Why 50 Ohms?

The lowest loss of a selection of cable diameters I presume. I'd think lower loss can be achieved with greater diameters (lower specific capacitance), but costs will be prohibitive.

That said, this isn't even the only place it turns up - there are timber sizes which are specified as one dimension and just "commonly known" to actually be a different one: https://www.thesprucecrafts.com/why-isnt-a-2x4-a-2x4-3970461

There are several articles about that in BSTJ, with pictures.

Consider a system in which we have raised 100 baseballs to a height of 10 centimeters. The amount of potential energy stored in this system is equivalent to the energy in a system with a single baseball raised to a height of 10 meters. We could say that these systems, while storing equivalent potential energy, would have a different kinetic impedance when the energy is released. Imagine how different it would feel to be laying underneath a blanket of baseballs dropped from a few centimeters versus standing underneath a baseball dropped from several stories! One would be uncomfortable, the other, barely survivable. This is the impact that mismatched impedance can have on an electrical device.

Like baseballs, electrons in their orbitals have a certain potential energy "voltage" relative to another nucleus. This energy is released (electrical current flows) when there is a conductive path for electrons at a higher potential to move to a lower potential. A system with 10 billion electrons at 1 volt potential has the same energy as a system with 10 million electrons at potential of 1000 volts, but the one with the higher voltage would have proportionally less current (fewer electrons) than the one with the lower voltage. We would describe these two systems as having different electrical impedance.

In practice, electrical impedance is more complicated than a simple ohms law exchange because of the wonderfully useful property that impedance varies with frequency in all natural materials. This makes analysis less straightforward, but allows us to build filters.

If "electrons with potential" seems abstract, it can be conceptually worthwhile to examine the many analogs to impedance matching in the mechanical realm which are easiest to see when a natural or convenient energy source is transformed into a more useful form. No energy is created (indeed, energy is lost to heat due to inefficiencies); only the impedance is transformed, as we see in the following examples:

An automotive transmission is just an impedance transformer, taking the engine's optimal energy output at 1500-2000 RPM at low force and delivering it to the wheels at a lower RPM with higher force. The transmission matches the output impedance of the engine to the impedance of the car on the road so the engine doesn't stall under too high a load or burn too much gas under too low a load.

A butter knife is an impedance transformer (transforming the low pressure in your hand across the large area of the knife handle into 100X the pressure across the tiny 1/100th area of the blade.) The knife matches the pressing impedance of your hand to the slicing impedance of the butter.

A nut-cracker is an impedance transformer that transforms your hand's low force over a long distance into a very high force over the very short distance required to crack the nut. It matches the impedance of your grip strength to the cracking impedance of the nut.

An electrical utility transformer is an impedance transformer, transforming high-voltage low current into low voltage, high current. It matches the impedance of the utility line to the impedance of your toaster.

A hydro electric dam usually takes the high cross sectional area and low speed water flow of a river and chokes it into a single point with a much lower cross sectional area and a much higher speed where it can drive a turbine. The dam transforms the impedance of the river to the impedance of the turbine.

Even an air conditioner compressor could be said to be an impedance transformer; the coolant starts at room temperature and at a regular volume. Compressing the coolant increases its thermal potential while decreasing its volume. When the higher potential energy is radiated out through the exchanger coils, the coolant is cycled back inside and decompressed, but since it has lost energy to radiation, it is cooler. The air conditioner matches the impedance of the coolant to the thermal radiation impedance of the exchanger coil.

With the exception of the compressor example, all of these devices are simple and passive: some gears, a knife, a lever, some coils, a funnel... impedance transformers are everywhere, doing the simple and passive task of matching the form of energy you have into the form that you need.

The analogue to two pieces of wire with different impedance is two pieces of glass with different index of refraction. What happens when you shine in light is a partial reflection at the boundary of the two pieces of glass, which also means less light gets through.

Impedance matching is just trying to match these indices of refraction to me. Sadly, I have no good analogue for a balun or impedance matching circuit.

I have no idea whether this is mathematically correct, but it gives me some intuition for why it matters.

Actually, it's easy to understand the basic concepts in pure DC. Let's see...

high impedance inputs - this is the easiest one to explain. A device with high-impedance input only allows a small, often negligible amount of current to flow into it, as if nothing is connected.

Imagine a 5-volt power source followed by an 10k ohm output resistor in series, and we put it in a blackbox. If you connected another 10k ohm resistor across the blackbox, you create a voltage divider, and the voltage across your resistor would be 5 x (10k / (10k + 10k)) = 2.5 volts, and a current of 0.25 mA flows.

But if you increase the value of your inserted resistor to, e.g. 10M ohm, now the 99.9% of the voltage is now across your resistor, and you get 4.995 V, and there is almost no current flows across the power source, as if nothing is connected, minimizing the "observation effect".

This allows you to measure the voltage without disturb it by drawing power from a circuit. Example:

ideal voltmeter - gives a voltage measurement.

ideal oscilloscope probe - just an ideal voltmeter that takes many measurements and display a line on a chart.

ideal ADC - just an electronic voltmeter.

ideal transistor switch and logic gate - output only depends on the voltage reading of the input, it doesn't absorb power.

ideal buffer and operational amplifier - measures the voltage at the input without affecting it, and creates a replica of this voltage by using its own power source.

impedence matching - This one is difficult if we're talking about reflections, S-parameters, and standing waves, but it's easy understand if we only talk about impedence matching for maximum power transfer. People often say that, to get the maximum power transfer, the impedance of the source and the load should be matched. But it's misleading. Theorem of Maximum Power Transfer is actually telling us how to choose our load, given a known source, not vice versa.

Imagine an ideal 5-volt DC voltage source. The maximum available power from such a power source is infinite, and only depends on the impedance of your load. By connecting an infinitesimal resistor, infinite power will be absorbed by the resistor.

Now imagine an ideal 5-volt DC voltage source followed by 10 ohm of output resistor in series. This is closer to a real power supply, or a signal generator, or a radio antenna, whose internal impedance sets an upper limit of available power.

What the Theorem of Maximum Power Transfer is telling is that, if we want to absorb the maximum available power out of this power source, our load must be equal to the source impedance, in this case, 10 ohm.

If our load has the same impedance as our source, the voltage across our load will be Vcc x R2 / (R1 + R2), or 5 x 10/(10+10) = 2.5 V, and the power is absorbed by our load is V^2 / R, or 0.625 W.

If our load is much smaller than the source, there will be more current but less voltage. A 1-ohm load will get a voltage of 5 x 1/(1+11) = 0.416 V, and absorb only 0.173 W.

If our load is much bigger than the source, there will be more voltage but less current. A 100-ohm load will get a voltage of 4.54 V, and absorb only 0.206 W.

It's simple if the following misconceptions are cleared.

* Misconception 1: If I have a 300-ohm load, the output impedance of the power supply should be matched to 300-ohm for maximum efficiency.

False. Maximum Power Transfer does not imply Maximum Efficiency. Efficiency is the ratio, Useful Power / (Useful Power + Wasted Power), and it has nothing to do with maximum power transfer.

Recall the example above, when a 100-ohm load is connected to a 10-ohm power source, the voltage across the load is 4.54 V, drawing 0.206 W. Meanwhile, the voltage across the internal impedance of the power is 0.46 V, drawing 0.021W. The efficiency is 91%. But when impedance is matched, 100-ohm load is connected to a 100-ohm power source, the efficiency is "only" 50%.

Often, the efficiency is the best when there's low power transfer. You don't waste a lot of power if you don't take away a lot of power.

* Misconception 2: If I have a 300-ohm load, (e.g. a headphone), the output impedance of the power supply (e.g. a headphone amplifier) should be 300-ohm for maximum power transfer.

False. Maximum Power Transfer is only about how to select our load to absorb maximum power when we have no control over a given source, not vice versa. In real-life, for non-RF circuit, if we have control over the source, the solution is to make the output impedance of the power source close to zero.

For a 5-volt DC source with a 300-ohm impedance, connected by a 300-ohm load, only 0.02W of power comes out because there's only 2.5 V across the load. But if we have the output impedance to be 0.1-ohm, 4.998 V is now across our load, and drawing 0.083 W of power.

This is called impedance bridging, and it's how power supplies and modern headphone amplifiers are designed - they are designed for maximum efficiency by minimizing their output impedance, while the headphones themselves only draw a little bit of power.

On the other hand, a radio receiver is a good example when we have no control over the source impedance, given the same antenna, it cannot be changed. In this case, the better our receiver is matched to the impedance of the antenna, the higher its sensitivity.

BTW, in RF-circuit, even if we have control over a signal source, we use matched impedance, in this case, the purpose is not to get the maximum power transfer, but for avoiding reflections, a completely separate issue.

* Misconception 3: 50-ohm coax cable has 50-ohm of electrical impedance.

False. The "50 ohm" and "75 ohm" for coax cable is its characteristic impedance, not electrical impedance.

This is when things get complicated and when we must think electrical signals as waves. But the other comment has a good explanation, so it won't be repeated. In short, when a wave hits an impedance discontinuity in its medium of propagation, some of the energy is reflected, so in an RF system, 50-ohm everywhere is desirable.

A small piece of lossless coax cable can be modeled as an inductor in series and a capacitor in parallel, it's the radio of the inductor and the capacitor that determines its characteristic impedance that we can match to avoid a discontinuity in the circuit, thus avoiding reflection. Or we can say that the characteristic impedance is the input impedance when the length of the lossless coax is infinite.

It's not an actual electrical impedance that we can measure by an ohmmeter. A 10-meter coax has an electrical impedance of less than 1 ohm.

Such as your speaker's voice coil, which is a fairly low-resistance length of thin copper wire commonly wrapped around a hollow paper core an inch or two in diameter and enameled in place. The paper form is cemented to the center of the suspended speaker cone, with the coil itself free to move back & forth within the matched gap of the speaker magnet.

When the audio signal (which is a series of waves, therefore an AC voltage) comes in on the wires, the coil moves the cone in response to the signal.

DC voltage is not allowed to pass to this kind of speaker since DC would just displace the cone in a single direction for the duration of DC voltage application.

But when you measure the resistance in Ohms of an individual _8-Ohm_ speaker, it is always less than 8 Ohms, sometimes between 6 & 7 Ohms because regular Ohmmeters are measuring DC resistance using a DC battery inside the Ohmmeter as a reference.

The DC resistance of the coil is no different than it would be if the whole coil was unwound, it basically depends on the length of the wire and the diameter of the wire, similar to the way precision wirewound resistors are available for lower Ohm service. Thinner wire and longer wire has higher resistance to current flow.

But when audio is passing through the coil alone in free air, IOW no magnetic or ferritic core is present, there will be slight additional resistance to the audio AC (over and above the fundamental DC resistance measurable with the regular Ohmmeter), simply because of the wire being shaped into a coil. This total working resistance to AC is what is referred to as impedance. When a ferritic or magnetic core is in place, there will be that much more of a challenge for the audio signal since it needs to then reverse polarity against the inertia of the magnetic material at whatever frequency it happens to be operating at. This magnetic bump in AC resistance is significant and causes a nominal 6-Ohm-resistance speaker to operate as a characteristic 8-Ohm-impedance audio load.

Speakers do work really well with much higher than 8-Ohm voice coils, provided a matching amplifier is available. Practically this is not done since the speaker wires from the power amplifier to the speakers in a high-power system would then be required to carry audio AC voltages as dangerously high as the line voltage coming in to the amplifier.

So using the everyday low-impedance speakers we are stuck with, a slightly poor connection having an extra 1-Ohm of unwanted resistance, due to corrosion for instance, will result in that connection dissipating a significant percentage of the power passing through it, as heat, in many cases accelerating a corrosive source of power loss.

Hardware and electronics was an important area in computer engineering, and many early hackers were hardware hackers. Today, the field has shifted as software development at large, but hardware and electronics is still an indivisible part of the hacking community. It can be seen from the fact that EECS is still taught at schools, the fact that embedded electronics is having increased popularity due to IoT, and the renewed interests within free and open source community.

I'll keep submitting more hardware/electrical articles which I find interesting.