As promised, after two installments on the basics of impedance and on the most common load-source relationships, we’re ready to conclude with the good stuff
So far, we’ve talked about impedances as if they were simple resistances, the same for DC and AC signals. I promised that it was more complicated than that, so here we go!
Y Z (and what about X)?
Scientists originally studied electricity using direct current (DC), but things got more complicated when they began to investigate the behavior of devices under alternating current (AC). They discovered that capacitors (also called condensers, hence “condenser mic”) and inductors (like transformers and guitar pickups) exhibited different “resistances” depending on the frequency of the alternating current. To distinguish this behavior from conventional resistance (as seen in resistors), they coined the term “impedance” and split it into a “real” part and an “imaginary” part. (If advanced math freaks you out, please skip to the next section.)
The real part of impedance is the resistance. It’s dependent on physical things like the length and diameter of the wire that makes up the winding of a transformer, and its value is constant regardless of the direction of current flow. The impedance of a resistor is just its resistance, and that’s part of why we’ve been able to talk about the two interchangeably in the first two parts of this series.
The imaginary part is the reactance, which is much like resistance but varies with frequency. By the way, the word “imaginary” comes from the fact that the mathematical behavior of reactance is easier to write out and explain if we make use of a quantity called i, which is the square root of –1. Numbers like i behave a little differently than ordinary numbers like 6 or 15 or 3.1416; they’re very good for helping to describe how AC circuits work, but if you actually ask a pocket calculator to tell you the square root of –1, it’ll crash. Try it and see!
In pure mathematics, the sum of these real and imaginary parts is called a complex number and it is written with the general form Z = X + iY, where X is the real part of the number, Y represents the imaginary part, and i is that pesky “square root of –1” that makes things so weird. Impedance is the only thing in electrical engineering that’s represented by a complex number, so it’s likely that this is the reason the letter Z was borrowed from the world of pure math to be its symbol.
So how does this affect audio?
We can sometimes get away with ignoring the reactive (complex) part of impedance because, compared to resistance, the other components of impedance are insignificant at audio frequencies. In other words, signals below 20 kHz are close enough to zero Hz (DC) for the purposes of figuring out impedance effects that we can consider impedances as simple resistances, just as we did in Parts 1 and 2.
Now, though, it’s time to look at situations where this isn’t the case any more. We’ll begin by considering the impedance of audio cables. Their inductance we can usually ignore, but we must often consider the effect of their capacitance.
In a piece of shielded cable, the conductors and shield form a capacitor that’s connected in parallel with whatever the cable is connecting. Cable capacitance is generally specified in picofarads per foot, with around 33 pF/ft being typical. The longer the cable, the bigger the capacitor. We use the term capacitive reactance to describe the “AC resistance” of such a capacitor.
The formula for capacitive reactance is the equation 1 (see images)... where f is the frequency of the signal under consideration and C is the capacitance.
At f=20 kHz, 150 feet of our typical cable looks like:
...or about 1600 ohms. When hung across a microphone with a nominal source impedance of 150 ohms, it will cause a loss of about 1 dB at 20 kHz. (See the earlier parts of this series for the math that gets us to this number.)
A look at the formula shows that the reactance increases as frequency decreases, so as we get into the more important part of the audio spectrum, the loss is pretty insignificant (remember from Part 1 that high load (input) impedances are what we want). The frequency response of the system (which includes the cable capacitance), however, will no longer be flat—the cable acts as a lowpass filter connected across our microphone. Mic cables of this length or greater are common in remote recording and large PA situations and we’ve learned to accept this small loss, though today’s trend toward perfection is leading us to putting mic preamps right at the stage and driving long cables from the low-impedance line level preamp output.
The effect of cable capacitance on frequency response becomes very significant, however, when we have a high-impedance source such as an instrument pickup. Remember our guitar pickup (from Part 2) that doesn’t like working into a load impedance lower than 50 kΩ? By loading the pickup with an impedance equal to the source, we’ll drop half the voltage across the pickup, leaving only half the original voltage (a 6 dB drop) going to our amplifier.
Rearranging the terms of the capacitive reactance formula to solve for capacitance knowing the frequency and reactance, we have the equation at right:
Given that 7 kHz is about the highest frequency we can expect to get out of even the most screeching guitar, we can calculate that a capacitor of:
...or 455 pF, will have a reactance of 50 kΩ at this frequency. At 33 pF per foot, that’s only about 14 feet of cable. It’s no wonder your guitar doesn’t sound very good with a 50-foot cable!
It’s common knowledge that high-impedance inputs are more susceptible to hum pickup than low-impedance inputs. You’ve probably demonstrated it to yourself entirely too often by having your electric guitar too close to an electromagnetic field.
A cable connected to a high-gain input stage makes a pretty good antenna, though it’s one with a fairly high source impedance. This “antenna” is one leg of a voltage divider, the other leg being the input impedance of the amplifier. If the input stage is high-impedance, you’ll have a fairly large percentage of the stray voltage appearing at the input of the amplifier. If the input stage, however, has a low impedance, the voltage developed across its input impedance will be quite low, reducing the amount of stray signal that’s amplified.
Many people associate all unbalanced interconnections with high-impedance circuits. It’s true that equipment with high-impedance inputs such as guitar amplifiers will always be unbalanced (for cost reasons); however, virtually all of today’s studio gear with unbalanced connections is low-impedance. If you’re having hum problems with unbalanced inputs and outputs in your studio, it isn’t because of impedance, it’s because of improper grounding and shielding.
Impedance and digital audio
In the world below 20 kHz, our concerns with impedance are mostly about avoiding loss of voltage or power. In the digital world, we’re connecting digital data in the form of signals in the radio frequency (RF) range, around 6 MHz. At these frequencies, the impedance of the cable itself as well as proper source and termination impedance becomes significant.
The electrical wavelength of a 1 kHz signal in a perfect conductor is about 186 miles, considerably longer than any cable in our studio is likely to be. At 6 MHz, however, the wavelength is on the order of 120 feet, and real-world cable slows things down a bit so it’s actually about 25% shorter. (Note: Acoustical wavelength, a bit less than 1 foot at 1 kHz, is an entirely different thing, the difference being that the speed of sound through the air is about 1100 feet per second, while electricity travels at the speed of light, 186,000 miles per second.)
If the cable is shorter than about 1/20 the wavelength of the signal we’re passing through it, the cable type isn’t very critical, but this means that we have to take some care with all but the shortest of our digital cables. If the source, the load, and the cable impedance aren’t accurately matched, reflections of the signal will occur within the cable. The these reflections, called standing waves, can have several detrimental effects, ranging from a small loss of power to a total loss of power (when the cable length is exactly a half wavelength and the standing wave gets “trapped”). Also, the constantly changing interaction between the standing wave and our desired signal causes a time instability known as digital jitter.
Did someone say cable impedance? At these frequencies, the capacitance and inductance of the wire becomes significant and can’t be ignored. RF Cable has what’s known as characteristic impedance, which is a function of its physical dimensions: the diameter of the wire, the diameter of the shield, the spacing between the conductor(s) and the shield, and the dielectric characteristics of the insulating material that separates the conductors from the shield.
If the source is 75 ohms, the destination is 75 ohms, and the cable’s characteristic impedance is 75 ohms, there will be no loss along the cable other than that resulting from the resistance of the wire—the cable won’t behave as a capacitor or an inductor and, up to a real-world limit, will pass all frequencies equally. Using cable of the wrong impedance or not matching the source and termination impedances can cause standing waves and frequency-dependent losses.
Fortunately, standards are well established for AES/EBU and S/PDIF digital audio interfaces so input and output impedance matching isn’t a problem. 75 ohm coaxial cable commonly used for video interconnections has been with us for over 50 years and is a good match for the S/PDIF impedance standard. The AES/EBU interface is 110 ohms, which is derived from the physical characteristics of the XLR connector that’s part of the AES/EBU interface specification. For about the first ten years of digital audio, we used microphone cables for AES/EBU interconnections, but as resolution got better, we began to notice the effects of improper impedance matching. Today, 110 ohm cable is available from several manufacturers, and, particularly with longer runs, can make a significant difference.
Word clock, a digital signal that’s becoming more important as digital systems grow larger, is another RF signal that needs care and feeding similar to that of digital audio transmission. While voltages and impedances have been standardized for S/PDIF and AES/EBU interconnections, word clock has yet to have an established standard. Most word clock sources (outputs) today are 75 ohms, but some are 50 ohms. How do you know? Look at the spec sheet (few specify this parameter), measure it yourself, or just assume that it’s 75 ohms.
Word clock inputs, however, can be somewhat ambiguous. Some have a built-in termination (almost always 75 ohms) and some are “bridging,” requiring that you add a terminator to properly load the cable and the source. Still others, the best kind, have a switch so you can choose whether that device’s word clock input is terminated or not.
When one word clock source must feed more than one device, you can sometimes get away with bridging the unit in the middle (no terminator) and terminating the far (away from the source) end of the cable with the proper impedance. If both inputs are terminated, however, you’ll be putting twice the expected load on the clock source. This will reduce its amplitude, possibly below the level at which one or both devices require in order to achieve lockup. If the device in the middle of the string is terminated and you leave the device on the end unterminated, that last leg of the cable run will have standing waves.
Digital interconnections really should be easier to deal with than this, and some day they will be. In the meantime, you may have to do some experimenting to optimize things in your own system.
Well, if you’re still awake, you should now have a better understanding of where some of our common terms come from, and what they mean to you and your system.
Mike Rivers (firstname.lastname@example.org) lives and works in Virginia, where he contemplates the ongoing evolution of audio technology.