Calculating the Impedance of Your Speaker Hookups
By hcadmin |
No More Math! Use This Excellent Excel Solution for All Your Impedance Calculations
By Jon Chappell
One of the great things about doing research on the Web is that you often find out stuff you weren’t necessarily looking for in the first place, but is valuable in another application. Case in point: I was researching the best way to hook up some lithium-polymer batteries to drive a high-powered motor for my radio-controlled airplane (a 60" P-51 Mustang), and I happened to stumble across a familiar, but out-of-context website: www.duncanamps.com. This was linked from one of the aeromodeling forums I frequent, and touted as “one of the best sources for software-based electrical-circuit calculators”— little apps that people make on Excel that provide easy answers to life’s little calculation problems.
Wait a minute. duncanamps.com for electric motors? Of course, the site in question was for amp info! They wouldn’t be talking about batteries, I thought. And sure enough, they weren’t. But what the site was talking about was the way to calculate the resistance in series and parallel circuits. You see, combining the source of power (batteries) follows the same principles as the receivers of power (the speakers); it’s just from the other end.
And that’s how I discovered this fantastic app. for calculating speaker hookup impedances. Since we guitarists never have to think about the power side (smart guys like Duncan Munro do that for us), we’re on the load side of the equation—usually in the form of hooking up speakers. The app. that Duncan amps offers is the most useful calculator (in the form of an Excel spreadsheet that you can download for free) for calculating power and impedance ratings for common amp hookups. Let’s look at the theory, and then you can forget all that and bookmark www.duncanamps.com for the easy way!
Ways to Hook Speakers Up
There are two ways to hook up speakers: in series and in parallel. At the most basic level, with two speakers, equal impedances in series doubles the total. Equal impedances in parallel halves the total. So when you have two speakers at 8 ohms apiece (guitar speakers are usually 4 or 8 ohms), hooking them in series (the positive terminal of the first speaker after the amp goes to the negative terminal of the second speaker) creates a circuit with 16 ohms. Hooking them in parallel (with their positive terminals all connected to each other and their negative terminals all connected to each other) creates a 4-ohm impedance.
The problems come with more than two speakers, or when the impedances are mismatched, or both. Then you need some math chops. So let’s look again at the nature of series and parallel.
Series is easier, from both a conceptual standpoint as well as a mathematical one. A series circuit is where everything is in a daisy chain, like the battery configuration in a long-handled flashlight. All the terminals are wired end to end, so that you have a positive connecting to a negative connecting to a positive, etc. (By the way, most small electronic devices have their batteries connected in series, so even if you see an arrangement of eight double-AAs in two rows of four each, they’re configured electrically like in our long-handled flashlight example.) Schematically, here’s how a series circuit looks, with simple flashlight batteries, and with two speakers.
Fig. 1. A series circuit for batteries and speakers. Each has a positive and negative terminal, which can be wired together.
When working with speakers, you hook up the positive and negative terminals the same way as batteries. Don’t be confused by the geometry. Electrically, the flashlight and the speaker cabs are wired the same—in series. This is similar to the way you daisy-chain pedals, with the output of one pedal feeding the input of the next one in line. And just as in a pedalboard, the outer most connections, are the ones that go somewhere else. In an amp, it’s a circuit, like “circle,” so the first-in-line negative terminal and the last-in-line positive terminal are both connected to the speaker output jack on the amp’s back panel.
When you want to compute the total resistance of the speakers in your chain (whether that’s a cabinet with its speaker terminals wired in series or a succession of separate speaker cabinets in a daisy chain), you simply add up the resistance, presented as ohms, and (usually) stamped on the back of the speaker. So if you have three 8-ohm speakers, your total resistance is 24 ohms. Mathematically that’s represented by this formula:
R total = R1 + R2 + R3 + …
Parallel connections hook all the positive terminals together and all the negative ones to each other, except for the first negative and last positive ones, which go to the source. Fig. 2 shows what this looks like with batteries and speakers.
Fig. 2. In a parallel circuit, all the like terminals are ganged to each other.
You can think of parallel as the “opposite” of series in one sense: the resistance decreases as you add more components. But it’s not simple subtraction (which would be the true opposite). In fact the formula is kind of complicated:
R total = 1/(1/R1 + 1/R2 + 1/R3 + …)
So let’s take the normal ohm values for speakers, which are 4 and 8. If you tackle the parentheses first, you realize right away that you can’t add denominators that aren’t the same, like 1/4 + 1/8. You have to convert them to the lowest common denominator (2/8 + 1/8 = 3/8).
Okay so, we solve for the parentheses, and then we invert the results, creating the reciprocal, because that’s what the front of the equation, 1/(…), tells us to do. We’re left with 8/3. That isn’t just a fraction, it’s a term that has to be solved. So we divide 8 by 3 and we get 2.67 ohms. That’s the impedance of a 4-ohm and 8-ohm speaker hooked up in parallel. And it’s not easy to intuit that number.
Whew. That’s a lot of work. Is there an easier way than the equation above? Yes! Easier, but not as elegant, as it requires more steps. The way to re-jigger that complex, fraction-laden formula is to do this:
So plugging in the numbers 8 and 4, we get 32/12. That equals 2.67. You still have to divide as your last step, but most people find this:
XY/X+Y is easier to deal with than 1/(1/X + 1/Y)
Deeper into the Parallel Universe
Things get even weirder with parallel when we combine speakers of differing impedances. And that is when, dear readers, we stop thinking and start surfing! Though the math still holds true, you can go to this fantastic calculator at www.duncanamps.com/technical/impedance.html
You can download and then use this calculator for several different speaker configurations, including series-parallel, which will give you different impedance values according to the way you hook up more than two speakers.
Check out the following four screen shots, taken from the separate sheets of the Duncan amps Excel application, and you’ll see why this is a cool program—simple but clever.
Fig. 3: Here is how a simple two-speaker hookup looks in series in the Excel program offered at www.duncanamps.com.
Notice that I’ve highlighted the cell located at E8 (Column E, Row 8). Whenever you select a cell in Excel, it shows you the formula used in that cell in the formula bar (also highlighted). Here, no surprise, it just adds the values of the cells (indicated by the column/row locations E6 and E7). The great thing about this spreadsheet is that it will show you the power distribution to each speaker (see E10 and E11) in addition to the load the speaker produces on the circuit. This can come in handy later on with more complex arrangements.
Fig. 4. This is a simple parallel circuit, again, with the key cell selected, which shows the formula (Excel-style) in the formula bar.
This is where the calculator starts to earn its money, if you’re plugging in numbers that are not the same values for the speakers. The trick of taking half of the value of two speakers of equal impedance works only with that scenario. The more complex formula for parallel will work for all parallel scenarios (different numbers of speakers, differing impedances among speakers).
Fig. 5. Here’s the spreadsheet configured with three speakers in parallel.
For easy math, I’ve left the speakers with the same impedance (all 8 ohms). But if I changed one of the speakers to 4 ohms, the value would be 2 ohms. Change two of the speakers to 4 ohms (leaving one at 8) and the value would be 1.6 ohms.
The next two screenshots show complex arrangements: more than two speakers and in the arrangement called series/parallel, which combines aspects of both serial and parallel circuits. Note that in these next two examples, we still have not used speakers of differing impedances. This not only keeps the math more intuitive, but it’s the most likely scenario you find yourself working with speakers of the same values.
Fig. 6. A series-parallel hook-up with Speaker A in parallel with B, and AB in series with C.
The arrangement above mixes serial and parallel by putting the A speaker in parallel, while leaving the B and C speakers in series. The result in the above series/parallel configuration is 12 ohms, which is less than 24 and closer to 16, or a two-speaker cabinet in series.
Fig. 7. A series parallel configuration with Speaker B in series with A, and AB in parallel with C.
The above arrangement is a slightly different series-parallel configuration than the one shown in Fig. 6. Here, the B speaker is placed in parallel, and the A and C speakers are in series. This produces a total impedance that is closer to the all-parallel hookup of 2.67 ohms, but higher than that dangerously low rating, and closer to the 8-ohm rating of a single-speaker cabinet or the 4-ohm rating of two speakers in parallel.
In both arrangement of the series/parallel configurations above, the range is closer to the original 8 ohms of the speakers used than if the circuit were entirely series (24 ohms) or entirely parallel (2.67 ohms). This gives us much more flexibility when trying to maximize the power of the amp, and run it safely.
Better Living Through Excel
The calculator takes you up to four speakers in series, parallel, and series/parallel with two configurations each. This should give you plenty of options to think about, and be sure to run numbers of different impedances to see what the results will bear. You really shouldn’t be running more than four speakers on one amp, because it creates power-management problems, but with this handy calculator, you can “run the numbers” of any likely scenario you’re likely to encounter—and learn about speaker hookups in the process.
Jon Chappell is a guitarist and the Senior Editor of Harmony Central. He has contributed numerous musical pieces to film and TV, including Northern Exposure, Walker, Texas Ranger, All My Children, and the feature film Bleeding Hearts, directed by actor-dancer Gregory Hines. He is the author of The Recording Guitarist: A Guide for Home and Studio (Hal Leonard), Essential Scales & Modes (Backbeat Books), and Build Your Own PC Recording Studio (McGraw-Hill), and has written six books in the popular Dummies series (Wiley Publishing).