Directional protection requires the setting of an appropriate Relay Characteristic Angle  (RCA) to define what direction the relay is "looking" to define half of the plane as the operating zone and the other half as the blocking zone.

The first training course I received on this back in 1982 provided a copy of "Sonnemann's paper" (and was referred to as such with highly revered status as you will see why).  It was published in an AIEE Transactions back in 1950 so the copy I was given itself seemed to be a "copy of a copy".  As you can see Sonnemann's paper gives an extensive mathematical analysis of various fault conditions and the different 90, 60 (#1 and #2) and 30 degree connections.

I would say "good luck with reading that in its entirety!!"  But try to get through it as much as you can.

However the subject deserves some additional explanation as follows which may "bridge the gap" / "fill in the missing pieces of the puzzle" for you, although in itself is not short either.

Definitions

There are two aspects to defining a Directional Relay application :

1. (Angle of the) Relay Connection
   The angle by which the current applied to the relay (at unity power factor) is relative to the voltage applied to the relay as the polarising voltage.
   e.g. Ia and Vbc is a 90° or Quadrature connection.
   
   
2. Maximum Torque Angle (MTA) / Relay Characteristic Angle (RCA)
   The angle by which the current applied to the relay must be displaced from the voltage applied to the relay to produce maximum torque
   e.g. If the setting is MTA/RCA = +30°, it means that it is operating on the MTA/RCA when the actual current is leading the polarising voltage by +30°.
   Hence, if it is a Quadrature connection Ia and Vbc,, when Ia is leading Vbc by +30°, therefore Ia is lagging Van by 60°, the system phase angle of Ia relative to Van is -60°  at MTA/RCA,. 

The discussion is in two parts - Phase Fault Overcurrent Relays and Earth Fault Relays.  Both sections use an example of a phase-to-earth fault.

Phase Fault Overcurrent Relay Application

(Angle of the) Relay Connection

I can recall in my early years being annoyed that this first term could easily and inadvertently be called “Relay Connection Angle” – whilst that is understandable, as an acronym, that can easily be confused with RCA being officially the Relay Characteristic Angle as the electronic relay equivalent of MTA!!

Considering this “Angle of the Relay Connection”, Sonnemann defines four possible connections of voltage and current to the relay.

The "old school" electromechanical rotating induction cup directional elements, e.g. the GEC CCD directional cup unit and CDD combined directional and overcurrent/earth fault, were physically "per phase" elements so you had to be very specific with the CT and VT wiring to get the right inputs to each element.

With modern multifunction devices with the 3 phase-to-neutral voltage connections and the three phase currents,  effectively every "connection" option is available to the directional element as required by setting. 

However if we consider what are the required inputs to a particular directional calculation for one phase, we can consider those inputs in the generic sense of a "connection". 

If we just consider the A‑phase directional overcurrent element, with

• zero degrees of the V_an phasor at the 12 o’clock position
  and
• unity Power Factor

 we get the following four connections to a directional element for the A phase.  

NOTE: at this stage we have not yet determined the required MTA/RCA for the required protection zone!

As most texts would agree, the most common and practical form is the 90° Quadrature Connection for various technical reasons of sensitivity as described by Sonnemann.  Whilst I have shown the direct physical connections for the various directional options, (numerical relays can make the changes by settings), the remainder of this dissertation is based on Quadrature Connection as per Figure 3 connections.

Maximum Torque Angle (MTA) / Relay Characteristic Angle (RCA)

MTA and RCA are the same thing.  You may find "oldies" like me refer to MTA, when we should really now be using RCA in most cases.

Maximum Torque Angle refers to electromechanical relays like the old GEC CCD and CDD where the cup mechanism literally rotated depending on whether the fault was seen in the operating zone.
The Torque refers to the force on the directional unit to swing one way or the other, which would then either allow or block the other relay function, i.e. a simple operating decision based on the fault being in the nominal range -90° to +90° of the MTA.  The “torque” was important as at low values of voltage, the operating zone would collapse inwards at the outer boundaries centred at the MTA.
However but even at say 80° from MTA, it would still swing in the ring direction.

RCA was introduced as the new term for electronic relays because there was no mechanical torque inside the relay.  This also means that the effect of low voltages on the ± degree range of the operating zone is much reduced. Nevertheless, the need to define the centre of the operating zone remains.

This second term of MTA/RCA is a little tricky in the wording used in the above definition:
“The angle by which the current applied to the relay must be displaced from the voltage applied to the relay to produce maximum torque”

In the Quadrature Connection, we know what voltage is applied, that is Vbc
So for a +30° RCA, we have to place a fault current phasor Ia’ at +30° relative to Vbc …. Note: it is NOT relative to Va
So MTA is when Ia’ is here

Figure 13

So what that means is the current Ia’ is effectively at 60 lagging from the Va phasor, i.e. it is 0.5 power Factor lagging

Figure 14

Hence MTA of +30° is achieved when the system Phase Angle = -60° i.e. phase current Ia lags phase voltage Vn by 60° i.e. at Power Factor 0.5 lagging.

There is some possibility of confusion due to the official MTA/RCA definition uses a current compared to a voltage, but it is not our usual sense of such relativity which is the phase angle of Ia with respect to Va,

This is somewhat evident in the text book “Protective Relays: Principles and Applications” by Blackburn and Domin ISBN 10-1-57444-716-5 and ISBN 13-978-1-57444-716-3.
In Table 3.1, connection number 5 as Ia and Vbc is referenced as a “90° -60°” connection
But you will see in the definitions in the preceding paragraph that the angle of -60° (and it is correctly a MINUS 60 in this definition) is
“the angle by which the system current lags the system voltage” 
i.e. they use PHASE ANGLE, not MTA/RCA to define the arrangement .. it is still an accurate identification of the arrangement but just using different definitions based on the usual terminology of current with respect to voltage as being the System Phase Angle from which we calculate Power Factor.

At the risk of being presumptuous of “redefining the entire world”, I would suggest to re-word the definition of MTA/RCA without reference to current, after all it is all about defining the location of the polarising voltage with respect to the applied voltage as follows:
“The angle by which the polarising voltage of the relay leads the applied voltage to the relay”

Figure 15

Now that would seem to make more sense that you get “maximum torque” when the connected current is in phase with Vpolarising.

Perhaps the following comparison examples will help clarify the terms.

Consider a Directional Overcurrent element with a 90° quadrature connection.
It has a certain pickup current magnitude that the red current phasor must exceed to operate .. that is the smaller semi-circular boundary.
The red phasor magnitude by definition cannot exceed the max fault current at that location .. that is the outer semi-circular boundary.
It also has a setting for the MTA/RCA .. the two examples are for +30° and +10° respectively (why 10° .. no reason, just to demonstrate!) .
The zones of operation are therefore bounded by ±90° of the respective MTA/RCA and referenced RELATIVE to the POLARISING VOLTAGE.
If the red current phasor falls with the green zone, the protection will operate.

However, as an alternative choice, if the polarising arrangement is not Quadrature, but Angle of Relay Connection is 30° (Ia and Vac), we get the same zones as follows by changing the MTA/RCA to suit:

If you compare the two blue background examples, you can see the same zone coverage for the two different Connections by choosing appropriate MTA/RCA to suit.

Similarly, the two fawn background examples are the same zone coverage according to the Connection Angle and the MTA/RCA to suit.

Earth Fault Relay Application

The trick there is that the zero sequence directional element needs V_o and I_n as the connected quantities.  Clearly there is no "angle of connection" as V₀ would be zero in a balanced load un-faulted condition.

The V_an voltage phasor will reduce in magnitude due to the fault to earth and shift slightly yielding the system voltage phasors and the resultant V₀ output of the delta winding VT as follows: 

Figure 26

With the extreme short circuit right at the VT terminals, Van would be zero so V₀ would be pointing straight down.

Also to note is that relative to the TF winding, I_a is flowing away from the TF and I_n is flowing towards the TF.  Hence In = –Ia.

I_n = 3 x I₀

Figure 27

The inputs to the directional earth fault element need to be V₀ and I₀

As per Figure 27, we can get V₀ from an open delta VT ... the output of that is by definition V_a + V_b + V_c = 3 x V₀

As for the zero sequence current, we can either use the direct neutral CT as per Figure 27, or the residual connection of the three line CTs connected in Holmgren, or a Core Balanced CT, or opposite of the A phase current since I₀ = I_n/3 =  -I_a/3.

In any case, we now see that the Zero Sequence Current as the directional element applied current is LAGGING the Polarising V₀ voltage.

Hence Earth Fault MTA/RCA would generally be expected to be NEGATIVE angle with respect to the Polarising quantity.

.

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