Headphones & amplifiers - output impedance, load impedance and frequency response

Oct. 20, 2020

Questions about headphones and headphone amplifiers are often related to the output impedance of the amplifier. Does high output impedance affect the sound of my headphone? Does an amp with low output impedance improve the sound?

When discussing headphone impedance it is easy to forget, that we are more interested in frequency response errors than low impedance. A headphone may sound a bit bass boosted with a 30-ohm headphone output compared to a "zero output impedance" amp, yet another is completely unaffected by output impedance. So the real question: what is the relationship between amplifier output impedance, headphone impedance response and response error?

Impedance

Impedance is similar to resistance, but extends the concept to alternating current circuits (resistance is a measure of the opposition of a circuit element to the flow of electric current). A low impedance load draws more current from a voltage source than a high impedance load. The output impedance of an amplifier determines how the voltage at the amplifier output changes with different loads. In headphone amplifiers a 10-50 ohm series resistor is placed after the amplifier for short circuit protection. The resistance of this resistor is the output impedance.

A perfect voltage source has an output impedance of zero ohms. An ideal load (so-called resistive load) has the same impedance at any frequency (impedance curve is a flat line).

Frequency response of a perfect voltage source is not affected by the load.
Frequency response of an ideal load is not affected by the source impedance.

Fortunately, a perfect voltage source ("zero ohm amp") or an ideal load is not required.

Impedance graph

Headphone impedance curves show wild variations: some headphones have a flat impedance curve and others have an uneven impedance curve with the resonance peak 3-5 times higher than the minimum. If the impedance is not constant but changes with frequency and the output resistance exceeds a limit, then the headphone's impedance response will affect the frequency response of the amplifier.

Headphone types and impedance graph

It seems that headphones with uneven impedance curve are mainly open dynamic headphones:

Closed dynamic headphones have almost flat impedance response.
(NAD Viso HP50, Sennheiser HD 280 Pro are exceptions).
Open dynamic headphones may have a flat or may have an uneven impedance response.
Planar magnetic headphones have flat impedance response.
(Audeze SINE is an exception).

(Note: old Innerfidelity measurement are available in the webarchive. I used some measurements from the archived database.)

A possible explanation for the difference: in closed headphones the ear pad is less porous, which result in less leakage, but the less porous ear pad comes with higher damping. In addition to that, small holes (vents) in closed headphones may add extra damping.

The two extremes: Sennheiser HD 598 (open) and Audio Technica ATH-M50x (closed) impedance response

Effect of output impedance on the frequency response

Many open dynamic headphones have a broad peak at the resonance frequency in the impedance curve. The amplitude of the peak can be five times larger than the nominal impedance. It means that the bass response of open headphones varies according to the value of the source impedance. An open headphone may sound 'bassy' with a high output impedance amplifier, while it may sound 'flat' with a low output impedance amp.

Amplifier responses with Sennheiser HD 598 open headphone
output resistance (Rout) as a parameter (normalized responses)

As I mentioned earlier headphones with flat impedance response don't show such variations. The Audio Technica ATH-M50x with a 80 ohm output resistance has only 0.3 dB more bass.

Damping factor: a game with numbers

Damping factor is the ratio of the load impedance and the source impedance. The relationship between the damping factor and the source impedance is very simple: lower the source impedance higher the damping factor. Damping factor is used excessively in speaker-amplifier calculations.

Damping factor is a game with numbers and there are better tools to describe a system response with variable load. First, we can calculate the (maximum) loss in decibels and decibel has a real acoustical meaning (which has more meaning: 0,83 dB loss or a damping factor of 10). In addition to that the damping factor is useless when a speaker or headphone has high mechanical or acoustical damping - which is the case with headphones with flat (or almost flat) impedance curve.

And there are other recurring problems with the damping factor: if a speaker has highly varying impedance curve, which would be the load impedance: the DC resistance, the nominal impedance (which is sometimes an arbitrary number) or the minimum impedance (similar to DC resistance).

It follows from the above that damping factor fails to describe many amplifier-speaker system response accurately and it does not allow us a better understanding of these system responses.

The 1/8 rule

As a rule of thumb the output impedance of the headphone amplifier may not exceed 12.5% (or 1/8) of the headphone nominal impedance. This guarantees that the error will be lower than 1 dB with any headphone. The problem with this rule is that it is an overkill for the vast majority of headphones. The Audio Technica ATH-M50x with a flat 38 ohm impedance curve will sound the same with a zero ohm source and with a 30 ohm source. So buying an amp with 5 ohm output impedance for this headphone is unnecessary (5 ohm is approx. 1/8 of 38 ohms). On the other hand, a Sennheiser HD-598 will sound different with source impedances higher than 10 ohms. The maximum recommended output resistance for the Sennheiser HD-598 is only 10 ohms.

Exact calculation of the loss, response error and output impedance

This section contains some math. Just skip the equations, if you are not interested in the calculations.

The frequency response of a headphone amplifier (at the headphone out) will be the result of the following:

Minimum impedance of the headphone
Maximum impedance of the headphone
Output impedance (output resistance, source resistance)

The min/max impedance and the output impedance clearly define the response error. At maximum impedance the voltage is highest and the loss is lowest. At minimum impedance the voltage is lowest and the loss is highest. The response error:

Loss at maximum impedance:

A = 20 * log (Z_max / (Z_max+Z_source))

Loss at minimum impedance:

B = 20 * log (Z_min / (Z_min+Z_source))

The frequency response error of the amplifier is the difference between the losses.

C = A - B

Of course we are interested in an output impedance value for a pre-defined response error (e.g. 1 decibel):

10^(dB/20) = Z_max * (Z_min+Z_source) / (Z_min * (Z_max+Z_source))
10^(dB/20) * Z_min / Z_max = (Z_min+Z_source) / (Z_max+Z_source)
Let A = 10^(dB/20) * Z_min / Z_max, then
A*Z_max+A*Z_source = Z_min+Z_source
A*Z_max - Z_min = Z_source - A*Z_source = (1-A)*Z_source
Z_source = (A*Z_max - Z_min) / (1-A)

The result:

Z_source = (A*Z_max - Z_min) / (1-A)
A = 10^(dB/20) * Z_min / Z_max
dB: response error

We can set the phase to zero in the previous calculations since at resonance frequency (Z_max) the phase of the impedance is zero and at Z_min the phase is close to zero (+-10 deg).

Unfortunately headphone specs don't specify the min/max impedance values, not to mention the max. recommended output impedance for a certain error.

Calculated output impedance values for some headphones

Maximum amplifier output impedance (R_amp) for 1 dB response error.

Open dynamic over-ear headphones:

Headphone	Z_min	Z_max	R_amp (1 dB)
Beyerdynamic DT-880	33	38	160
Focal Clear	63	330	10
Grado PS1000	33	58	11
Philips Fidelio X2	35	45	33
Sennheiser HD 598	61	284	10

Closed dynamic over-ear headphones:

Headphone	Z_min	Z_max	R_amp (1 dB)
Audio Technica ATH-M50x	37	39	x
Meze 99 Classics	20	23	100
Sennheiser HD 280 Pro	60	155	13
Sennheiser HD 569	30	37	41
Shure SRH840	44	56	45

The maximum possible response error with the Audio Technica ATH-M50x is 0.4 dB, so there is no a matching output impedance for 1 dB error.

Output impedance and nonlinear distortion

So far we have assumed that the headphone transducer motion is linear. But what happens when the speaker reaches its limits and the motion becomes nonlinear? Is there any benefit of a low impedance source?

There are many types of distortions in speakers, but we are interested in those that are 'reflected' in the speaker impedance. Changing the magnetic field or the suspension changes the 'back EMF' (voltage induced by the motion of the voice coil) and as a result the resonance peak in the impedance will be compressed and shifted upward in frequency. Nonlinear magnetic field and nonlinear suspension results in nonlinear speaker impedance around the resonance frequency.

Without any sophisticated analysis:

The output impedance has zero effect on the overall distortion if the headphone has a flat impedance response.
By applying the "one decibel limit" or the "1/8 rule", the effect of output resistance on nonlinear distortion can be ruled out.

After running a low frequency simulation of an 8-Ohm speaker with nonlinear magnetic field (B = B(x)):

Increasing the output resistance from zero, distortion in the voltage increases and distortion in the current decreases.
If a loudspeaker or headphone driver is driven by a perfect voltage source (output impedance is zero), distortion in the voltage is zero and distortion in the current is maximum. (This makes sense with a voltage source.)
If a loudspeaker or headphone driver is driven by a perfect current source (output impedance is infinite; in practice it means that output impedance is much higher than the load resistance), distortion in the current is zero and distortion in the voltage is maximum. (This makes sense with a current source.)
Neither current nor voltage is a good indicator of distortion performance when the output impedance is not zero or not infinite. Current is a good indicator only, if the driver is powered by a voltage source. For performance analysis complete acoustical simulation or measurement is required.

Summary

The standard 30 ohm output impedance (which is not a standard, just widely used in integrated audio) works well with the vast majority of headphones. Some open dynamic headphones may require amplifiers with output impedance values between 4 and 8 ohm, but reducing the output impedance below 4 ohms will not improve the response even for headphones with wildly varying impedance curve.

Csaba Horváth

This article was last modified on Dec 8, 2023.