Steady Electric Currents

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Intelligent Wave Systems Laboratory

SEOUL NATIONAL UNIVERSITY

Dept. of Electrical and Computer Engineering

Sunkyu Yu

Dept. of Electrical and Computer Engineering Seoul National University

Steady Electric Currents

Introduction to Electromagnetism with Practice

Theory & Applications

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Steady Electric Currents: Idea & Assumption

• Electrostatics: Electronic phenomena with electric charges at rest

• What happens when the charges in motion constitute current flows?

• Remember that Electromagnetics that we studied is “classical”

➔ All the phenomena are reasonable & conceivable!

(Conservation of charges, there is no sudden emergence of charges, …) (In quantum mechanics, there are many counterintuitive results…)

• The charges in motion actually result in “time-varying” fields

➔ But, at this stage, we neglect such temporal variations

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Intelligent Wave Systems Laboratory

SEOUL NATIONAL UNIVERSITY

Dept. of Electrical and Computer Engineering

Current Density & Ohm’s Law

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Conduction Currents – Drift Motion of Electrons/Holes

No Field: Random behaviors at the Fermi velocity (you’ll learn in solid-state physics)

Applying Field: Net flow due to the induced force

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Dept. of Electrical and Computer Engineering

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Current Density

Q Nq n s t

 = u e   

The amount of charge passing through the surface

charge “carrier”

volume density

velocity surface normal vector

surface area

time

Current = The time-rate of change of charge

n

I Q Nq s Nq

t

 =  =   =  

 u e u s

(Volume) Current Density

Nq 

= =

J u u Why volume (despite Δs)?

➔ The current flow exists for 3D volumes

I =  S J  d s

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Mobility & Conductivity

General (Volume) Current Density (Multiple carriers: electrons, holes, ions, …)

i i i i i

i i

N q 

=  = 

J u u

 e

= −

u E

For ideal drift, the “average” drift velocity is directly proportional to the electric field (you’ll learn in solid-state physics)

For electrons,

Mobility > 0

( Ne 0)

   

= − = = − 

J E E

e e h h

 = −   +  

In semiconductors:

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Dept. of Electrical and Computer Engineering

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Ohm’s Law



=

J E

(Microscopic or Point Form) Ohm’s Law

I 1 V

= R

(Macroscopic) Ohm’s Law

For ideal drift, the “average” drift velocity is directly proportional to the electric field

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Example: Homogeneous Conductor



=

J E

S

I d JS

d ES

 

=  =



J s E s

12 1 2

V = − V V = El

2

1

2 1

P

V − = − V  P E  d l

V 12 S

 

= = 1 = l

=

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Dept. of Electrical and Computer Engineering

9

Example 019

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Series & Parallel

R l

 S

=

Series ~ Increase of Length!

Parallel ~ Increase of Area!

1 2

R = R + R

1 1 1

= +

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Dept. of Electrical and Computer Engineering

Kirchhoff’s Voltage Law

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Necessity of External Sources for Steady Electric Currents

C  = d 0

 E l

C d 0

  =

 J l

For an isolated circuit (without energy exchange with its environment),

a steady current cannot be maintained in the same direction

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Dept. of Electrical and Computer Engineering

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Electromotive Force (emf)

Electromotive Force

(emf) ➔ Voltage Source

2 1

in out

1 2 0

C  = d  + d  = d

 E l  E l  E l

E in

E out

2 1

in out

1 d 2 d

−  E  = + l  E  l

2 1

in out

1 d 2 d

 = −  E  = + l  E  l

Electric Battery, Electric Generator, Solar Cells, …

Allowing potential drop

from (+) to (–) electrode

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Kirchhoff’s Voltage Law



=

J E

2 1 1

in out

1 d 2 d 2 d RI

 = −  E  = + l  E  = + l   J  = l

j k k

j k

 = R I

 

C  = d 0

 E l

It is just …

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Dept. of Electrical and Computer Engineering

Kirchhoff’s Current Law

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Equation of Continuity

Charges do not suddenly appear or disappear (at least in classical physics)

out S V

dQ d

I d dv

dt dt 

=  J  s = − = − 

S  d =   V dv

 J s  J

d 

  = − J

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Dept. of Electrical and Computer Engineering

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Kirchhoff’s Current Law

d 0 dt

  = − J  =

For steady currents

J: Solenoidal or Divergence-Free Field

S  d = 0

 J s

j 0

j

I =



Kirchhoff’s Current Law

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Non-Steady Current Example: Charges inside Media



  = D

In a simple medium:

0 r



  = E  

d dt

 

  =   = − J E

 d 

   = − 0

d   

+   = e

⁰ ^r

t



  = −  

d dt

  = − J  What is the

result of these

equations?

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Dept. of Electrical and Computer Engineering

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Non-Steady Current Example: Charges inside Media

0

r

t

e



  = −  

Good Conductor (Large σ) Perfect Conductor (σ ~ ∞)

 ~ 0

Instantaneously

Insulator (σ ~ 0)

 0

Exponential Decay from

 0

“Almost” Preserving

0 r

  

= 

Relaxation Time

1/e decay time

In copper, τ ~ 1.5 × 10

^–10

(ns): negligible Local variation of

charge at a bulk

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Sunkyu Yu

Dept. of Electrical and Computer Engineering Seoul National University

Steady Electric Currents

Introduction to Electromagnetism with Practice

Theory & Applications

(21)

Dept. of Electrical and Computer Engineering

2

Remind: Current Density

Q Nq n s t

 = u e   

The amount of charge passing through the surface

charge “carrier”

volume density

velocity surface normal vector

surface area

time

Current = The time-rate of change of charge

n

I Q Nq s Nq

t

 =  =   =  

 u e u s

(Volume) Current Density

Nq 

= =

J u u Why volume (despite Δs)?

➔ The current flow exists for 3D volumes

I =  S J  d s

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Remind: Mobility & Conductivity

General (Volume) Current Density (Multiple carriers: electrons, holes, ions, …)

i i i i i

i i

N q 

=  = 

J u u

 e

= −

u E

For ideal drift, the “average” drift velocity is directly proportional to the electric field (you’ll learn in solid-state physics)

For electrons,

Mobility > 0

( Ne 0)

   

= − = = − 

J E E

e e h h

 = −   +  

In semiconductors:

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Dept. of Electrical and Computer Engineering

4

Remind: Ohm’s Law



=

J E

(Microscopic or Point Form) Ohm’s Law

I 1 V

= R

(Macroscopic) Ohm’s Law

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Remind: Kirchhoff’s Voltage Law



=

J E

2 1 1

in out

1 d 2 d 2 d RI

 = −  E  = + l  E  = + l   J  = l

j k k

j k

 = R I

 

C  = d 0

 E l

It is just …

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Dept. of Electrical and Computer Engineering

6

Remind: Kirchhoff’s Current Law

d 0 dt

  = − J  =

For steady currents

J: Solenoidal or Divergence-Free Field

S  d = 0

 J s

j 0

j

I =



Kirchhoff’s Current Law

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Joule’s Law

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Dept. of Electrical and Computer Engineering

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Remind: Drift Motion of Electrons/Holes

Macroscopic Drift Microscopic Collison

Electric Field Energy to Atomic Thermal Vibration

q

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Power from Charge Carrier Drifts

0 0

lim lim

t t

w q

p q

t t

 →  →

  

= = = 

 

E l

E u

Power provided by a charge carrier

i i i i

dP  N q  dv

=  

  

E u

Total power provided by all the charge carriers in a volume dv

Is it valid to measure the power delivered by carriers

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Intelligent Wave Systems Laboratory

Dept. of Electrical and Computer Engineering

10

Remind: Charges inside Conductors

0

r

t

e



  = −  

Good Conductor (Large σ) Perfect Conductor (σ ~ ∞)

 ~ 0

Instantaneously

Insulator (σ ~ 0)

 0

Exponential Decay from

 0

“Almost” Preserving

0 r

  

= 

Relaxation Time

1/e decay time

In copper, τ ~ 1.5 × 10

^–10

(ns): negligible Local variation of

charge at a bulk

Now, let’s consider another relaxation time

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Microscopic Derivation of Ohm’s Law: Collision

t = 0

Collision

u

0 ₀

q t

+ m E u

Because the collision is random, u

₀

does not

contribute to the average electronic velocity

^avg

q q

m t m

= E =  E u

Assume the average collision time τ

2

i i i i

i i i i i i

i i i i i

q N q

N q N q t

m m

= 

^avg

=  E =  

J u E

2 2

N q  N q 

 

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Dept. of Electrical and Computer Engineering

12

Validity of Estimating Power

p = q E u 

Power provided by a charge carrier

I. Power is exerted on the system

II. But charge carriers maintain their average velocity III. Then, where does the exerted power go?

IV. It is delivered to atomic vibrations ➔ Thermal Energy!

t = 0

Collision

u

0 ₀

q t

+ m E u

Assume the average collision time τ

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Joule’s Law

i i i i

dP =    N q   dv =  dv

  

E u E J

dP

dv =  E J

P =  V E J  dv

Power Density under steady-current conditions

Joule’s Law

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Dept. of Electrical and Computer Engineering

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Joule’s Law for Circuit Theory

P =  V E J  dv

Joule’s Law

L S

S L S L S L S L

P =   E J  dlds =   EJdlds =   J Edlds =  Jds  Edl = IV

E & J parallel Continuity Equipotential Surface

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Joule Heat is critical in Electronic Microprocessors…

Impact of Temperature on Intel CPU Performance Written on October 28, 2014 by Matt Bach

https://www.hardwaretimes.com/amds-ryzen-5000g-apus-are-up-to-2x-more-power-efficient -than-intels-11th-gen-rocket-lake-s-cpus-in-multi-threaded-workloads/

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Intelligent Wave Systems Laboratory

Dept. of Electrical and Computer Engineering

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Joule Heat is critical in Electronic Microprocessors…

Historical Evolution of “Electronic” Microprocessor

Joule heat, Noise, …

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Boundary Conditions

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Dept. of Electrical and Computer Engineering

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C  = d 0

 E l

C d 0

  =

 J l

 =  =  J

E O

Irrotational Electric Field

Governing Equations for Steady Current Density

d dt

  = − J 

S V

d d dv

dt

 = − 

 J s 

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Remind: Strategy for Boundary Conditions

Δh

C

 = d 0

 E l 

S

D  d S = Q

I. Boundary includes “different” materials ➔ Integral forms are proper II. Stokes ➔ “Closed Loop” across materials

Gauss ➔ “Closed Surface” across materials

III. Loop measures tangential fields & Surface measures normal fields

Medium 2

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Dept. of Electrical and Computer Engineering

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Remind: Analyzing Boundary Conditions

Δh

0 1 2

0

h t t

C

 d

_{ =}

= E  − w E  = w

 E l 

S

D  d S = e

ⁿ²

 ( D

¹ⁿ

− D

²ⁿ

)  =  S 

^s

S

E

_2t

E

_1t

D

_2n

D

_1n

Δh → 0 to characterize the “boundary”

Medium 2

Medium 1

e

_n2

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Remind: Boundary Conditions: Electrostatics

Δh 1 t 2 t

E = E e n 2  ( D 1 n − D 2 n ) =  s

E

_2t

E

_1t

D

_2n

D

_1n

Normal Fields

Medium 2

e

_n2

Tangential Fields

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Dept. of Electrical and Computer Engineering

22

Boundary Conditions for Current Density

1 2

t t

J J

 =  n 2 ( 1 n 2 n )

J J d

dt

 − = −  e

Normal Fields Tangential Fields

1 n 2 n

J = J

If there is no external source…

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Example 020

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Dept. of Electrical and Computer Engineering

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Example 020

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Resistance

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Dept. of Electrical and Computer Engineering

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Estimating Resistance

L L

S S

d d

R V

I d  d

−  − 

= = =

 

 

E l E l

J s E s

S S

L L

d d

C Q

V d d



 

= = =

−  − 

 

D s E s

E l E l

R 1

C



=  RC 

=  Related to operation speed

High σ is desired in many cases

: Graphene, Superconductor, …

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Example 021

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Dept. of Electrical and Computer Engineering

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