# Why is the voltage wave magnitude constant along a transmission line that is perfectly matched to a load?

I think I understand reflected waves and standing waves on transmission lines, but why is a traveling wave voltage magnitude constant on a perfectly matched transmission line? Why doesn't have peaks? I would expect to a decrease to zero in between peaks racing by what ever point is probed, dependent about the phase velocity. I get that there isn't a reflected wave for a perfect match.

If the transmission line is perfectly matched (or its length is infinite), there is no reflection.

Therefore the voltage at any point x, and any time t is (assuming losesless line): $$V(x,t) =A\cdot \sin{(\phi + 2\cdot\ \pi\ \cdot ( f\cdot t - \frac{x}{\lambda}))}$$ where λ is the wavelength, and Φ is the phase.

If you choose any x point, you can rewrite the voltage as: $$V(t) =A\cdot \sin{(\theta + 2\cdot\ \pi\ \cdot f\cdot t )}$$

As you can see, the amplitude of that sinewave is A, regardless the particular x.

• thanks for your answer but my question isn't really mathematical, it's more conceptual – Andrew Davis Sep 8 '17 at 11:40
• Ok. Assume there is no reflection, so we don't have to deal with stationary waves. Assume also that the line is loseless. In ANY point you'll get an oscillating wave, but it's amplitude will remain constant, i.e. it will oscillate from +A to -A. Think of a traveling sine wave: if you stop at any point, and you measure the instantaneous value, you'll measure of course a sinewave, but the amplitude (A) won't change. And A will be constant regardless the particular position you're considering. – next-hack Sep 8 '17 at 12:00

You understand that for a standing wave in a T-line, the amplitude depends where you are on that T-line.

Now the wave is not standing but travels. So pick a point on the T-line and you see the waves passing by. They all have the same maximum amplitude, right ?

I could move to the left or right on the T-line but I would still see the same as the waves are all the same and they keep passing by.

Why would there be any amplitude peaks ? For that to happen the wave and its amplitude has to be influenced by something. For the standing wave it is the reflected wave combining with the original wave resulting in a combined wave which appears to stand still.

Note that a wave cannot stand still, it is just the vector-sum of two (or more) waves which appear like the wave stands still.

A T-line which is perfectly matched looks like an infinitely long T-line to a wave that is travelling in it. The wave experiences no outside influence/interference because there is none. So it will have and continues to have a constant amplitude.

At the end of the T-line the wave's energy is dissipated in the load (assuming a resistive load), the wave is no more, it is converted into heat. So it cannot reflect and it will not interfere with the other waves.

• As the waves pass by won't their amplitude decrease then increase relative to a certain physical point? Otherwise isn't the signal DC? I.e won't the voltage a point change continuously? And therefore change continuously along the line? – Andrew Davis Sep 8 '17 at 11:46
• youtube.com/watch?v=mh3o8gUu4AE In this video the dude is using an animation to describe standing waves, it shows a traveling wave with reference points that rise and fall as the wave travels through – Andrew Davis Sep 8 '17 at 11:48
• No the amplitude stays the same, the amplitude is multiplied by a sinewave and the sinewave varies between -1, 0, +1 but the amplitude remains constant ! Perhaps you're assuming that when at a certain point you measure 0.5 that then the amplitude must be 0.5 but that isn't so. The amplitude is not the momentary value at one point but rather the maximum of the wave. The amplitude can be 1 and the sine 0.5, these multiply and give 0.5. – Bimpelrekkie Sep 8 '17 at 12:08
• So are you saying i am confusing instantaneous magnitude with amplitude? – Andrew Davis Sep 8 '17 at 13:16
• yes...at any point the amplitude is what it is along the sinewave at that time, but the amplitude of the sine wave is the full amplitude is traverses over a period (or half a period). for ex. v= A*sin(wt). A is the amplitude, reaching its peak when sin(wt)=1, so the amplitude of the sine wave is A, even though at any time t it may be less than that as the oscillation completes a full cycle w=2 *pi. – jrive Apr 9 at 18:19