Electronics Labs - Digital Electronics

Zhenyu Ye

14-Nov-16 1

The Art of Electronics by Horowitz and Hill – Chapter 8

Equivalent Logic Circuits

October 24, 2016 Digit Electronics, Zhenyu Ye 2

Circuit A

Circuit B

Half Adder

October 24, 2016 Digit Electronics, Zhenyu Ye 3

Boolean AlgebraBoolean algebra is the branch of algebra in which the values of the variables are the truth values:true and false, usually denoted as 1 and 0. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the basic operations of Boolean algebra are n conjunction and denoted as 𝐴 ∧ 𝐵, 𝐴 $ 𝐵n disjunction or denoted as 𝐴 ∨ 𝐵, 𝐴+ 𝐵n negation not denoted as ¬𝐀, �̅�

October 24, 2016 Digit Electronics, Zhenyu Ye 4

Boolean Algebra –Truth Table

October 24, 2016 Digit Electronics, Zhenyu Ye 5

x y 𝐱 $ 𝒚 𝒙 + 𝒚0 0 0 01 0 0 10 1 0 11 1 1 1

𝐱 𝒙-0 11 0

Boolean Algebra – Secondary Ops.

NAND n 𝑥 $ 𝑦 = �̅� + 𝑦1

NORn 𝑥 + 𝑦 = �̅� $ 𝑦1

October 24, 2016 Digit Electronics, Zhenyu Ye 6

x y 𝒙 $ 𝒚 𝒙- + 𝒚-0 0 1 11 0 1 10 1 1 11 1 0 0

x y 𝒙 + 𝒚 𝒙- $ 𝒚-0 0 1 11 0 0 00 1 0 01 1 0 0

Boolean Algebra – Secondary Ops.

n Exclusive OR (XOR)𝒙⊕ 𝒚 = (𝒙 + 𝒚) $ (𝒙 $ 𝒚)

October 24, 2016 Digit Electronics, Zhenyu Ye 7

x y 𝒙⊕ 𝒚 (𝒙 + 𝒚) (𝒙 $ 𝒚)0 0 0 0 11 0 1 1 10 1 1 1 11 1 0 1 0

Boolean Algebra – De Morgan’s Laws

n Associativity of OR x + 𝑦 + 𝑧 = 𝑥 + 𝑦 + 𝑧n Associativity of AND x $ 𝑦 $ 𝑧 = 𝑥 $ 𝑦 $ 𝑧n Commutativity of OR x + 𝑦 = 𝑦 + 𝑥n Commutativity of AND x $ 𝑦 = 𝑦 $ 𝑥n Distributivity of AND over OR

x $ 𝑦 + 𝑧 = 𝑥 $ 𝑦 + (𝑥 $ 𝑧)n Distributivity of OR over AND

x+ 𝑦 $ 𝑧 = 𝑥 + 𝑦 $ (𝑥 + 𝑧)

October 24, 2016 Digit Electronics, Zhenyu Ye 8

Equivalent Logic Circuits

October 24, 2016 Digit Electronics, Zhenyu Ye 9

Circuit A

Circuit B

Half Adder

October 24, 2016 Digit Electronics, Zhenyu Ye 10

Advanced Labs – Brownian Motion

Zhenyu Ye

14-Nov-16 11

Brownian Motionn Brownian Motion is the random motion of particles

suspended in a fluid (a liquid or a gas) resulting fromtheir collision with the fast-moving atoms ormolecules in the gas or liquid.

n https://upload.wikimedia.org/wikipedia/commons/6/6d/Translational_motion.gif

n https://upload.wikimedia.org/wikipedia/commons/5/51/Brownianmotion5particles150frame.gif

14-Nov-16 12

Brownian Motion

14-Nov-16 13

Reproduced from the book of Jean Baptiste Perrin, Les Atomes, three tracings of the motion of colloidal particles of radius 0.53 µm, as seen under the microscope, are displayed. Successive positions every 30 seconds are joined by straight line segments (the mesh size is 3.2 µm)

Brownian Motion

14-Nov-16 14

𝑚𝑑9𝑥𝑑𝑡9 = −𝛼

𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎

Brownian Motion

14-Nov-16 15

𝑚𝑑9𝑥𝑑𝑡9 = −𝛼

𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎

𝑚2𝑑9𝑥9

𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡

9= −

𝛼2𝑑𝑥9

𝑑𝑡 + 𝑥𝐹(𝑡)

Brownian Motion

14-Nov-16 16

𝑚𝑑9𝑥𝑑𝑡9 = −𝛼

𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎

𝑚2𝑑9𝑥9

𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡

9= −

𝛼2𝑑𝑥9

𝑑𝑡 + 𝑥𝐹(𝑡)

Define 𝛽 = DEF

DG

𝑚2𝑑𝛽𝑑𝑡 − 𝑚

𝑑𝑥𝑑𝑡

9= −

𝛼2 𝛽 + 𝑥𝐹(𝑡)

Brownian Motion

14-Nov-16 17

𝑚𝑑9𝑥𝑑𝑡9 = −𝛼

𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎

𝑚2𝑑9𝑥9

𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡

9= −

𝛼2𝑑𝑥9

𝑑𝑡 + 𝑥𝐹(𝑡)

Define 𝛽 = DEF

DG

𝑚2𝑑𝛽𝑑𝑡 − 𝑚

𝑑𝑥𝑑𝑡

9= −

𝛼2 𝛽 + 𝑥𝐹(𝑡)

𝑚2𝑑𝛽𝑑𝑡 − 𝑘I𝑇 = −

𝛼2 𝛽

Brownian Motion

14-Nov-16 18

𝑚𝑑9𝑥𝑑𝑡9 = −𝛼

𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎

𝑚2𝑑9𝑥9

𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡

9= −

𝛼2𝑑𝑥9

𝑑𝑡 + 𝑥𝐹(𝑡)

Define 𝛽 = DEF

DG

𝑚2𝑑𝛽𝑑𝑡 − 𝑚

𝑑𝑥𝑑𝑡

9= −

𝛼2 𝛽 + 𝑥𝐹(𝑡)

𝑚2𝑑𝛽𝑑𝑡 − 𝑘I𝑇 = −

𝛼2 𝛽 𝛽 =

2𝑘I𝑇𝛼 + 𝐴𝑒L

MGN⇒

Brownian Motion

14-Nov-16 19

𝑚𝑑9𝑥𝑑𝑡9 = −𝛼

𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎

𝑚2𝑑9𝑥9

𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡

9= −

𝛼2𝑑𝑥9

𝑑𝑡 + 𝑥𝐹(𝑡)

Define 𝛽 = DEF

DG

𝑚2𝑑𝛽𝑑𝑡 − 𝑚

𝑑𝑥𝑑𝑡

9= −

𝛼2 𝛽 + 𝑥𝐹(𝑡)

𝑚2𝑑𝛽𝑑𝑡 − 𝑘I𝑇 = −

𝛼2 𝛽 𝛽 =

2𝑘I𝑇𝛼 + 𝐴𝑒L

MGN

𝑥9 =2𝑘I𝑇𝛼 𝑡 =

2𝑘I𝑇6𝜋𝜂𝛼 𝑡

⇒

Brownian Motion

14-Nov-16 20

𝑚𝑑9𝑥𝑑𝑡9 = −𝛼

𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎

𝑚2𝑑9𝑥9

𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡

9= −

𝛼2𝑑𝑥9

𝑑𝑡 + 𝑥𝐹(𝑡)

Define 𝛽 = DEF

DG

𝑚2𝑑𝛽𝑑𝑡 − 𝑚

𝑑𝑥𝑑𝑡

9= −

𝛼2 𝛽 + 𝑥𝐹(𝑡)

𝑚2𝑑𝛽𝑑𝑡 − 𝑘I𝑇 = −

𝛼2 𝛽 𝛽 =

2𝑘I𝑇𝛼 + 𝐴𝑒L

MGN

𝑥9 =2𝑘I𝑇𝛼 𝑡 =

2𝑘I𝑇6𝜋𝜂𝛼 𝑡 𝑟9 =

4𝑘I𝑇𝛼 𝑡 =

4𝑘I𝑇6𝜋𝜂𝛼 𝑡

⇒

⇒

Brownian Motion

14-Nov-16 21

Brownian Motion

14-Nov-16 22

Brownian Motion

14-Nov-16 23

Brownian Motion

14-Nov-16 24

𝑟9 =4𝑘I𝑇𝛼 𝑡 =

4𝑘I𝑇6𝜋𝜂𝛼 𝑡

𝜂 = 8.90×10LX Pa $ 𝑠𝑎 = 1.1 𝜇𝑚

Brownian Motion

14-Nov-16 25

𝑟9 =4𝑘I𝑇𝛼 𝑡 =

4𝑘I𝑇6𝜋𝜂𝛼 𝑡 ⇒ 𝑘I =

𝑀𝑆𝐷𝑡

6𝜋𝜂𝑎4𝑇 ~1.1×10L9a𝐽/𝐾

𝜂 = 8.90×10LX Pa $ 𝑠𝑎 = 1.1 𝜇𝑚

Advanced Labs - Zeeman Effects

Zhenyu Ye

14-Nov-16 26

Experiments in Modern Physics – A. Melissinos Chapter 6

Modeling of Hydrogen Atoms

14-Nov-16 27

n Schrodinger equation in 1926

i! ∂∂tΨ!r, t( ) = −!2

2m∇2 +V !r, t( )

⎡

⎣⎢

⎤

⎦⎥⋅Ψ

!r, t( )

Ψ!r( ) = 1

r⋅ χ l r( ) ⋅Ylm θ,φ( )

En = −e2

!c⎛

⎝⎜

⎞

⎠⎟

2mec

2

2n2

m = 0,±1,!,±l

L = l(l +1)! Lz =m!

l = 0,1,!,n−1n =1,2,!

See Adv.Lab.2

Electron Spin

14-Nov-16 28

S = s(s+1)! s = 12

1925: G.Uhlenbeck, S.Goudsmit

Sz =ms! ms = ±12

𝐽=𝐿+𝑆

𝑚g=𝑚h +𝑚i

Electron Spin

14-Nov-16 29

S = s(s+1)! s = 12

Ag Shell Structure: 2, 8, 18, 18, 1

1925: G.Uhlenbeck, S.Goudsmit

Sz =ms! ms = ±12

Stern-Gerlach Experiment 1922

𝐽=𝐿+𝑆

𝑚g=𝑚h +𝑚i

Electron Spin

14-Nov-16 30

S = s(s+1)! s = 12

Ag Shell Structure: 2, 8, 18, 18, 1

1925: G.Uhlenbeck, S.Goudsmit

Sz =ms! ms = ±12

Stern-Gerlach Experiment 1922

Bohr magneton 𝜇I =jℏ9N

𝑔m = 1

𝑔n = 2

𝜇h = 𝑔m𝑚m𝜇I

𝜇i = 𝑔n𝑚n𝜇I

𝐸p,Nr,Ns = −𝑒9

ℏ𝑐

9𝑚j𝑐9

2𝑛9 + 𝜇h𝐵 + 𝜇i𝐵

L, S and J

14-Nov-16 31

2S+1LJ

541.6nm

S=1, L=0, J=1

S=1, L=1, J=2

𝐽=𝐿+𝑆

𝑚g=𝑚h +𝑚i

L, S and J

14-Nov-16 32

𝑔g =𝑔i $ 𝑆 + 𝑔h $ 𝐿

𝑆 + 𝐿

2S+1LJ𝐸 = 𝐸Ivw + 𝜇g𝐵

∆𝐸 = ∆(𝑔g𝑚g)𝜇I𝐵

541.6nm

𝐽=𝐿+𝑆

𝑚g=𝑚h +𝑚i

𝜇g = 𝑔g𝑚g𝜇I

L, S and J

14-Nov-16 33

ΔJ=±1, Δmj=0, ±1

2S+1LJ

541.6nm

𝐽=𝐿+𝑆

𝑚g=𝑚h +𝑚i

𝑔g =𝑔i $ 𝑆 + 𝑔h $ 𝐿

𝑆 + 𝐿

𝐸 = 𝐸Ivw + 𝜇g𝐵

∆𝐸 = ∆(𝑔g𝑚g)𝜇I𝐵

𝜇g = 𝑔g𝑚g𝜇I

Zeeman Effect Lab

14-Nov-16 34

Polarizer

14-Nov-16 35

Interference Filter

14-Nov-16 36

=𝜆4

Fabry-Perot Etalon

14-Nov-16 37

𝑘𝜆 = 2𝑑𝑐𝑜𝑠𝜃 = 2𝑑 1 − 𝑠𝑖𝑛9𝜃 ≈ 2𝑑 1−𝜃9

2 ≈ 2𝑑 1−𝐷~9

8𝑓9

Wave-length Shift Calculation

14-Nov-16 38

∆𝜆 =𝜆9

2𝑑𝐷~a9 − 𝐷~99

𝐷~L�9 − 𝐷~99=𝜆9

2𝑑𝐷~99 − 𝐷~�9

𝐷~L�9 − 𝐷~99

14-Nov-16 39