Damping and Attenuation - Waves - Lecture Slides, Slides of Microwave Engineering and Acoustics

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DAMPING & ATTENUATION

How do we describe, or model, energy dissipation and damping?

What would a traveling waveform look like if k were complex?

ψ ( x , t ) = Ae

i − ω t + k c ( x )

k c

= k ± i κ ( k , κ real, positive)

Exponentially damped or growing traveling waveform - former

corresponds to energy dissipation or damping.

0

1

2

ψ

-2 0 2 x

ψ Left

( x , t ) = e

i − ω t + k 1 ( x )

k 1

− k 2

k 1

• k 2

e

i − ω t − k 1 ( x )

Right

( x , t ) =

2 k 1

k 1

• k 2

e

i − ω t + k 2 ( x )

R

ψ

k 1

− k 2

k 1

• k 2

T

ψ

2 k 1

k 1

• k 2

Reflection and transmission coefficients exactly as before, but now

that k values are complex, there is (i) a decrease in amplitude in

the direction of travel, and (ii) a phase shift introduced into

transmitted and reflected waves, both displacement and force.

R

F

k 2

− k 1

k 1

• k 2

T

F

2 k 2

k 1

• k 2

For light damping, the most important effect is the amplitude

decay over long distances. The change in magnitude of R and T

relative to no damping is quite small.

ψ( x , t ) = Ae

i (^) ( − ω t + kx )

2

ψ ( x , t )

∂ x

2

v

2

∂ψ ( x , t )

∂ t

v

2

2

ψ ( x , t )

∂ t

2

New, friction term proportional to material/field velocity in our

model.

Γrepresents magnitude of frictional term - what are dimensions?

v is still velocity of propagation.

will solve the equation but ONLY if k is complex! And complex k

means attenuation, which is what happens if friction is present!

Let’s see …….

ω

2

• i ωΓ = v

2

k

2

real and imaginary parts are separately zero

2

1 + i

= v

2

k

2

Light damping - Γ/ω is << 1

ω 1 + i

= v ⎡Re ( k ) + i Im ( k )

Re( k ) = ±

ω

v

Im ( k ) = ±

2 v

Choose signs so that wave propagates in desired direction

and loses amplitude in the direction of travel.

ω 1 + i

1 / 2

= v (^) ⎡Re (^) ( k ) + i Im (^) ( k ) ⎣

Re( k ) = +

ω

v

Im( k ) = +

2 v

ψ( x , t ) = Ae

i (^) ( − ω t + kx )

ψ ( x , t ) = Ae

i − ω t +

ω

v

x + i

Γ

2 v

x

⎛

⎝

⎜

⎞

⎠

⎟

ψ ( x , t ) = Ae

−

Γ

2 v

x

e

i − ω t +

ω

v

x

⎛

⎝

⎜

⎞

⎠

⎟

0

1

2

ψ

-2 0 2 x

Important aspect is the loss of energy (represented by the

amplitude) in the direction of travel

Left

( x , t ) = e

i − ω t + k 1 ( x )

Z

1

− Z

2

Z

1

+ Z

2

e

i − ω t − k 1 ( x )

Right

( x , t ) =

2 Z

1

Z

1

+ Z

2

e

i − ω t + k 2 ( x )

F

Right

( x , t ) =

2 Z

2

Z

1

+ Z

2

e

i − ω t + k 2 ( x )

F

Left

( x , t ) = e

i − ω t + k 1 ( x )

Z

2

− Z

1

Z

1

+ Z

2

e

i − ω t − k 1 ( x )

Your job is to work out how all this applies to the electrical

co-ax cable system.

In the generic wave equation, how are the generic parameters

v and G related to the properties of the cable?

How do we express the complex k and complex Z in terms of the

properties of the cable?

How much resistance would you cable have to have to account for

the attenuation you measured? What would the impedance have to

be to account for the velocity you measured (and the condition of

zero reflectance)? Do you have any other information to

corroborate your findings?

2

ψ ( x , t )

∂ x

2

v

2

∂ψ ( x , t )

∂ t

v

2

2

ψ ( x , t )

∂ t

2

Right

( x , t ) =

2 Z 1

Z 1

• Z 2

e

i − ω t + k 2 ( x )

2 Z

1

Z

1

+ Z

2

2 μ 1

1 + i

1

2 ω

μ 1

1 + i

1

2 ω

• μ 2

1 + i

2

2 ω

2 μ 1

1 / 2

• i

μ 1

1 / 2

1

2 ω

μ 1

1 / 2

• μ 2

1 / 2

• i

μ 1

1 / 2

1

2 ω

μ 2

1 / 2

2

2 ω

2 Z

1

e

i φ 1

Z

1

+ Z

2

e

i φ 12

A closer look at the effect of damping on the transmission coefficient:

• Damping removes energy from the wave and is evident over

large distances

• Amplitude decay in direction of travel is represented by a

complex k vector

• Reflection and transmission coefficients for ψ and d ψ/ dx are

worked out in a similar fashion to the case for no damping.

The magnitudes are not affected very much for light damping,

but there can be a phase shift that is different from the

undamped case

• Light damping is our focus
• Mathematical representations of the above

DAMPING & ATTENUATION - REVIEW