circuit faraday ppt

Author: Carlo Andrea Gonano

Co-author: Prof. Riccardo Enrico Zich

Politecnico di Milano, Italy

AP-S 2015,

July 19-25, Vancouver, BC, Canada

Insights on the possibility of

circuit representation for single

Maxwell’s Equations

Contents 1. Introduction

2. Free space Maxwell’s eq.s

3. Applications & Fundamental variables

9. Conclusions

10. Question time

4. Charge conservation

5. From E and B to EM potentials

6. Circuit Faraday’s Law

7. Circuit Lorentz’s Gauge

8. Maxwell-Ampére set

2 C.A. Gonano, R.E. Zich

Introduction CIRCUITS VS FIELDS

• Circuit models are often used to describe an EM system with lumped elements


• No need to solve the complete set of Maxwell’s Equations

However, circuit models have some limits:

• The electric field E should be conservative in order to define a potential V

0loop

jiVVE

That’s implicitly stated by the Kirchhoff Voltage Law (KVL)

1V

3V

2V

NV

...

CR

LEV

• Discrete-space with nodes and edges

Free-space Maxwell’s eq.s CIRCUITS VS FIELDS


t

E

cJB

E

e

e

T

2

0

0

0

1

t

BE

BT

0

• Circuit Theory is usually regarded as an approximation of full Electro- Magnetic theory. For example, KCL is associated to Charge Conservation.

• In this work we examine the possibility to rigorously rephrase free-space Maxwell’s Eq.s in circuit form

Maxwell’s Eq.s in free space

For that purpose, we are going to adopt EM potentials V and A instead of magnetic field B

In other words, for each fundamental relation we want to find its equivalent circuit element

Applications WHY SHOULD WE REPHRASE MAXWELL’S

EQUATIONS IN CIRCUIT FORM?


• Need to council circuit models with EM field theory

• Circuit models are often used as a design tool by engineers, e.g. for the project of antennas, metamaterials etc.

• Integral, lumped variables and discrete space

• Suitable for numerical implementation in EM solvers, e.g. for Finite-Volume Method (FVM)

Developing a useful, conceptual design tool for engineers

Fundamental variables


HOW TO PASS FROM FIELDS TO CIRCUITS?

• In Circuit Theory and Maxwell’s Eq.s different variables are used

• Circuit variables are usually defined on nodes or on edges

• Maxwell’s variables are fields, defined on a continuous space:

e

E

B

eJ

electric and magnetic fields

charge and current densities

Node variables

jQnet charges

jVpotentials

scalar fields

Edge variables

jkicurrents

jkVvoltages

vector fields

We have to transform fields in their circuit equivalents

Charge conservation (I)


0

e

Te Jt

dQj

ejE , dSJni

kj

E

T

jkE

,

t

E

cJB

E

e

e

T

2

0

0

0

1

• Integrating the eq. on a domain j, we get:

• The eq. for charge conservation can be deduced from Maxwell’s:

LET’S REARRANGE IT IN CIRCUIT FORM!

0

dJt

j

e

Te 0

dSJndt

jj

e

T

e

• Now we can define some lumped variables:

net charge on node j current flowing from node j to node k

Charge conservation (II)


0

e

Te Jt

01

,

,

N

k

jkE

jEi

dt

dQ

jejE VolQ ,

jkEjkjkE JSi ,,

This eq. can be regarded as an Extended KCL

• More compactly:

• Substituting in the integral eq., we obtain the circuit equivalent:

• Let’s note that the net charge QJ can be considered as stored on a plate or half-capacitor, facing the empty space

• No need for constitutive relations (till now)

Magnetic field B is for loops (I) How can we model E and B fields with circuits?


xEVE

• The electric field E can be easily associated to a voltage VE on an edge jk

xExdEV jk

x

x

T

jkE

j

k

,

The magnetic field B is associated to nodes or to edges?

Neither of them!

• In fact, the magnetic field B and its flux F(B) are related to whole, closed loops.

• Remember that B is not a vector but a pseudovector or a tensor

• For example, in a 2D circuit ring the B field is not associated to any node or edge

Magnetic field B is for loops (II) So, how can we reproduce the magnetic effects?


0 BT

AB

xdAdSBn TT

D

2

• The magnetic field B can be replaced by the vector potential A

• In order to identify an oriented loop we need at least 3 nodes j, k, l

• The flux F(B) is equal to the circuitation G(A) for every loop

• Instead of B field, A is a vector field, it can be interpreted as a current

density iA and it is defined on edges rather than on loops

From E and B to EM potentials

Lorentz’s Gauge


t

BE

BT

0

EVt

A

AB

02

0

A

c

V

t

T

t

E

cJB

E

e

eT

2

0

0

0

1

e

e

JAt

A

c

Vt

V

c

0

2

2

2

2

0

0

2

2

2

2

0

1

1

• It’s now convenient to re-write Maxwell’s Eq.s in terms of EM potentials

Faraday’s set

Wave equations

• Wave equations for V and A are derived choosing Lorentz’s Gauge:

We’ll focus the attention on Faraday’s and Lorentz’s eq.s

Circuit Faraday’s Law (I)


EVt

A

j

k

x

x

T

jk xdVV

j

k

x

x

T

jkAA xdAiL

,

jkEjk

jkA

jkA VVdt

idL ,

,

,

• The vector Faraday’s law is:

j

k

x

x

T

jkE xdEV

,

• It can be integrated along an edge jk of length x

j

k

j

k

x

x

Tx

x

T xdEVxdAt

• We can define some lumped circuit variables:

• The circuit Faraday’s Law will so look:

EM inertia

potential difference

voltage (e.m.f. )

Let’s analyze it more in detail

Circuit Faraday’s Law (II)

No need for loops!

C.A. Gonano, R.E. Zich

kjjk VVV

jk

T

jkE xEV

,

jk

T

jkAA xAiL

,

xLA

0

1

AiA

0

1

EVt

A

jkEjk

jkA

jkA VVdt

idL ,

,

,

• With a short-hand notation:

• Equivalent inductance:

• “Aether” current:

EM inertia

potential difference

voltage (e.m.f. )

• The e.m.f VE is modeled through a dependent source, since it could be generated by external currents

13

Circuit Lorentz’s Gauge (I)


02

0

A

c

V

t

T

01

,

,

N

k

jkA

jAi

dt

dQ

• The Lorentz’s Gauge has the same form of a Continuity Eq.:

0

e

Te Jt

• The circuit equivalents will have the same form, too:

01

,

,

N

k

jkE

jEi

dt

dQ

How can we obtain that kind of extended KCL?

• Integrating the Lorentz’s Gauge, we get:

02

0

dSAncdVt

jj

T

jjAjA VCQ ,, • The charge QA could be linked to the

potential V through a capacitance CA:

01

,,

N

k

jkA

j

jA idt

dVC

Circuit Lorentz’s Gauge (II)


01

,,

N

k

jkA

j

jA idt

dVC

j

j

jA xS

VolC 00,

dVVol

Vjj

j

1

dSAnS

i

kj

T

jkA

0

,

1

average potential on node j

current flowing from node j to node k

• We can define some lumped variabiles:

• After some calculations, we finally obtain the circuit Lorentz’s Gauge:

capacity for node j

02

0

dSAncdVt

jj

T

02

0

A

c

V

t

T

• That’s equivalent to the KCL for a capacitor CA connected to the ground (V=0)

Faraday-Lorentz circuit grid


• The circuits for Faraday’s set and Lorentz’s Gauge can linked together:

• That circuit grid is just for EM potentials V and A

• The e.m.f VE sources are coupled to electric charges QE and currents iE through Gauss Law and Maxwell-Ampére equations.

Synthetic results


Lorentz’s Gauge

EVt

A

AB

02

0

A

c

V

t

T

Faraday’s set

01

,,

N

k

jkA

j

jA idt

dVC

GF

jkEjk

jkA

jkA

JKLJKL

VVdt

idL

AB

,

,

,

)()(

Circuit equivalents for some Maxwell’s equations

Maxwell-Ampére set (I)


What about Gauss and Maxwell-Ampére equations?

Which is the circuit equivalent for Maxwell-Ampére eq.?

• Wave equations for V and A are derived choosing the Lorentz’s Gauge

t

E

cJB

E

e

eT

2

0

0

0

1

e

e

JAt

A

c

Vt

V

c

0

2

2

2

2

0

0

2

2

2

2

0

1

1

Wave equations

• Charge conservation follows from them, and so the Extended KCL

• Wave propagation Trasmission Line model

However, Transverse EM waves require a special treatment, since they invoke the concept of proximity among edges

Maxwell-Ampére set (II)


In fact, Maxwell-Ampére Law is not a pure topological law: it requires a metric

However, we have already faced the problem*:

Finding it’s circuit equivalent is too complex to be treated in this context…

(*) C. A. Gonano, and R. E. Zich, “Circuit model for Transverse EM waves”, Antennas and

Propagation Society International Symposium (APSURSI), 2014 IEEE, pp. 613–614, 2014

Conclusions

• The magnetic field B is associated to loops, not to nodes or edges


MAIN APPLICATIONS

• Circuit description also for non-conservative fields and EM waves in empty space

• Target: finding the circuit equivalent for each free-space Maxwell’s Eq.

• Use of EM potentials V and A

• Circuit equivalents for: Charge cons., Faraday Law, Lorentz’s Gauge

• Design tool for the project of antennas, metamaterials etc.

• Implementation in Finite Volume Methods

That’s all, in brief…

THANKS FOR THE ATTENTION.

QUESTIONS?


Extra details • Net charge and capacitors

• “Is there an aether?”

• The magnetic field is not a vector

• Constitutive relations


• Lorentz’s force

• Gabriel Kron’s model

23

Net charge and capacitors

• Normally, none of circuital devices can accumulate charge, as stated by Kirchhoff Current Law at every node

HOW TO DESCRIBE A LOCAL NET CHARGE?

• Though, a real capacitor is made of two plates, each with net charge

• …and current can’t flow across them! 01

N

n

nE i

dt

Qd

01

N

n

ni

Classic KCL


i 1

i 2

...

i N

Qe

So, a plate allows to store a net charge on a circuit node

CHARGE CONSERVATION

• The classic capacitor is made of two plates connected by a insulator

121,2,2

1VVCQQ EE

constitutive equation

J.C. Maxwell

24

“Is there an aether?”


P.A.M. Dirac, “Is there an aether?”, Nature, vol. 168, pp. 906–907, 1951

Why should we use the EM potentials instead of E and H?

• J.C. Maxwell originally (1865) wrote his equations in terms of “Electromagnetic momentum”, that is vector potential A

“It is natural to regard it [A] as the velocity of some real physical thing. Thus with the new theory of electrodynamics we are rather forced to have an aether”

• Easy analogy with mechanics and fluid-dynamics

• Need in Quantum Mechanics

P.A.M. Dirac

• Aharonov-Bohm experiment (1956): EM potentials are not a mere math. construction

R.P. Feynman

• ..and many physicist like them!

25

The magnetic field B is not a vector


• The magnetic field B is not a “true” vector, but a pseudovector

• In fact, it does not respect usual vector reflection rules

- C. A. Gonano, and R. E. Zich, “Cross product in N Dimensions - the doublewedge product”,

Arxiv, August 2014

• In a wider, ND view, B is a matrix or tensor

- C. A. Gonano, Estensione in N-D di prodotto vettore e rotore e loro applicazioni,

Master’s thesis, Politecnico di Milano (2011).

ijjiij AAB //

For further details, see also:

26

Constitutive relations


WHAT ABOUT MAXWELL’S EQUATIONS IN MEDIA?

• You can notice that the Faraday’s set is left unchanged, so the circuit for the EM potential V and A will be the same

t

DJH

D

free

free

T

t

BE

BT

0

Maxwell’s Eq.s in media

• For the other set of equations, you need some constituive relations

Ohm’s Law: EJ Efree

ES

xR

Efree VR

i 1

Usually the Constituive Relations are related to the quantum forces inside the material, as in the Drude-Lorentz model

27

Lorentz’s force


• However, if we talk about force we should also introduce the concept of a moving mass inside the circuit.

WHAT ABOUT THE LORENTZ’S FORCE?

Ok, actually we need it in order to complete the EM field theory

EE JBEf Lorentz’s force

• Usually the effects of the Lorentz’s Force are reproduced by mean of some constitutive relations for the material.

yQp E

EQ

EQ

tieE

0

Example:

Drude-Lorentz model

28

Gabriel Kron’s model (I)


G. Kron, "Equivalent circuit of the field equations of

Maxwell-I.” Proceedings of the IRE 32.5 (1944): 289-299.

The idea of equivalent circuits for Maxwell’s eq.s is not completely new …

• Gabriel Kron, electrical engineer, developed a complex model with 3 networks:

Gabriel Kron (1901 – 1968)

“Now the above field equations of Maxwell define just such a multiple stationary network, a combination of electric, magnetic, and dielectric networks”

In our model we used EM potentials instead of B field, avoiding the need for orthogonal networks and magnetic monopoles

• In Kron’s model the magnetic fields B and H are intepreted as currents in an orthogonal network

29

Gabriel Kron’s model (II)


Kron’s model is quite elegant, though it looks a bit complex…

circuit faraday ppt

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