Laboratory 6: Wave propagation in lossy media SOLVED

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Laboratory 6: Wave propagation in lossy media
In this laboratory, we will explore wave propagation in lossy media, including liquid water and silicon.
For a medium with complex permittivity 𝜖௘ = 𝜖′ − 𝑗𝜖′′ and permeability 𝜇 = 𝜇଴, the propagation constant 𝛾 and
the intrinsic impedance 𝜂 are given by:
𝛾 = 𝛼 + 𝑗𝛽 = 𝑗𝜔ඥ𝜖௘𝜇 𝜂 = ඨ
𝜇
𝜖ୣ
As you will soon learn, the loss tangent tan(𝛿஽) is a useful parameter for describing the degree of loss per unit
wavelength of propagation distance, and is given by,
𝑡𝑎𝑛(𝛿஽) =
𝜖′′
𝜖′
Recall further that the wavelength in the medium is given by 𝜆 = 2𝜋/𝛽 and the attenuation per unit length in
[dB/m] is given by 𝐴/ℓ = 8.69 𝛼 .
1. Wave propagation in pure water
The normalized complex permittivity 𝜖(𝑓)/𝜖଴ of pure water is very well modelled by the function,
𝜖(𝑓)
𝜖଴
= 𝜖(∞) +
𝜖(0) − 𝜖(∞)
1 + 𝑗2𝜋𝑓𝜏
over the frequency range 0 Hz < f < 300 GHz where the parameters 𝜖(∞) = 5.2 , 𝜖(0) = 78.3 and 𝜏 = 8.3 𝑝𝑠 at T
= 25 C. The origin of this behaviour is the orientational response of water molecules to electric fields. The
permeability 𝜇 = 𝜇଴.
Write a function that calculates the normalized complex permittivity 𝜖(𝑓)/𝜖଴ of pure water for a frequency f:
[ eps_norm_H20 ]= purewatereps(f)
Calculate the complex permittivity over a frequency range 1 kHz < f < 300 GHz with frequency distributed on a
logarithmic scale, as setup for example by f = 10.^[3:0.05:log10(300e9)] . Plot the normalized real
component, 𝜖′/𝜖଴, and the normalized negative imaginary component, 𝜖′′/𝜖଴, of the normalized permittivity using
a logarithmic frequency scale, using semilogx(f,real(eps_norm_H20),f,-imag(eps_norm_H20))
for example.
Calculate and plot (on logarithmic frequency scale) the loss tangent tan(𝛿஽).
Is the “dielectric constant” really a constant?
At what frequency is 𝜖′′/𝜖଴ a maximum?
№ 1: Show your results to the teaching assistant.
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Calculate the propagation constant 𝛾 = 𝛼 + 𝑗𝛽 versus frequency f.
Calculate the attenuation per unit length 𝐴/ℓ in [dB/m] versus frequency f and plot on a log-log scale. This can be
done using the command line loglog(f,norm_lambda) for example.
Calculate the wavelength ratio λ/λ0 versus frequency f, where λ is the wavelength in pure water and λ0 is the
wavelength in vacuum. Plot λ/λ0 versus the logarithmic frequency scale.
Review your plots carefully.
How should one choose frequency f if one wants to transmit an electromagnetic wave with minimum attenuation
through pure water?
At what frequency is the attenuation per unit length 𝐴/ℓ = 3 dB/m ?
Is the wavelength 𝜆 in pure water larger or smaller than the wavelength 𝜆0 in vacuum at f = 1 GHz ? In other words,
does pure water shorten or lengthen the wavelength of an EM wave in the microwave frequency range?
№ 2: Show your results to the teaching assistant.
2. Wave propagation in sea water
Outside of a strictly controlled laboratory environment, water contains ions from dissolved salts of varying
concentration. The normalized complex permittivity 𝜖(𝑓)/𝜖଴ of sea water is very well modelled by the function,
𝜖(𝑓)
𝜖଴
= 𝜖(∞) +
𝜖(0) − 𝜖(∞)
1 + 𝑗2𝜋𝑓𝜏 +
𝜎/𝜖଴
𝑗2𝜋𝑓
over the frequency range 0 Hz < f < 300 GHz where the parameters 𝜖(∞) = 5.2 , 𝜖(0) = 78.3 and 𝜏 = 8.3 𝑝𝑠 at
the temperature T = 25 C, as before, and the conductivity 𝜎 = 4 S/m for typical ion concentrations in salt water.
Note that sea water differs from pure water by the addition of ionic conductivity, seen in the last term. The
permeability 𝜇 = 𝜇଴.
Write a function that calculates the normalized complex permittivity 𝜖(𝑓)/𝜖଴ of sea water for a frequency f:
[ eps_norm_salt_H20 ]= seawatereps(f)
Calculate the complex permittivity over a frequency range 1 kHz < f < 300 GHz with frequency distributed on a
logarithmic scale, as in the previous exercise.
Plot the real component 𝜖′/𝜖଴ and the negative imaginary component 𝜖′′/𝜖଴ of the normalized permittivity versus
frequency f. You may want to plot the permittivity components on a log-log scale due to the divergence in 𝜖′′/𝜖଴
at low frequencies.
Calculate and plot (on logarithmic frequency scale) the loss tangent tan(𝛿஽).
Why is 𝜖′′/𝜖଴ ∝ 1/𝑓 at “low frequencies” f ?
What is the approximate range of frequencies 𝜖′′/𝜖଴ ∝ 1/𝑓 ?
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№ 3: Show your results to the teaching assistant.
Calculate the propagation constant 𝛾 = 𝛼 + 𝑗𝛽 versus frequency f.
Calculate the attenuation per unit length 𝐴/ℓ in [dB/m] versus frequency f and plot on a log-log scale.
Calculate the wavelength ratio λ/λ0 versus frequency f, where λ is the wavelength in pure water and λ0 is the
wavelength in vacuum. Plot λ/λ0 versus frequency. A log-log scale will be useful here.
Review your plots carefully.
How should one choose frequency f if one wants to transmit an electromagnetic wave with minimum attenuation
through sea water?
At what frequency is the attenuation per unit length 𝐴/ℓ = 3 dB/m ?
Which medium, pure water or sea water, has a greater attenuation per unit length 𝐴/ℓ ? Does your answer
depend on frequency?
What is the wavelength 𝜆 in sea water at f = 1 kHz ? How does this compare to the wavelength 𝜆0 in vacuum at f
= 1 kHz ?
№ 4: Show your results to the teaching assistant.
Optional: Fresh water suitable for aquatic life has a typical conductivity 𝜎 = 0.02 S/m. How does the attenuation
versus frequency of fresh water compare to sea water?
Consider the implications of your results for the interaction of electromagnetic waves with saline water, in the
context of communications, radar, microwave heating and biomedical imaging.