# Relationship between frequency response, headphone impedance and amplifier output impedance

*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 of my headphone?

When discussing headphone impedance it is easy to forget, that we are more interested in frequency response errors than impedance. A 30 Ohm headphone output may have a huge effect on the response with 'headphone X', but little effect with 'headphone Y'. So the real question: what is the relationship between amplifier output impedance, headphone impedance response and the 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. Usually there is a 10-30 ohm series resistor after the headphone amplifier for short circuit protection, which defines the value of the output impedance.

A perfect voltage source has an output impedance of zero ohms. A perfect (so-called resistive) load has the same impedance at any frequency.

- The frequency response of a perfect source is not affected by the load.
- The frequency response of a perfect load is not affected by the source impedance.

Fortunately, we don't need a perfect source (zero ohm amp) or a perfect load.

### Impedance response of headphones

The impedance response of headphones shows 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 of a headphone 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. Or in other words: the frequency response of amplifiers with 'low' output resistance is not affected by the headphone impedance and the frequency response of headphones with flat impedance curve is not affected by the amplifier output resistance.

### Headphone types and impedance response

I felt that there is something more in the impedance curve, so I went to the deleted innerfidelity measurement page (available in the webarchive) and examined the impedance curve of some popular headphones and compared the different types. It seems that headphones with uneven impedance curve are mainly open dynamic headphones. I get these results:

- Closed dynamic headphones have almost flat impedance response.

Exceptions: NAD Viso HP50, Sennheiser HD 280 Pro. - Open dynamic headphones may have a flat or may have an uneven impedance response.
- Open dynamic headphones from Focal, Grado, Sennheiser have a peak in their impedance response.
- Open dynamic headphones from AKG, Audio Technica, Beyerdynamic, Philips, Shure have an almost flat impedance response.

- Planar magnetic headphones have flat impedance response.

Exception: Audeze SINE.

A possible explanation for the difference: in closed headphones the ear pad is less porous, which result in more bass boost (less leakage), but the less porous ear pad has higher acoustical resistance hence the damping is higher in closed headphones.

### 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 'bass heavy' with a high output impedance amplifier, while it may sound 'flat' with a low output impedance amp.

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 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 an 50 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 the 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.

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:

_{max}/ (Z

_{max}+Z

_{source}))

Loss at minimum impedance:

_{min}/ (Z

_{min}+Z

_{source}))

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

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:

_{source}= (A*Z

_{max}- Z

_{min}) / (1-A)

A = 10^(dB/20) * Z

_{min}/ Z

_{max}

dB: response error

We can do the above calculations with zero phase 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? The result is not what I expected.

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.

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

- When the speaker driven by a perfect voltage source (output impedance is zero), then the distortion in the voltage is zero and the distortion in the current is maximum. (This makes sense with a voltage source.)
- When the speaker driven by a perfect current source (output impedance is infinite), then distortion in the current is zero and the distortion in the voltage is maximum. (This makes sense with a current source.)
- Increasing the output resistance from zero the distortion in the voltage increases, and the distortion in the current decreases.
- The output impedance has insignificant effect on the total (electrical + mechanical) distortion. A low output impedance can't reduce nonlinear distortion in headphones and speakers.

The first two are evident (even without simulation), the third still doesn't need a complete acoustical simulation only electrical.

### 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 Horvath*