Capacitor microphones require a power source. This can be supplied either as a battery in the mike handle, or down the cable from the recorder.
In practice, you are likely to come across these three main methods of supplying power to the microphone.
The simplest method - you put a battery into the mike and switch it on. A microphone doesn't take much power and a 1.5V battery is usually fine. The ATR55 and MKE300 use this method. The advantages are low capital cost, and a well-defined power source spec under the control of the manufacturer. Until the battery runs down, that is, which is the disadvantage of this method, particularly as it takes a particular discipline to switch the mike off after use and to make sure you always have a fresh replacement to hand. This disadvantage is particularly severe in a professional context, so you will seldom find this powering method in higher-end mikes. The microphone will start to lose sensitivity (your signal will get softer) and may get more distorted or noisy as the battery runs down. It is nice to have some indication of a good battery on switch-on. This usually takes the form of a LED lighting momentarily on switch-on if the battery is good.
The microphone is powered from the recorder. Electret capsules lend themselves very well to this method, which was popularised by Sony Minidisc recorders in the 1990s, but is common on many consumer items, such as PC sound cards which use a variant of this on their inputs. You can read a technical explanation but essentially the advantage is you only have one battery to worry about, which is the recorder battery. The recorder already has a battery level indicator, and the extra drain of microphone is very small compared to the recorder load.
The disadvantage of this method is that the voltage and current levels are not defined in a specification. Older MD recorders tended to supply more power than Sony's recent HiMD range (at least the MZ-NH700 and RH1 I tested) and it is possible for the mic and recorder to be mismatched so the microphone does not perform at its best. The high signal handling capacity of the microphone is limited by the low plug-in-power voltage of 1.5 to 5V. This is not usually and issue with wildlife sound recording, but can easily be a problem with music recording, particularly for the concert taper. Because of these issues, some people use a separate battery box (see the technical explanation for details), giving better large signal handling at the cost of getting the disadvantages of a microphone battery to worry about.
Professional systems usually power the microphone using P48 phantom power, where the balanced signal cables (XLR pins 2 and 3) carry the +ve rail and the screen (XLR pin 1) the -ve. Unlike plug-in-power, this is specified in a standard (part of IEC 61938)
The standard also specifies the voltages and maximum current drains. Not all equipment is capable of supplying the maximum mic current drain of 10mA specified (the spec used to call for a maximum mic drain of 2mA) but this does not seem to be a problem for the sort of microphones commonly used in wildlife sound recording.
If you don't have equipment capable of supplying P48 power you can still use P48 mics, with an appropriate outboard phantom power supply. You can find details of how to go about doing that on this page. These usually are XLR through units, presenting the output as a male XLR, usually but not always without the P48 power, and a female XLR for the mic with P48 power. A small DC-DC converter inside generates the 48V from a 12V input or internal batteries.
As well as 48V P48 phantom power, IEC 61938 specifies a 12V variant, P12. There was in the past a 24V variant but this was not widely adopted. P48 is more common than P12, though the latter finds favour on some location equipment. You should only use P48 mics with 48V power, and P12 mics with 12V power. Some mics, like the Sennheiser K6 and K6P with the ME range, are specified to operate from 12V to 48V, so they can be used with P48, P24 or P12.
You may come across an older system called T-powering, which put the power supply in series with the signal cables. This is not compatible with P48 powering or anything else. You should make sure you match the T-powered mic with the corresponding T-powered power supply.
Some mic manufacturers and most valve microphones use a proprietary high voltage power supply, once again mic and supply need to be matched. Digital microphones (where the signal is digitised at the microphone and the digital signal is sent down the microphone cable) seem to be trending towards a low-voltage high-current phantom powering standard (AES 42-2001) using 10V at 250mA or more. This may become an issue in wildlife recording situations where long runs of cable of more tha 100m are sometimes encountered, due to voltage drop in the cable.
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