Saturday, December 17, 2016

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 Ohms 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?

5 comments:

  1. I wanted to drop you a line to thank you for writing this blog. It is a worthwhile contribution to the hobby.

    Regards, Jon

    ReplyDelete
  2. Thanks Jon, despite the page views, few ever take the time to leave a note!

    - N

    ReplyDelete
  3. Excellent explanation.Thank you for taking the effort to pull this together. This is possibly the best content to read and then move to spkr design books and relevant forums.

    ReplyDelete
  4. This is amazing, thank you. I have been looking at how to build and think of speaker design. Your posts have been helpful. I have not gone over all the posts, so I don't know if you ever provide a full math/engineering approach, but I look forward to it if you do.

    ReplyDelete
    Replies
    1. Thanks so much for your kind words.

      There are lots of good books and online sources for the math part. I wanted to kind of help bridge the gap between those with basic DC skills rather than cover the same ground. I will leave that hard stuff to them.

      Best,

      Nigel

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