# patch and dipole antenna

#### legyptien

Hi

Ok I have a contradiction in my mind:

Patch antenna is called a leaky wave resonant cavity because of the magnetic slots which radiates. We all agree on the fact that they are standing waves with clearly identified max and min as a characteristic of a specific mode inside this cavity.

1) My question is if we have standing waves, in my mind we would have a max reflection (=1) on the magnetic walls so how comes we have leakage if we have full reflection on the wall ??

2) If we have NOT full reflection on the wall so it means there is no standing wave in the cavity right ? the max and mins are moving ?

Thanks

#### studiot

There certainly seems to be a contradiction, but I'm not sure if it's not in your terminology.
In fact I'm not sure what you are asking.

At any cavity wall you can only have nodes of standing waves.

Anti nodes are only possible in (across) an opening.

#### legyptien

If we have a transmission line missmatched at one of its end, we will have a partially reflected wave. How is the distribution of current along the transmission line ? My main question is the maximum and minimum of current will move along this line or not ?

I have an opinion but I dont wanna influence you so dont read the following unless you know the answer.

To me since there is a partially reflected wave, the max and min should move. We do have a standing wave but we have to add to it a progressive wave which is bringing the movement to the distribution.

#### studiot

Sorry, this is no clearer.

Cavities and transmission lines are different structures. Cavities are resonant, transmission lines are not.

A wave travelling in an infinite transmission line has no reflections and no standing waves.

If a finite line is terminated with its characteristic impedance there will be no reflections.

If a finite line is unterminated or terminated with, as you put it, a mismatched impedance, then there will be standing waves. You do not get a say in the position of the nodes and antinodes in a standing wave, that is determined purely by the length of the line. In fact that method is used to determine the position of a break along a line. Look up Time Domain Reflectometry and Lecher Lines.

All this pre supposes a fixed frequency or wavelength.

The maxima, minima and zeros of a travelling wave do not have a fixed position. They pass through every point in the line of travel with a fixed phase relationship and a positional spacing set by the wavelength.

#### legyptien

Cavities and transmission lines are different structures. Cavities are resonant, transmission lines are not.
Thanks for your answer. If you agree we forget about cavity for now. I have to understand the non resonant thing first.

"If a finite line is unterminated or terminated with, as you put it, a mismatched impedance, then there will be standing waves. You do not get a say in the position of the nodes and antinodes in a standing wave, that is determined purely by the length of the line. In fact that method is used to determine the position of a break along a line. Look up Time Domain Reflectometry and Lecher Lines."

I agree that there will be a standing wave but I would like to know the distribution of the current along the transmission line in this case. The result isn t a standing wave to me, it is a standing wave plus a propaagating wave. we agree ?
My english is not my first langage, I have trouble to understand : you do not get a say however I don t think that the length will determine the position of nodes or nulls but it will determine the number of nulls and nodes to me but may be I m wrong...

I will Look up Time Domain Reflectometry and Lecher Lines.

thanks

#### studiot

OK, this is advanced electricity and magnetism.

The differential equation of (wave) motion has the soltuion below.

v_subs is the isntantaneous voltage, the expressions with the A coefficient are the incoming wave and the expressions with The b coefficient are the reflected wave. Phi is the phase angle between the incoming A wave and the reflected B wave.

Can we talk about these equations?

Sorry I can't get the figure to display in full, only as a thumbnail.

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#### legyptien

OK, this is advanced electricity and magnetism.

The differential equation of (wave) motion has the soltuion below.

v_subs is the isntantaneous voltage, the expressions with the A coefficient are the incoming wave and the expressions with The b coefficient are the reflected wave. Phi is the phase angle between the incoming A wave and the reflected B wave.

Can we talk about these equations?

Sorry I can't get the figure to display in full, only as a thumbnail.
Thanks for your answer. My math is good enough I think .
I forgot to say that I would like to stay in the lossless case since its not gonna change the concept in the non resonnant case (=transmission line). In the cavity this assumption will change the quality factor but we said we forget about that...

This is what I wrote:

Vi*sin (wt - kx ) + Vr*sin (wt + kx ) = Vr*(sin (wt - kx) + sin (wt + kx)) + C*sin (wt + kx) with Vi = Vr + C

Vr*(sin (wt - kx) + sin (wt + kx)) represent a standing wave if we use the Simpson formula...

ok in my mind I beleive they are 2 steps: 1) the math , 2) the physical interpretation.

Just to troubleshoot where we wont agree can you tell me if you agree with step 1 then step 2 please ?

Thanks

#### studiot

Thus far your math is correct so keep going and perhaps we will arrive at the point of your question.

#### legyptien

Thus far your math is correct so keep going and perhaps we will arrive at the point of your question.
Ok actually I finished with the math. This is how I see things:

- Whatever the reflection at the load, we will always have part of the amplitude of the incident wave which will "combine" with the reflected wave to form a standing wave with nodes and nulls fixed in the transmission line of course. However it doesn't mean that the drawing (representation or animation) of the total voltage along the line is a standing wave. Actually it is a superposition of standing wave and a propagating wave based on the equations that I wrote and that we agreed on. We can note that the amplitude of the reflected and the standing wave depends on the incident wave amplitude and the reflection coefficient.

We can ask our-self few questions:

1) Since the result is not a standing wave but a combination of standing wave and propagating wave, what is the speed of the nodes/nulls in this case ?

2) At which speed is transmitted the power (yes I think that the speed of the nodes/nulls are different than the speed of the energy) ?

3) What is the quantity of power which is transmitted ?

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By trying to answer to question 2 and 3, we notice that :

a) In the case of propagating wave, the speed of nodes/nulls is the same as the speed of transmission of energy.

b) The difference between the speed of transmission of power and the speed of nodes/nulls can be seen in the standing wave case (full reflection: reflected coefficient = 1). In the case of standing wave, the speed of nulls/nodes is zero (we all agree on that) but actually the power flows from the generator to the load AND the same quantity of power flows from the load to the generator. To me its not the same thing at all as if we would have considered a "standing power". I dont believe in such a concept at least in this case.

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By trying to answer to question 1 (which drove me crazy for 5 months now), we notice that :

a) If we have a case which is very close to standing wave, where the C value of my derivation is very low compared to Vi (or Vr), the nodes and nulls will definitely move/oscillate slightly around its original value (the one obtained by its standing part (Simpon formula of my previous derivation)). The speed of this move/oscillation depends on the amplitude of the transmitted wave (in other words it depends on C in my equations). In the perfect case (full reflection) no transmitted wave exist and the nodes/nulls are not moving anymore.

b) The speed of the nodes/nulls increase little by little as reflection as load becomes weaker until the moment for which this is no reflection, then the nulls/nodes move at the speed of light (if the dielectric in the transmission line is the air). We are now in the propagating case only. It would be interesting to quantify mathematically the evolution of the speed of the nodes/nulls from 0 (standing wave case = full reflection) to the speed of light (propagating wave = no reflection). I didnt quantify that yet but I plan to do that when I have time.
Before I thought that there is a discontinuity of speed between the standing wave case and the case of "small" transmitted wave. Anyway there is no discontinuity of any speed (neither the nodes/nulls speed nor the speed of propagation of power).

I think Im done with the explanation of what I think. Before writing that it wasnt so clear but it became clearer and clearer by writing it... Let me know what you think about it.

I forgot to say that I put myself in the simplest case: lossless line, resistive load (real coefficient of reflection) and non radiative transmission line since Quasi-TEM propagation with fringing fields are the reality for micro strip line but let's stay in the simple case just to understand...

PS: I attached the propagating wave case with its model of cascaded parallel self and capacity and I tried to display the voltage and current at t=0 for example to see what happens for the energy in the simple case of propagation. The resonant circuits will resonate and a transmission of energy will go from self to capacity then from capacity to self. The speed of transmission (of nodes or energy) is given by 1/LC of course. The same graph can be done with 4 curves (partially reflected case).

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#### studiot

Quick thoughts

Stationary wave have no speed.
That is why they are called stationary waves.

Stationary waves do not transport energy, beyond a node.

Since we are ignoring dispersion, your C term is zero unless there is transmission at the end boundary.

If the reflection coefficient is 1 then the reflected wave is the same amplitude as the incident.

The reflected wave always has the same frequency as the incident.