Why do cables specify 75 Ω impedance when impedance is a two-dimensional quantity?

Question

As I understand it, impedance has an imaginary and a real part. To be able to perform any useful calculations, one needs to know both. Cables and speakers usually only say like "8 Ω" or "75 Ω". Is that purely resistive? If so, what is so special about 75 Ω that all coaxial cables have that resistance, and why are speakers are usually 2, 4, or 8 Ω?

Answer 1

The 75-ohm impedance associated with a transmission line is a so-called characteristic impedance, and it describes the behavior of high speed effects on the line. In particular, if you transmit a fast¹ pulse or fast sine wave over a transmission line, the voltage wave and current wave traveling over the line will have a 75-volt-per-ampere (i.e. ohm) ratio.

This value is useful when designing high speed systems that avoid reflections and ringing-- while an impulse travels smoothly along matched transmission lines, if the impedance suddenly changes then a reflection will occur, much like a window reflects a bit of light at the air-glass interface. These reflections can be a major practical problem that degrades signal integrity in communication systems and may cause damage to high power radio transmitters when not properly handled.

As it happens, 75 and 50 ohm lines are common values that got standardized for their various applications and can be manufactured with relative ease. The actual values are a consequence of fundamental physical constants, the cross sectional geometry of the cable, and the materials used in its fabrication.

If your characteristic impedance is reactive, you are also modeling the losses on the line; this loss is often instead modeled as attenuation per distance in dB/m or similar, which is more convenient for practical calculations than a complex impedance.

On the other hand, speaker impedance is a more vague, approximate value given to estimate power handling and amplifier requirements. Reactive component aside, it's highly frequency dependent anyway and is more of an average (see example charts at https://www.aperionaudio.com/blogs/aperion-audio-blog/the-truth-about-speaker-impedance)

¹ fast enough that the transmission line is long relative to the rise time of the signal. At audio frequencies, this kind of high frequency behavior only shows up when your transmission line is on the order of multiple km long.

Answer 2

Why do cables specify 75 ohm impedance when impedance is a 2 dimensional quantity?

Cable characteristic impedance is a big subject

When a coax cable is specified as being 50 Ω or 75 Ω it relates to its characteristic impedance and not any cable resistance (that you can measure with a DMM). In fact it's the square root of the ratio of the distributed inductance divided by the distributed capacitance: -

$$Z_0 = \sqrt{\dfrac{L}{C}}\hspace{1cm}\text{(at high frequencies)}$$

And, that ratio has the units of ohms (despite not being related to any real resistance element). A more general formula that covers all frequencies is this: -

$$Z_0 = \sqrt{\dfrac{R+j\omega L}{G+j\omega C}}\hspace{1cm}\text{(at low and high frequencies)}$$

At sub-audio frequencies the above formula is also purely resistive: -

$$Z_0 = \sqrt{\dfrac{R}{G}}\hspace{1cm}\text{(at sub-audio frequencies)}$$

At around mid-range audio frequencies, the above formula becomes this: -

$$Z_0 = \sqrt{\dfrac{R}{j\omega C}}\hspace{1cm}\text{(mid-range audio frequencies)}$$

And, at those mid-range audio frequencies, the characteristic impedance is more "reactive" than at any other point in the spectrum. Graphically: -

Taken from this answer. The cable approximates the standard 600 Ω telephone cable.

But, we don't usually specify characteristic impedance at audio or sub-audio frequencies; we are interested in its value at high-frequencies (above 100 kHz usually).

At this point you may be confused and wonder what characteristic impedance is all about?

_{If you had an infinitely long piece of idealized 75 Ω coax and, applied a 12 volt supply at the near-end; the current that will flow will be 12/75 amps (160 mA). In other words, with an infinitely long cable, 160 mA will continue to flow while ever the input of 12 volts is presented.}

_{Of course, in the real world, an infinite idealized coax doesn't exist but, for a 100 m length of coax, we could apply a short 12 volt pulse (say 100 ns in duration) and we'd expect to see a short current pulse of 160 mA. It gets more complicated after that but, in essence, that is what characteristic impedance is all about.}

_{If you want to know how it can be calculated from 1st principles read this answer.}

what is so special about 75 such that all coaxial cables have that resistance, and all speakers are usually 2,4, or 8 Ohms?

I've explained the cable characteristic impedance.

Speakers are just simple loads and, they will have an impedance specified in the middle of the audio range (usually 1 kHz). That impedance will contain a resistive and reactive element but, we usually don't go into that much detail because, an amplifier is only concerned with driving an impedance.

Answer 3

Speakers and cables are different beasts:

• Yes, a reasonable cable has a purely real wave impedance: The imaginary part is the loss.
• With speakers it's a bit more complicated, but the moment you feed them in an in-phase manner, you should see the real impedance.
  That's very much in compliance with what you've learned about reactive power, apparent power and effective power (Blindleistung, Scheinleistung und Wirkleistung).

If so, what is so special about 75 such that all coaxial cables have that resistance,

Not all cables have 75Ω impedance – in fact, typically mostly consumer-grade antenna cables and cables which are only coaxial for noise reasons, not to actually transport RF, have 75Ω; lab-grade and internal cabling more commonly has 50Ω, and once you do anything on a PCB, or anything with powers that require stiff waveguides, you more re less design things that fit your problem, instead of using what's there.

However, standardizing of fixed cable, connector and waveguide characteristics makes sense, or else everyone would need to reinvent the coaxial connector anytime they want to connect two devices. So, 50Ω for most applications, 75Ω for low-loss long distance transport over low-cost cabling have established themselves.

Now, speakers, again, different business. Impedance of a speaker wildly varies across its frequency range. You should probably treat 2, 4, 8, 16, 32 and 64Ω (the latter two mostly in headphones) as "this is the low-frequency, high-power operating point you should roughly (as in: power of two) design your amplifier for", not as actual impedances.

Answer 4

Quite a few people have already talked about cable impedance, so I'm only going to talk about loudspeaker impedance.

Let's start with what the rated impedance of a loudspeaker actually refers to. A typical loudspeaker will have an impedance curve something like this:

The tall peak there is at the free air resonant frequency of the driver, generally referred to as F_s. The rated impedance is the value at the first minimum above that first maximum.

That leaves the question of why values of 2, 4, and 8 ohms are common. First of all, I'd note that at least in my experience, the most common values are probably 4, 8, and 16 ohms. 2 (and 1) Ohm have become a bit more common in car audio recently, largely because cars have fairly low supply voltage (12 Volts, nominally) but can deliver utterly massive current (more about that below).

16 Ohm Speakers

Back in the days of vacuum amplifiers, 16 ohms was pretty common. With a vacuum tube amplifier, that amplifier itself typically has a fairly high output impedance, so you use a transformer to match the amplifier's output to the speaker, and it's generally easier to design that transformer for somewhat higher load impedance.

8 Ohms

But then specsmanship entered the picture. Although it generally hurt frequency response and distortion, an amplifier designed for a 16 Ohm load could usually produce more power into an 8 Ohm load. So, last year's (obviously anemic) 10 watt amplifier became this year's (obviously much better) 18 watt amplifier, without having to waste any money on redesigning the amplifier itself at all.

About then, the Institute of High Fidelity (IHF) set some standards for amplifier specs, and at least the better quality vendors made at least some attempt at following it, so rating at still lower impedances mostly stopped, at least for home audio¹.

There are a few vacuum tube amplifiers (e.g., Julius Futtermans) that drive 8 Ohm loads without a transformer, but these didn't come into use until well after 8 Ohms had become the de facto standard for home audio loudspeakers.

4 Ohms

4 Ohm speakers were (and still are) mostly used in car audio. Nominal supply voltage in a car is only 12 volts, so an 8 Ohm load is limited to 12²/8 = 18 watts. Reducing the load to 4 ohms allows around 36 watts instead.

Of course, there are designs that exceed that, but you need something like a high power charge pump to do it. But given a low supply voltage, the simplest way to increase power was to reduce the load impedance, so 4 Ohms became pretty much the standard.

Even Lower Impedance

Over the past few decades, car audio competitions have become common. One common form is simply measuring maximum volume ("SPL competition"). For this, quite a few have taken the logic above even further, so they've gone to 2 Ohm and even 1 Ohm loads. Quite a few use other strategies as well, such as massive capacitor banks so they can deliver utterly ludicrous power for a short time. For example, one is rated at 17 kilowatts (I'm tempted to make a joke about it doubling as an arc welder, but somebody would probably think that was a good idea).

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1. Before that, some ratings were pretty outrageous. Especially cheaper vendors frequently gave peak power instead of RMS, so for a sine wave it was approximately RMS x 1.414. Not satisfied with that, some invented the term "music power" which was simply taking the measured maximum, and multiplying by 2. Then, of course, those were combined to give "peak music power", which was the measured maximum RMS power time 2.828. Then "to keep from confusing the consumer", some rounded that up to simply triple the measured power. Quite a few also based their measurements on what they could produce for only a very short period of time, but would cook the amplifier if you tried to draw that much for more than a few milliseconds. The IHF standards helped with a much of that, and the FTC later stepped in with even more rigid (and enforceable) requirements.

Answer 5

In some cases, the impedance of a circuit or component may be purely resistive, meaning that there is no reactance present and the impedance is equal to the resistance. In this case, the value given, such as "8 Ω" or "75 Ω" is the resistance of the circuit or component.

The values of 2, 4, and 8 Ω are common for speakers because they are convenient values that are easy to work with in audio systems.