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Prof. Ji Chen

Adapted from notes by Prof. Stuart A. Long

Notes 5 Poynting Theorem

ECE 3317

Spring 2014

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Poynting Theorem

The Poynting theorem is one on the most important in EM theory. It tells us the power flowing in an electromagnetic field.

John Henry Poynting (1852-1914)

John Henry Poynting was an English physicist. He was a professor of physics at Mason Science College (now the University of Birmingham) from 1880 until his death.

He was the developer and eponym of the Poynting vector, which describes the direction and magnitude of electromagnetic energy flow and is used in the Poynting theorem, a statement about energy conservation for electric and magnetic fields. This work was first published in 1884. He performed a measurement of Newton's gravitational constant by innovative means during 1893. In 1903 he was the first to realize that the Sun's radiation can draw in small particles towards it. This was later coined the Poynting-Robertson effect.

In the year 1884 he analyzed the futures exchange prices of commodities using statistical mathematics.

(Wikipedia)

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Poynting Theorem

BE

tD

H Jt

From these we obtain:

BH E H

tD

E H J E Et

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Poynting Theorem (cont.)

BH E H

tD

E H J E Et

Subtract, and use the following vector identity:

H E E H E H

B DE H J E H E

t t

We then have:

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Poynting Theorem (cont.)

J E

B DE H J E H E

t t

Next, assume that Ohm's law applies for the electric current:

B DE H E E H E

t t

2 B DE H E H E

t t

or

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Poynting Theorem (cont.)

2 B DE H E H E

t t

From calculus (chain rule), we have that

1

2

1

2

D EE E E E

t t t

B HH H H H

t t t

2 1 1

2 2E H E H H E E

t t

Hence we have

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Poynting Theorem (cont.)

2 1 1

2 2E H E H H E E

t t

2 2 21 1

2 2E H E H E

t t

This may be written as

or

2 2 21 1

2 2E H E H E

t t

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Poynting Theorem (cont.)

Final differential (point) form of Poynting theorem:

2 2 21 1

2 2E H E H E

t t

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Poynting Theorem (cont.)

Volume (integral) form

2 2 21 1

2 2E H E H E

t t

Integrate both sides over a volume and then apply the divergence theorem:

2 2 21 1

2 2V V V V

E H dV E dV H dV E dVt t

2 2 21 1ˆ

2 2S V V V

E H n dS E dV H dV E dVt t

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Poynting Theorem (cont.)

Final volume form of Poynting theorem:

2 2 21 1ˆ

2 2S V V V

E H n dS E dV H dV E dVt t

For a stationary surface:

2 2 21 1ˆ

2 2S V V V

E H n dS E dV H dV E dVt t

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Poynting Theorem (cont.)

Physical interpretation:

2 2 21 1ˆ

2 2S V V V

E H n dS E dV H dV E dVt t

Power dissipation as heat (Joule's law)

Rate of change of stored magnetic energy

Rate of change of stored electric energy

Right-hand side = power flowing into the region V.

(Assume that S is stationary.)

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Poynting Theorem (cont.)

ˆS

E H n dS power flowing into the region

ˆS

E H n dS power flowing out of the region

Or, we can say that

Define the Poynting vector: S E H

Hence

ˆS

S n dS power flowing out of the region

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Poynting Theorem (cont.)

ˆS

S n dS power flowing out of the region

Analogy:

ˆS

J n dS current flowing out of the region

J = current density vector

S = power flow vector

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Poynting Theorem (cont.)

S E H

E

H

S

direction of power flow

The units of S are [W/m2].

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Power Flow

The power P flowing through the surface S (from left to right) is:

ˆS

P S n dS

surface S

n̂

S E H

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Time-Average Poynting Vector

, , , , , , , , ,S x y z t E x y z t H x y z t

*1Re E H

2S t E t H t

Assume sinusoidal (time-harmonic) fields)

, , , Re E , , j tE x y z t x y z e

, , , Re H , , j tH x y z t x y z e

From our previous discussion (notes 2) about time averages, we know that

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Complex Poynting Vector

Define the complex Poynting vector:

We then have that

*1S E H

2

, , , = Re S , ,S x y z t x y z

Note: The imaginary part of the complex Poynting vector corresponds to the VARS flowing in space.

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Complex Power Flow

The complex power P flowing through the surface S (from left to right) is:

surface S

n̂

ˆP SS

n dS

*1S E H

2

Re P

Im P

Watts

Vars

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Complex Poynting Vector (cont.)

What does VARS mean?

2 21 1VARS 2

4 4

2

V

m e

H E dV

W W

E

H

VARS consumed Power (watts)consumed

ˆReSS

n dS Watts flowing in

ˆImSS

n dS VARS flowing in

Equation for VARS (derivation omitted):

The VARS flowing into the region V is equal to the difference in the time-average magnetic and electric stored energies inside the region (times a factor of 2).

Watts + VARS

V

S

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Note on Circuit Theory

Although the Poynting vector can always be used to calculate power flow, at low frequency circuit theory can be used, and this is usually easier.

ˆS

P E H z dS

Example (DC circuit):

V0 R

S

PI

z

0P V I

The second form is much easier to calculate!

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Example: Parallel-Plate Transmission Line

At z = 0:

w

hx

y

,

z

I

+-V

x

y

E

H

w

h

0V V jkzz e

0I I jkzz e

0V 0 V

0I 0 I

The voltage and current have the form of waves that travel along the line in the z direction.

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0VˆE( , ,0)x y y

h

0IˆH , ,0x y x

w

At z = 0:

*1S E H

2

*

0 01ˆS

2

V Iz

h w

Example (cont.)

w

hx

y

,

z

I

+-V

x

y

E

H

w

h

(from ECE 2317)

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Example (cont.)

*

0 0V I1ˆS

2z

h w

0 0 0 0

ˆP S Sh w h w

f zz dx dy dx dy

*

0 0V I1P

2f whh w

*0 0

1P V I

2f

Hence

w

hx

y

,

z

I

+-V

x

y

E

H

w

h