The IDMT relay, or rather IDMT element of a relay is the fundamental to the protection of all power systems in various different applications.  Certainly below 66 kV it is extremely common as the principle protection system (a.k.a. but a poor choice of term "primary" protection as all protection protects the power system primary - the reference is between primary and back up form of protection)  

IDMT curves have benefits for matching the maximum time before tripping to the thermal withstand of the equipment, i.e.. to allow maximum duration of operation before you have to trip to prevent damage. 
An example of "special" inverse curve operation is motor protection where the IDMT curve is specifically set with two true thermal curves representing starting the motor from cold, or starting or running the motor "hot" after it has been running for some time.  These motor protection devices also incorporate negative phase sequence into the thermal model due to the much lower thermal withstand and much higher damage caused by NPS.
Another example is matching the thermal withstand of transformers which are of course in general are permanently energised in a "hot" state.

However IDMT also serves to provide the fastest possible protection at any point whilst also providing graded "back up" protection for faults further downstream which could be seen by both relays as above their respective but different pick up settings.

Over the years (decades) the technology of IDMT operating devices has changed but the terminology often harks back to the original electromechanical technology pre 1980's, and hence can appear to be somewhat unusual.  

IDMT - what does it mean?

An instantaneous relay or a Definite Time relay has a fixed operating time once current exceeds the setting e.g. if the setting is "1", then regardless of whether the current is "2" or "20", the operating time is the same.

An Inverse Definite Minimum Time (IDMT) function has decreasing operating time if higher values of current occurs above the pick up setting.
i.e. if the setting is "1", it will have a very long operating time at low current above "1" and very fast operating time when the current is "20".
Therefore the inclusion of the term “Inverse” makes total sense.

The slope of that inverse nature is defined in several different characteristic terms, and which are may be slightly differently defined in different Standards.

A fuse has a purely inverse nature characteristic based on I²t thermal affect on teh wire melting. 
The so-called Extremely Inverse characteristic roughly corresponds to the fuse I²t characteristic, so we often find EI characteristics either side of fuses to achieve "good" grading.

But why “Definite Minimum Time for relays?
Well, consider that old electromechanical rotating disc relay … the laws of physics gets to a point where the disc just cannot spin any (appreciably) faster, no matter how much higher the current goes, so it has a Definite Minimum Time to operate.
This is in contrast to fuses where the wire just melts faster and faster as the current is higher. 
Because there are literally millions of such relays probably still in service, when electronic relays came about they kept the same DMT philosophy – and by coincidence that, in some respects, made things hugely easier for the electronics engineers and programmers.

So IDMT characteristics of all technologies will have inverse characteristic up to a certain current, and then just go definite time.

Hence the term : IDMT 

Plug setting, Pick up setting - what does that mean?

“plug” refers to the old electromechanical induction disc relays (e.g. CDG shown below) where the current setting was made by pulling out a metal plug/pin and inserting it in a different hole.

As technology changed, the Plug became rotary knobs, miniature “DIP” switches and then ultimately setting via a menu on the relay and/or via a communications port.
      

The current setting can be expressed in different ways:
e.g. consider a 5 A rated relay (for connection to a 5 A secondary rated CT)
If the electronic relay “knows” the CT ratio is say 200/5, it could be set as 300 A (primary)
Any technology relay could be set in terms of secondary amps e.g. set at 7.5 A (secondary)
It could be set in terms of % of rated current of the relay e.g. 150%
All three are the same setting, just expressed in different ways.

But if the CT ratio is actually 250/5, the setting would be 300 A, 6 A or 120% respectively.

Plug Setting Multiplier, Multiple of setting - what does that mean

PSM simply means the actual current magnitude with respect to the relay setting whatever technology the setting is made.
PSM= (Actual current)/Setting

If the setting is 300 A primary and actual current is 1200 A gives PSM = 4

If the CT is 200/5 and the setting is 7.5 A
1200/(200/5) = 30 A CT output
PSM = 30 / 7.5 = 4

If the setting is 150% of rated
1200/200 = 600%
PSM = 600% / 150% = 4

Time Dial Setting, Time Multiplier Setting

The Standards define operating curves at a certain magnitude of current with respect to the plug setting.
e.g. a Standard Inverse so-called 3-second curve has an operating time of 3 seconds at 10 x setting and 10 seconds at 2 x setting.

In the electromechanical relays, you could reduce that operating time by turning a dial on the relay.  The dial would move the backstop (the resting position of the disc with current below setting) towards the trip contact.  This reduced the amount of rotation, and hence time, it would take for the disc to rotate to operate the trip contact.  Since the speed fo teh disc was proportional to the current, the time dial value applied to the whole characteristic regardless of current magnitude.

The term "Time Dial Setting" ( TD or TDS) was also referred to as Time Multiplier Setting (TMS).  The value was from 0 to 1.  So a standard 3-second operating time at 10 times plug setting could be reduced to say 0.3 seconds with a TMS of 0.1.

Origin of the Types of Curves

Different Standards have similar but slightly different curve definitions.

Whilst the Standards have specific formulas to define the curves, it is useful to understand the origin of the various curves themselves.

Again harking back to electromechanical days before the Standards such as BS 142, the manufacturers created their proprietary induction disc movement which all operated same laws of physics to operate.  However they used their own coils, their disc material and their brake magnets.  As such their characteristics were different.
In order to provide some standardisation, the curves of the various manufacturers were "averaged" to establish the particular standard shape.. 
They were not based on formulas as such.
The accuracy bands along the curve reflected the various proprietary deviations that would still meet the Standards.

In the process of updating of Standards, the curves were converted into formulas as we know today.

Drawing IDMT or DT curves

Relay technology

Electromechanical devices inherently obey physical laws related to the analogue value of current directly.

The first electronic relays of course were electronic components (resistors, capacitors, inductors, transistors and diodes) working with analogue signals.
So you could calibrate easily because peak of a sinewave was 1.414 x the r.m.s.

So now the relay manufacturers have to apply various filters.
Some may have used the average over a half cycle (average of a full cycle is 0).
Some may operate on the basis of the peak signal.
Some may have used full wave rectified signal.
.

In the early 1980's along comes digital devices.  Digital can mean two different things

1. purely being positive or negative or being above or below a certain value (simple 1 or 0 at any instant)
2. creating a hexadecimal number representing the value at a certain sampling period - typically 80 or 96 samples per cycle, even faster for certain vendors and/or applications.
   Or they could operate on the r.m.s. calculated over half or full cycle based on thos esamples.

These digital relays used various means to provide their operating mechanism (no apology for being a "GEC man" in my early days - the other vendors had other types) e.g.:

• The revolutionary Micromho SHNB distance relay used a sequence comparator based on the relative sequence of the zero crossings of the current c.f. the polarising signal.
• The revolutionary "world's first microprocessor based relay" MCGG, and later KCGG for example used a logarithmic look up table to determine the operating time based on that hexadecimal number.
• The revolutionary  MBCH Transformer Differential used a "gap detector" with the analogue value given a value of "1" above the certain threshold magnitude and "0" below and then determining how long it was "0". 

In the late 1980's these digital devices also started to provide some form of communications interface to SCADA.

Then in the 1990's, digital went one step further  and became fully numeric where an actual algorithm formula (such as provided in IEC 60255‑151) to determine the operating time of the relay.

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