EMII Ch.10 Notes

3

8

8

ω=2×10 rad/sec

f=31.8 MHz

T=3.14 10 sec

8u=c=3×10 m/sec

8

8

ω 2×10β= = =0.67 rad/m

u 3×102

0.67 9.426m

4

8 T=3.14 10 sec

5

9.426m

0.67 rad/m 0.67 rad 1 m

6.28 rad ?? m

6

In free space, H = 0.1 cos (2 X 10^8 t - kx) ay A/m. Calculate(a) k, λ, and T(b) The time t1 it takes the wave to travel λ/8(c) Sketch the wave at time t1.

x

8

8

8

8

8

8

8

8

1

1

propgation direction is +a

(a)

ω=2×10 rad/sec

u=c=3×10 m/sec (free space)

ω 2×10k=β= = =0.67 rad/m

u 3×10

1 2πω=2πf=2×10 , T= = =31.42ns

f 2×10

λu ,λ (3×10 )(31.42ns) 9.426m

(b)

λu

(λ / 8) (9.42 / 83×10

wave

T

T

tt

8

)3.9

3×10ns

7

8 9

y

y

1

(c)

H=0.1cos(2 10 3.9 10 0.67 )

=0.1cos(0.78 0.67 )

for t=0 H=0.1cos(0 ) 0.1cos(0 0.67 )

for t=t H=0.1cos(0.78 ) 0.1cos(0.78 0.67 )

figure at t1 is the same as at t but shifted by 0.78=pi/4

x

x

x x

x x

a

a

9.426m

( , ) sin( )z

o xE t z E zte a

z

2 2

f rad

f u m

t

T

u fT

: attenuation costant (Np/m)or (dB/m)

2

1 12

Wave attenuates in amplitude by a factor e-az

l

1

Anattenuation of 1 neper denotes a reduction to e of the original value whereas an increase

of 1 neper indicates an increase by a facto

neper

r

/

e

s

of

m

How far?This is often formulated ts the question:-

how far does the wave propagate before its magnitude is reduced by e-1?this is called the skin depthe-1 is approximately 0.3

In copper, the wave is down to 36% of its initial strength in the first 6.62mm.i.e. light really doesn't get far into copper!

Perfect dielectric:

σ=0 α=0 δs=∞

Perfect Conductor:

σ=∞ α=∞ δs=0

18

Reflection of a plane wave

(1) Normal Incidence

The proportion of the incident wave that is reflected or transmitteddepends on the constitutive parameters (ε,µ, σ) of the two media involved.

1 i r

1 i r

2 t

2 t

E =E +E

H =H +H

E =E

H =H

19

2 1 r

2 1 i

E= =

E

Reflection coefficient

t2

1 2 i

E2= =

E

Transmission coefficient

(1) =1

(2) 1 1 , 0 | | 1 , 0 2

(3) If 1 total reflection

(4) ( & ) are dimensionless and may be complex.

20

2 1 =0 =1

(No reflection )=(total trasmission)

1

21

1

medium(1):

lossless( =0) 2

medium(2):

perfect conductor( = )

2 45o

2 1 r

2 1 i

E= =

E

t2

1 2 i

E2= =

E

total reflection

Comment: fields in a perfect conductor must vanish.

1 0

1 and =2

2

Compared to 2

22

1

medium(2):

lossless( =0)2

medium(1):

perfect conductor( = )

1 45o

2 1 r

2 1 i

E= =

E

t2

1 2 i

E2= =

E

total reflection with phase shift

2 0

1 and =0

3

23

1 and =0

0 and =1

1 and =2 total reflection

total reflection with phase shift

(No reflection )=(total trasmission)

24

A standing wave

It consists of two traveling waves (Ei and Er) of equal amplitudes but in opposite directions.

1 | | 1= , | |

1 | | 1

SS

S

(1) 0 | | 1 1

(2) SWR:standing wave ratio

(3) S is dimensionless (always expressed in dB)

(4) If S=1 | | 0 & 1 reflected wave(totally transmitted)

(5) If S= | | 1 transmission (totally ref

S

No

No

lected)

25

1 and =0

0 and =1

1 and =2 total reflection

total reflection with phase shift

(No reflection )=(total trasmission)

=S

=1S

=S

26

2 1 r

2 1 i

E= =

E

t2

1 2 i

E2= =

E

1 | |=

1 | |S

271 1 , 1r 1 , 4r r

28

z

x

y

29

z

x

y

2 188.5 4

o

o

1 377

30

1 2

2 1

2 1

2

1 2

(a) 188.5 , 377 4

377 188.5= = 1/3

377 188.5

2 2(377)= = 1.33

188.5 377

o

o

31

8tt

i

8

t

8

1 8

2

8

t

2

2

1/3 , 1.33

EE (1.33) 5cos(10 )

E

E =1.668cos(10 )

1 3 10 rad/s3 10

4 0.667

E =1.668cos(10 0.667 )

z

z

o o

o o

z

t y

t y

t y

a

a

a

32

8

t

toto

2

r

0.667

(note it fro

E =1.668cos(10 )

| E | 1.668|H |

m figur

0.0265377

H 0.0265cos

e) ,

( )( )

, k z E x H

z

E

y

k y

t y

t z

a a a a a a a

a

a

a

33

34

1

2

4 (a) =754

= 95.44 37.570.1

(1 ) 4 (1 )4

o

o

oo

o

o

j j

2 1

2 1

95.44 37.57 754= = 0.8186171.08

95.44 37.57 754

1 | | 1 0.81861 0.23 33.56 , 10.0253

1 | | 1 0.8186

o

o

S

35

8

1

2

22

2

8

2

1

2

2 2 22

2

2

2 2 22

2

5 4 7.5 10

2 2(37.57) 75.14

tan(2 ) tan(75.14) 3.77

(3.77)(4 )(7.5 10 ) 0.1 (as given)

0

1 1 6.021 Np/m 2

1 1 7.822

o o

o

o

65 rad/m

36

rr

i

8

r

oror

1

8

r

5 171.08

5 17

E (b) | E | (0.8186)(50) 41

E

E 41sin(7.5 10 )

E 41H 0.05428

754

H 0.05428sin(7.5 10 )1 ).08 (

y

z

t x

t x

a

a

37

tt

6.02

6

i

8

t

otot

2

8

t

.02

E (c) | E | (0.23)(50) 11.5

E

E 11.5 sin(7.5 10 7.8265 )

E 11.5H 0.1205

95.4

H 0

33.56

3.1205 sin(7.5 10 7.8 3.56 3 .2 5 56 )7 7

x

x

y

z

e

e

t x

t x

a

a