How Do EM Waves Behave in Transmission Lines with Varying Frequencies?

[unscientific]

Homework Statement

(a) Find the capacitance per unit length and find the inductance per unit length of a transmission wire above a infinite conducting plane. I used the image method

(b) Show that EM waves can propagate, and find their
speed and the characteristic impedance of the line.

(c) Find the characteristic impedance for no reflected power?

(d) The frequency of incoming wave multiplied by 4, find fraction reflected power and the phase difference between the incident and reflected waves.

Homework Equations

The Attempt at a Solution

Part (a)

Part (b) and (c)
I got an imaginary value for the impedance, not sure if it's right. It implies there is a ∏/2 phase difference between voltage and current?

Part (d)

For part (d), I'm not sure why there would be a phase difference? If it is a short circuit, then there would be a phase change of ∏ (Vr/Vi = -1) if it is an open circuit then it would be completely reflected. What has ramping up the input frequency got to do with the phase difference between incoming and reflected wave?

 

[TSny]

V should be the potential difference between the two actual conductors, not between the wire and its image. Likewise, the flux $\Phi$ should be for a region between the plane and the wire.

L’ is the ratio of the absolute value of the flux and the current. So, it should be positive. The characteristic impedance should be a real, positive number for a lossless transmission line.
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Think about the condition to have no reflection at the point of connection of the infinite transmission line with the $\lambda /4$ transmission line so that all the power can be delivered to the load. This link might help
http://en.wikipedia.org/wiki/Quarter-wave_impedance_transformer

 

[unscientific]

V should be the potential difference between the two actual conductors, not between the wire and its image. Likewise, the flux $\Phi$ should be for a region between the plane and the wire.

L’ is the ratio of the absolute value of the flux and the current. So, it should be positive. The characteristic impedance should be a real, positive number for a lossless transmission line.
----------------------
Think about the condition to have no reflection at the point of connection of the infinite transmission line with the $\lambda /4$ transmission line so that all the power can be delivered to the load. This link might help
http://en.wikipedia.org/wiki/Quarter-wave_impedance_transformer

By the way, the answer to part (c) is Z_{in} = \frac {Z^2}{Z_L}, right?for l = λ/4, the additional matching network need to be added is:

Z_{in} = \frac {V_{(-l)}}{I_{(-l)}} = \frac {V_i e^{jkl} + V_r e^{-jkl} } {V_i e^{jkl} - V_r e^{-jkl} - V_r e^{-jkl} }Z = \frac { (Z_L + Z)e^{jkl} + (Z_L - Z)e^{-jkl} } {(Z_L + Z) e^{jkl} - (Z_L - Z)e^{-jkl} } Z

For l = λ/4 , Z_{in} = \frac {Z^2}{Z_L}

For part (d):
As the input frequency is increased by 4 the input wavelength divides by 4, now it would imply that the matching line length becomes \frac {λ}{4} → λ

This results in a mismatch, causing a reflected wave. I have found that the new input impedance Z'_{in} = 0

This is equivalent to a short circuit, and I have found the new \frac {V_r}{V_i} = -1 so the phase difference is ∏.. and all power is reflected!

 

[TSny]

By the way, the answer to part (c) is Z_{in} = \frac {Z^2}{Z_L}, right?

Yes. You can simplify this using the relationship given between Z and Z_L

Z_{in} = \frac {V_{(-l)}}{I_{(-l)}} = \frac {V_i e^{jkl} + V_r e^{-jkl} } {V_i e^{jkl} - V_r e^{-jkl} - V_r e{-jkl} }Z = \frac { (Z_L + Z)e^{jkl} + (Z_L - Z)e^{-jkl} } {(Z_L + Z) e^{jkl} - (Z_L - Z)e^{-jkl} } Z

OK, but some typing errors appear in the denominator of the middle expression.

For part (d):
As the input frequency is increased by 4 the input wavelength divides by 4, now it would imply that the characteristic impedance becomes \frac {λ}{4} → λ

The length of the matching line becomes λ.

This results in a mismatch, causing a reflected wave. I have found that the new input impedance Z'_{in} = 0

I get a different answer. Can you show your calculation?

 

[unscientific]

Yes. You can simplify this using the relationship given between Z and Z_L
OK, but some typing errors appear in the denominator of the middle expression.
The length of the matching line becomes λ.
I get a different answer. Can you show your calculation?

When l = λ, e^{jkl} = e^{j \frac{2∏}{λ} λ} = 0 so Z_{in} = \frac {0}{0} Z

 

[TSny]

What is the value of e^j2$\pi$ ?

Note e^jθ is never zero. It's a point on the unit circle in the complex plane.

 

[unscientific]

latest calculation below.

 

[TSny]

It should be = 1.

Yes.

That gives 2Z_L.

I don't get that result for Z_in.

 

[unscientific]

Yes.
I don't get that result for Z_in.

Sorry, I got Z_{in} = Z_L

So on the left side, we have impedance of \frac {1}{4} Z_L and on the right side we have impedance of Z_L.So \frac {V_r}{V_i} = + \frac {3}{5} (No phase difference)

So proportion of power reflected = (\frac {V_r}{V_i})^2 = \frac {9}{25}

As for phase difference, I don't think there's a phase difference? it should be zero. (Anyway all cases include 0 or ∏)

 

[TSny]

I think that's right.