Crossover Basics - Impedance

In my first post on the subject of Crossover Basics we talked about the electrical response. How various types of high and low-pass filters changed the voltage that the speaker drivers experience. We glossed over the issue of impedance entirely. Understanding this will help you better grasp what's going on.

Introduction to Speaker Impedance

Anyone who has examined speaker specifications has seen an impedance rating, such as 4 or 8 Ohms. If you see speaker measurements from Stereophile or SoundStage you would have come across charts showing complicated details about impedance and phase. Here is the impedance chart from the ported LM-1. This first chart shows the combined effects of the drivers, crossover components as well as the cabinet and port.

Ignore the thin blue line, and focus on the thick one instead. Notice the chart axis. Across the X (bottom) we have the frequency scale, across Y (left side) we have Ohms.  So the first thing you learn is that the nominal 4 or 8 Ohms you are used to seeing are kind of a summary, and sometimes an outright lie. No speaker is exactly 4 or 8. These are guides to help match speakers and speaker counts to amplifiers. Personally I would rate the LM-1 as an 8 Ohm speaker, though it dips below this above the bass. The triple hump you see is indicative of a 2-way, ported loudspeaker. The two humps at left are caused by the port, while the hump at right, just below 2kHz  is where the high and low-pass filters meet.

You might have picked up on the fact that we are using Ohms to measure impedance (sometimes called Z). What makes impedance different from resistance is that it is frequency dependent, and it has an angle. This angle is the difference between the voltage and current. We'll ignore this for now and focus on the magnitude in Ohms. Impedance is what we use to understand the effects of capacitors, coils and speaker drivers which are electrically complicated little beasts.

One very important point to note is that even though the tweeter and woofer sections are arranged in parallel, the impedance of the drivers does not actually appear in parallel. That is, a pair of 8 Ohm drivers wired in parallel would normally result in an impedance of 4 ohms, but in this "perfect" system the actual impedance stays very close to 8 Ohms at all frequencies.

Let's talk a little more about the LM-1, above. From left to right, the first two humps are classic indicators of a ported loudspeaker. From the bottom up to around 1,000 Hz the impedance is entirely that of the woofer. The peak around 1,800 Hz occurs where the low and high pass filters overlap, and after about 5kHz, the impedance is entirely a result of the tweeter.

In the sections that follow we'll go over each section in detail.

Effects of High and Low Pass Filters

We'll use the original example of a first-order crossover for all of our discussions.

To refresh your memory, this chart shows us how first order filters behave. The red trace is the electrical signal a tweeter would see, while the yellow is the woofer (1 kHz is actually a very low frequency for most tweeters, let's pretend it's OK). Black is the summed response of the tweeter and woofer. While it is smooth and straight we don't care about it right now.

This is strictly the voltage and electrical view of what would be measured at the inputs of the "ideal" 8 Ohm tweeter. We are, of course, missing the acoustical results, but we cover in the post Crossover Basics - Driver Response.

The question we answer in this post is: "How do we achieve this frequency dependent change in response?" Let's take a look at the original crossover first:

There is a capacitor in series with the tweeter, 20uF. We talked about it "blocking" low frequencies. This is achieved by an increased impedance. That is, as the frequencies drop, the Z (impedance) of C1 goes upwards. Let's take a look at the impedance of the tweeter circuit, with the driver and the driver + capacitor.

Series Voltage and Impedance

Keep this rule in mind: The voltages across each component in a series circuit is proportional to the resistance of each component. We're dealing with impedance, and complicated but for the sake of simplicity we ignore it here. This simple rule-of-thumb is enough to explain the behaviors without delving into reactive impedance calculation.

High-Pass Impedance

In the chart below the red line represents our tweeter impedance. Again, this is theoretically perfect 8 Ohms. No drivers are like that in rea life but we need it simple so we can better see what the high-pass filter is doing.  The blue line however represents the entire tweeter section. That is, C1, S1.

Unfortunately at 100 Hz the impedance is cut off but it's close to 80 Ohms. If C1 is 80Ohms we can then estimate the relative voltages across C1 and R1. It's only an estimate because we're not taking phase into account, but it's going to be very close. So we estimate the voltage at S1 to be 8 / 80 = 0.1 of the total. So at 100 Hz the tweeter gets approximately 10% of the voltage. At 10 kHz Z ~= S1, meaning our tweeter gets nearly all the voltage, and C1 is acting like a short circuit.

With those two points (100 Hz and 10 kHz) in mind, now the following graph should make sense. It plots the voltage across C1 and S1 as the frequency varies. Assume a 4V input voltage.

As we calculated, at 100 Hz the voltage across S1 is about 4V x 0.1 = 0.4V

This shifting of impedance and voltage between the filters and the drivers is similar regardless of the order of the filter. It's also mirror-imaged for the woofer.

From Voltage to Decibels

Let's calculate the dB drop at the crossover point of 1 kHz. The peak tweeter input is 4V. At the crossover point it is approximately 2.8V. So, we calculate using the rule:

dB = 20 log ( Vout / Vin ) = 20 log ( 2.8/4 ) = -3 dB

In other words, at the crossover point the input to the tweeter is 3 dB less than the amplifier output. Simple, right? let's do this again for 100 Hz:

 20 log ( 0.4/4) = 20 log ( 0.1 ) = -20 dB

So at 100 Hz, the voltage to the tweeter is -20 dB below the amplifier's output. A very good thing since tweeters are easily damaged by low frequency signals.

Low-Pass Impedance

The same and complementary effect is happening at the woofer side of this example:

In the woofer side, by around 100 Hz L1 has no meaningful contribution to the impedance magnitude (i.e. Ohms) so nearly 100% of the input voltage is at the woofer. By 10 kHz however the impedance of the coil is approximately 80 Ohms. Estimating 8/80 = 10%.

Additional Exercises

Using XSim, and "ideal" drivers, create 2nd order filters and compare the impedance of the high and low pass sections. Compare the electrical response and impedance curves. Combine them and examine the impedance bump where they meet.

Using a second order filter, examine why the 2nd component works without shorting out the entire speaker. For instance, if a high pass filter, examine why the coil (L) works. What would the impedance be if the coil alone was there, but not the capacitor?

Crossover Basics - The Zobel

Introduction to the Zobel

A Zobel network is used to flatten a driver's impedance (usually a woofer or mid-range), therefore making the filter (usually low-pass) more effective. There are many kinds of Zobel networks, but for speakers the simplest and most common Zobel circuit consists of a capacitor and resistor which are wired in parallel to a driver.

C1,R1 on the left are an example of the Zobel network. Their values are chosen to minimize the impedance rise of a driver above the resonant frequency.

Speaker Impedance

If you aren't quite sure what this means, please visit my blog post on Crossover Basics - Impedance before reading more.

Introduction to Series Circuits

Take a look at a very simple series circuit. It consists of two resistors in series (one after another) with the amplifier. The circuit is closed by the ground points (the downward facing triangles).

As before, this may be something you like to play with, so I encourage you to grab a copy of XSim and try this out along with the Peak Voltage chart.

What's important to understanding here is that the voltage across the resistors will be proportional to the resistance offered.  So

Vr1 = Vin * R1/(R1 + R2)
Vr2 = Vin * R2/(R1 + R2)

And of course:

Vr1 + Vr2 = Vin

That is, the voltage across R1 and R2 must add up to the input voltage.

Think of Ohms as elephants that eat volts. The more elephants, the larger portion of the incoming voltage they eat.  Yes, this is a very silly analogy.  Still, we can do some quick math. The total resistance is 100 Elephants (hah!) or 100 Ohms. R1 has only 10 elephants, so it gets 10% of the incoming voltage, whatever that voltage may be. R2 has 90% of the Elephants, so it takes 90% of the incoming voltage. There's a lot more to circuit analysis, but this is the bare minimum to understanding Zobel networks. Hopefully you'll be intrigued and learn more on your own.

Woofer Impedance

So with a little background under your belt you are now ready to look at a typical woofer. We'll use the LM-1 woofer, the Peerless 830991. You should know that impedance graphs will change once a driver is in the cabinet, especially if the cabinet is ported, so the data I present here will be different than a specification sheet which measures the driver in free-air.

You have heard the term "coil" used interchangeably with "inductor." Which is correct, but you may not have thought about the term "voice coil" in the same context. The voice coil is the part of the speaker that will electrically connect to your amplifier and produces the magnetic force which moves it against the magnetic field of the permanent magnet. We won't get too much into this, but suffice it to say that above the resonant peak the voice coil behaves like that of any other inductor, specifically it has both a resistive element (DC Resistance, or Re) and an inductive element (Le). These combine to give us the woofer's electrical impedance (Z) at any given frequency.

In the chart below we will compare the impedance of the woofer in a sealed cabinet (red) with a woofer that has a Zobel network applied (blue).

Let's ignore what happens below 200 Hz. That's a topic beyond this posting, and it's also far below our likely filter points. From about 200 Hz to 400 Hz the woofer is purely resistant, and we have the minimum around 6.6 Ohms, but what happens to the right? That's correct, suddenly more voltage-eating elephants arrive! The inductive qualities of the voice coil become more and more important and overwhelm the well behaved resistance. Compare the peak difference of the two impedance charts, all the way at the right. The blue line represents a woofer compensated with a Zobel network. The impedance never goes above 7 Ohms, while the normal un-compensated woofer goes to over 30! That's more than 4:1 difference. This increase impedance is going to compete with the low pass filter and make it behave in ways we probably don't want.

For clarity we'll leave behind the LM-1 schematic and create a new one, with two identical woofers and 2nd-order low-pass filters set to 4 kHz. Of course, this is not how a real speaker would be designed, we are just using this to see exactly how a Zobel circuit works. The Zobel consists of C1 and R1. 

The first thing we should do is examine the transfer function of the two filter sections:

As you can see, S2 is behaving like we expect a low pass filter to work. S1 however is having a very difficult time getting to the right slope. At 5kHz the output is almost 10 dB higher than we want it to be. That's a big deal. Eventually the impedance (elephants) on the low pas filter take over, but they don't reach our desired behavior until past 20 kHz, definitely not good enough for us. Let's take a look at the final outcome, below:

How Does a Zobel Work? 

Above we discussed how serial components work in a circuit. You may feel a little tricked because while we learned enough to understand why coil inductance needs to be compensated for, we never talked about how a parallel circuit works, which is what a Zobel is. A parallel circuit has one unique property:

The apparent impedance of a parallel section is never more than the smallest impedance.

If we imagine C1 as a short, then no matter what S2 rises to, the impedance will never go above R1, or 8.2 Ohms. It can be less than that, but never more.  In a parallel circuit you calculate the apparent impedance like so:

Rtotal = 1 / (  (1/R1) + (1/R2) + (and so on and so forth)  )

Things are more complicated because we are actually calculating impedance, but you get the picture.

Let's do some quick, Dr. Leach style of analysis on the components in a Zobel. At very low frequencies, C1 behaves like an open circuit, essentially removing R1 and C1 from meaningful contributions to the system impedance. Remember we mentioned that impedance cannot rise more than the smallest value? So at low frequencies C1 is so large that S2 becomes the limit on impedance. You can see the impedance below 500 Hz or so is barely affected. At high frequencies, C1 rapidly decreases until it evectively becomes a short, putting R1 in parallel with the driver, S2 and limiting the absolute maximum impedance to 8.2 Ohms. In this case we don't reach 8.2 until well-past 20kHz but it would eventually reach that point once the woofer's impedance was high enough. Also past our point of concern. If we can limit the impedance from 6.6 Ohms to 7  Ohms then we have a much more stable impedance curve than before, and that's good enough.

Do I Need a Zobel?

That's a tricky question! So let's examine this woofer and it's output. If you wanted to cross it over at 4kHz I would think the Zobel was mandatory, however if you were going to set your crossover frequency at 2kHz or lower I would say not really. The LM-1 uses it, but the effect of the Zobel is small, and benefits the phase response so I leave it in. It is possible a very similar sounding LM-1 could be built without a Zobel and with different choices in the filters left behind. The best chart to look at to see if a Zobel matters in your circuit is the transfer function chart.

It is very rare, but not unheard of, that a tweeter needs a Zobel because their voice coils are relatively tiny and therefore don't have a lot of inductance. The most common exception to this rule is with ribbon tweeters. The ribbon itself is not inductive but the entire assembly often include matching transformers. Transformers are coils .... and coils are inductive... see where this goes? :)

The real point to the Zobel is to make things better in the area you need the filter to behave at it's best. That's usually up to about -20 to -30 dB. Beyond that if your slope isn't perfect we no longer really care. There's no audible difference between -60 and -67 dB for example.

An important consideration in choosing a Zobel or not is that they are not free. The more parts in a system, the more expensive, the more chances of failures or parts being out of specification. If this is a personal project, no problem, it's all experience. However if you are building for mass production eliminating unnecessary components is the final stage before committing a design to the factory. 

Placement

In most cases, you want to place a Zobel closest to the driver. Put anything else such as padding resistors, filters, etc. before it. While the order of serial components does not matter, the order of parallel components does. Leaving the Zobel last prevents unexpected consequences.

There is a rare exception, when you must equalize a driver (usually a tweeter) by adding inductance. In which case you want the equalizing circuit closest to the tweeter, then the Zobel, so the Zobel can also control the EQ's impedance. I'll write more about this later in a section on handling difficult tweeters. 

The Secret Uses of the Zobel

Many will rely on on-line calculators to determine the right values for a Zobel network, and that's fine, but be aware that the absolute values can be tweaked. The main benefit of this tweaking is to gently nudge the phase charts one way or the other, helping you get near-perfect matching between two drivers you otherwise might not have. This is where having a tool like Xsim to simulate your tweaks comes in super handy.

Exercises

The data for the Peerless 830991 is contained in the LM-1 XSim files here.  Feel free to take it and modify it to help you complete these exercises.

The Peerless 830991 has an Re of around 6.6 and Le of around 0.330mH. Try simulating this in Xsim using a resistor and coil in series. Compare your impedance curve with the red impedance curve, above. What's the biggest difference you see?

Try using an online-calculator to create a Zobel for this driver. Tweak the capacitor and resistor values. Can you do better than the on-line calculator?

Using the complete LM-1 schematics, compare the woofer response with and without the Zobel. Is it a big difference? Can you fix the LM-1 so it no longer needs a Zobel? What difficulties did you encounter?Pay attention to the phase matching as well as the frequency response.

Use XSim to calculate the power through the Zobel resistor if the amplifier is set to 100 Watts. This is wasted power. Is it worth it?

Stereophile - Data Part II

This is a follow up to a previous article on what makes a speaker great to the editors at Stereophile.

I thought that was the end of that discussion, but thanks to an article written in 2008 but recently republished by the good Dr. Joseph D'Appolito there is more. Those who don't follow speaker design and measurement will not know D'Appolito literally wrote one of the most cited books on speaker measurements, in addition to having a configuration named after him.

In the article published by audioXpress D'Appolito shares an interaction with Stereophile head honch John Atkinson (JA). JA did something I though was pretty interesting mathematically, but I call bullshit on his message. He claims he analyzed a number of speakers and compared them to those which would make the recommended components according to frequency response and that most were perfectly neutral. D'Appolito states:

[John Atkinson] defined the standard deviation (SD) from flat response over the frequency range of 170Hz to 17kHz as a criterion for judging flatness of frequency response.

Further:

Of the 15 speakers with an SD of 1dB or less, 14 were added to the list by Stereophile reviewers.

But take a look at two speakers Stereophile raves about in my previous post. FAR from neutral as defined above. Then take a look at the hatchet job they did to the Crystal Cable Minissimo Diamond here.

So, bunk. I personally don't care what John Atkinson likes. If he likes the B&W diamonds above all others that's fine with me. But to call them neutral, or try to sell them as the reference against which other speakers are too dull or bright is shilling.

Crossover Basics - Driver Response

The Decibel or dB

Decibels (dBs) are a curious way to measure electrical and acoustic energy. Curious, and terribly convenient! For us, we use relative electrical dBs to discuss how filters work, and absolute acoustic dBSPL to measure speaker output.

When discussing the effects of a filter on a signal, we'll use relative dBs. That is, there's no set standard, but we talk about something being +4dB or -18 dB. This is useful because we can map this to speaker outputs no matter the volume settings. It is how we will discuss how a filter works, without worrying about the absolute output levels.

On the other hand, when we discuss the acoustical outputs we'll use dBSPLs which are in absolute terms, but using a set input level. Don't worry too much if this is confusing, we'll make it more clear as we go along.

The LM-1 Crossover Revisited

As mentioned in my first installment on Crossover Basics, the effects of a crossover filter are additive to the speaker driver.

We are going to use the LM-1 crossover and focus on the tweeter response in detail. Let's refresh your memory about the crossover, here it is on the left.


We'll focus on the tweeter filter section. This includes C1, L1, R1, and R2. The woofer section will seem neglected by comparison, but we cover it in more detail in other blog posts, including the Zobel.

Let's go over the transfer function. That is, how the voltage at the tweeter is different from the amplifier output because of the crossover. 0 dB means there was no change, the input and output are the same. The woofer response (in red, below) is almost exactly 0 dB until around 700 Hz when the low-pass filter kicks in. The tweeter on the other hand is more complicated. Let's discuss.

Anytime you see a chart this clean, you can be sure you are NOT looking at acoustical measurements. The blue line is tne tweeter filter's response. Except for the level shifting, this seems like something straight from my previous post on Crossover Basics. First, notice the level of the tweeter. It has been "padded" or "lowered" 6 dB below input. This is accomplished by the 4.2 Ohm R1. R1 is effective at all frequencies. Everything gets shifted down about 6 dB because of it. It's not exactly always constant, but let's pretend it is for right now, which is very close to true.

In addition to the padding there is a high pass filter reducing the midrange and bass at about 8 dB/octave below 2kHz. We discuss pads by an absolute number, like "6 dB" because it's effect is constant at all frequencies but we talk about high and low-pass filters with rates. In this case, 8 dB/octave means every time you cut the frequency in half, you will loose 8 dB. This is the actual "high-pass" section at work. This is C1,L1,R2. Notice that after about 4 kHz the high pass filter effectively stops working. It's as if it wasn't there anymore.  Above this level the only parts still involved in the high frequency response are the tweeter and R1. 

Putting it All Together

The point of this post is that these changes are not in isolation, but rather in combination with the driver so let's take a look at how the padding resistor and thigh high pass filter combine withe the acoustical response of the driver to produce the final outcome.

Notice the scale is now different. We are now looking at dBSPL, or sound pressure dBs. It is most common to take the frequency response measurements of a driver at 2.83 Volts input with the microphone at 1 meter distance. As you can see, below, this particular tweeter outputs about 90 dB at 2.83 volts above 4kHz or so.  2.83V is a common reference standard because at 8 Ohms this is about 1 Watt.

The top black line represents the tweeter with no filter at all. The green line represents the tweeter with just R1 added. It's not exactly 6 dB down everywhere due to the tweeter's impedance curve, but it's close enough for us! You'll learn more about this in the next post which covers the Zobel. The red line represents the addition of the high pass filter section, C1, L1 and R2. You can see it pivots around 3 kHz.

By carefully selecting the filter knee (-6dB point) and it's Q, or steepness we can get a little bit of EQ thrown in for free. Take a look at the original response (black) at around 2 kHz. You see the broad bump centered there? The bump is pretty much gone thanks to the high pass filter. We have not only added the high pass filtering, but we also tamed a little over-activeness int he tweeter without increasing the part count.

Padding

In the chart below you can see the final LM-1 design in red, vs. the a redesign without R1:



It may not be obvious from this, especially since this author likes to use far-field as his reference, but the LM-1 without padding would shriek.

In designing a crossover, I find it easiest to start low and work my way up. The low pass filter will reduce the sensitivity of the woofer at the crossover point. After this, we must adjust the tweeter to match and then add the high pass filter.

The total amount of padding (dB loss) depends on a number of things, including:

• Innate woofer efficiency
• Woofer low pass filter and baffle step compensation
• Innate tweeter efficiency
• Tweeter high pass filter

Unfortunately there is no simple, accurate way to go from a manufacturer's sensitivity specs to appropriate filter design.

The crossover designer must balance all four of these issues at the same time which is why in-cabinet measurement and simulation are so important. I encourage you to grab the LM-1 simulation files and attempt this for yourself.

Also note, that doing the reverse, padding the woofer, is generally discouraged because the power dissipation needed to lower a woofer a few dB is pretty large and requires big resistors and will waste a larger amount of amplifier energy. If your tweeter is too insensitive you probably need to change tweeter or woofer. It is pretty rare to find any design that does not require any tweeter padding.

Summary

With this posting, you now have learned:

• How crossover filter's add to driver output to create the combined effect of both. 
• How you can use leverage a high pass filter to also work as an EQ for you. 
• Why tweeters usually have resistors to pad them down. 

In my next post, Crossover Basics - The Zobel,  we'll go over the LM-1 woofer response but spend particular attention on the often misused or misunderstood circuit, the Zobel.

Cheers!