)5 ศ㔝࠿ࡽ2 ศ㔝ࢆ㑅ࡧゎ⟅ࡍࡿࡇ࡜㸬㑅ࢇࡔศ㔝ẖ࡟ゎ⟅⏝⣬1 㒊ࢆ⏝࠸ࡼ㸬ࡲࡓ㸪኱ၥ୍ࡘ࠶ࡓࡾ୍ᯛࡢゎ⟅⏝⣬ࢆ⏝࠸ࡼ㸬
Select 2 fields out of the 5 fields and answer the questions. Use a set of answer
sheets for each of the fields you selected and use an answer sheet for each question.

(3) Solving Poisson's equation in the depletion layer of a semiconductor, the electrostatic potential ( )is given as follows.
( )=

2

( − )

Here, is the distance from the oxide-semiconductor interface, is the depletion layer thickness,is the impurity concentration in the semiconductor, and ( ) = 0 is assumed at the depletion layerand substitute it into the formula in (2) to obtain an expressionedge = . Express usingfor .