<p style="text-align:justify"><span style="font-size:14px">A directional relay measures the angle between current and voltage vectors. Torque, in a directional relay, can be expressed as,</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(T\propto \phi_v\hspace{0.1cm}\phi_i\hspace{0.1cm}sin\hspace{0.1cm}\alpha\)</span>,</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">But as&nbsp;<span class="math-tex">\(\phi_i\hspace{0.1cm}\propto\hspace{0.1cm}I\)</span>,</span></p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(\phi_v\hspace{0.1cm}\propto\hspace{0.1cm}I_v\)</span>&nbsp;and&nbsp;<span class="math-tex">\(I_v\hspace{0.1cm}\propto\hspace{0.1cm}V\)</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(T\propto VI\hspace{0.1cm}sin\hspace{0.1cm}(90-\phi)\)</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(T\propto VI\hspace{0.1cm}cos\hspace{0.1cm}\phi\)</span>&nbsp;(Refer figure 7.23).</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16253871031625387103.jpg" style="height:652px; width:800px" /></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16253871251625387125.jpg" style="height:931px; width:800px" /></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The characteristic of a directional relay, using this torque equation, is shown in figure 7.24. In this figure current is shown as reference vector (X axis). Voltage vector can lie in any of the four quadrants. At an angle <span class="math-tex">\(\phi=0^0\)</span>,&nbsp;torque is maximum [for V = rated voltage of the relay (usually 110 V) and I = rated current of the relay (1 amp or 5 amp)]. This angle (which is zero in figure 7.24) is known as Maximum Torque Angle (MTA) or characteristic angle of the relay. The line on which the torque is maximum is known as Maximum Torque Line (M. T. Line). At&nbsp;<span class="math-tex">\(\phi=+\hspace{0.1cm}90^0\hspace{0.1cm}or\hspace{0.1cm}-\hspace{0.1cm}90^0\)</span>&nbsp;,&nbsp;torque is zero. The line on which torque is zero, is known as zero torque line. For any angle less than +90<sup>0</sup> and more than - 90<sup>0</sup>, the torque will be positive as shown by &ldquo;Tripping Zone&rdquo; in figure 7.24. For the angle&nbsp;&nbsp;<span class="math-tex">\(\phi\)</span>&nbsp;&nbsp;more than + 90<sup>0</sup> and less than - 90<sup>0</sup>,&nbsp;<span class="math-tex">\(cos\hspace{0.1cm}\phi\)</span>&nbsp; will be negative and hence the relay cannot operate. This zone is shown by &ldquo;Blocking Zone&rdquo; in figure 7.24. Thus, the relay measures angle between V and I, irrespective of magnitudes of V and I. Hence directional relay is a phase comparator relay.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Certain minimum voltage V<sub>min</sub> (shown in figure 7.24) is required at any angle &Oslash;&nbsp;(from - 90<sup>0</sup> to + 90<sup>0</sup>) to make the relay operative. As, if V = 0, torque will be zero and hence the relay cannot operate. At fault point, V = 0. Hence, if fault occurs immediately after relaying point (in an operative direction), voltage measured by the relay (V = I Z) is zero and hence the relay does not operate. This means that some minimum voltage V<sub>min</sub> is required to make the relay operative. In electromechanical relays, actuating mechanism has to compensate for frictional torque and a slight spring bias. Hence, this voltage is comparatively larger than that in case of static or numerical relays. The value of this voltage (at rated current and MTA) is known as directional sensitivity and is usually expressed in terms of % of V<sub>R</sub>, rated voltage. It is usually 1 % in electromagnetic relays.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">As V = I Z, minimum voltage requirement in directional units mean minimum impedance of line. This means that certain minimum length (as Z is proportional to line length) of line from relaying point is not protected, howsoever the direction of fault current is correct (operative). This length is known as dead zone as shown in figure 7.24.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Directional overcurrent relay is an AND logic of a directional relay and an overcurrent relay. Hence the relay operates only if direction of fault current is correct (operative) and current is more than plug-setting of the relay. If fault current flows in a direction contrary to operative direction, the relay does not operate even if current is more than its plug-setting.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">As such, the characteristic angle (or MTA),&nbsp;<span class="math-tex">\(\theta=0^0\)</span>&nbsp;(as shown in the figure 7.24) has no meaning. When fault occurs, the angle between V and I vector is to the tune of 70<sup>0</sup> to 90<sup>0</sup> depending on fault location. Hence, directional relays should provide maximum torque at this angle. Therefore, the MTA of the directional relay should be of the same order. Torque of the directional relay, in this case,</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(T\propto VI\hspace{0.1cm}cos\hspace{0.1cm}(\phi-\theta)\)</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16292797431629279743.jpg" style="height:555px; width:800px" /></p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16253872381625387238.jpg" style="height:627px; width:800px" /></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The characteristic of such a directional relay is given in figure 7.25(b). If&nbsp;<span class="math-tex">\(\theta\)</span>&nbsp;=&nbsp;90<sup>0</sup> and for a fault location such that&nbsp;<span class="math-tex">\(\phi\)</span>&nbsp;=&nbsp;75<sup>0</sup>,&nbsp;<span class="math-tex">\(T\propto VI\hspace{0.1cm}cos\hspace{0.1cm}15\)</span>.&nbsp;</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Hence about 96 % of maximum torque is available at relaying point. Therefore, usually MTA is adjusted at 90<sup>0</sup>. Exact angle can be calculated by using fault calculations discussed earlier and can be set in numerical relays. Wide range of adjustment is provided in numerical relays (in steps of 1<sup>0</sup>).</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">MTA can be set at 30<sup>0</sup>, 60<sup>0</sup> or 90<sup>0</sup> by suitable connections of CTs and PTs. As 90<sup>0</sup> connection is widely used, it will be discussed here. Quantities fed to directional relays in case of 90<sup>0</sup> connection are tabulated below:</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<table cellspacing="0" class="MsoTableGrid" style="border-collapse:collapse; border:none; margin-left:76px; width:900px">
	<caption>
	<p><span style="font-size:18px"><strong>Table 7.9: Quantities to be Fed to Phase Units of Direction Relays</strong></span></p>
	</caption>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:top; width:125px">
			<div style="text-align:center"><span style="font-size:28px">Relay</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:top; width:121px">
			<div style="text-align:center"><span style="font-size:28px">Current</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:top; width:104px">
			<div style="text-align:center"><span style="font-size:28px">Voltage</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; vertical-align:top; width:125px">
			<div style="text-align:center"><span style="font-size:28px">R phase</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:top; width:121px">
			<div style="text-align:center"><span style="font-size:28px">I<sub>R</sub></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:top; width:104px">
			<div style="text-align:center"><span style="font-size:28px">V<sub>YB</sub></span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; vertical-align:top; width:125px">
			<div style="text-align:center"><span style="font-size:28px">Y phase</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:top; width:121px">
			<div style="text-align:center"><span style="font-size:28px">I<sub>Y</sub></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:top; width:104px">
			<div style="text-align:center"><span style="font-size:28px">V<sub>BR</sub></span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; vertical-align:top; width:125px">
			<div style="text-align:center"><span style="font-size:28px">B phase</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:top; width:121px">
			<div style="text-align:center"><span style="font-size:28px">I<sub>B</sub></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:top; width:104px">
			<div style="text-align:center"><span style="font-size:28px">V<sub>RY</sub></span></div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<p>&nbsp;</p>

<p style="text-align:justify"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16253872981625387298.jpg" style="height:856px; width:800px" /></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The vector diagram showing 90<sup>0</sup> connection is illustrated in figure 7.26. The diagram is drawn with unity power factor consideration. As vector I<sub>R</sub> leads vector V<sub>YB</sub> (similar is the relation between I<sub>Y</sub> - V<sub>BR</sub> and I<sub>B</sub> - V<sub>RY</sub>) by 90<sup>0</sup>, the connection gets the name 90<sup>0</sup> connection. When fault occurs I<sub>R</sub> will lag V<sub>R</sub> by approximately 90<sup>0</sup> providing maximum torque [as I<sub>R</sub> (under fault condition) will be in phase with V<sub>YB</sub>].</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Now, if there is R-Y fault immediately after relaying point, R-phase and Y-phase units will sense a fault. For R-Y fault V<sub>RY</sub> = 0 and as fault has occurred immediately after relaying point, V<sub>RY</sub> measured by the relay is zero. But the voltages fed to R and Y units are V<sub>YB</sub> and V<sub>BR</sub> respectively which are not zero. (Similar argument holds good for Y-B fault and B-R fault also). Hence, there is no dead zone in case of L-L faults if 90<sup>0</sup> connection is used.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">A directional earth-fault unit is fed with residual current and open delta (residual) voltage. As V<sub>R</sub> + V<sub>Y</sub> + V<sub>B</sub> = 0 under normal condition, under the conditions of L-L-L faults and in case of L-L fault, directional earth-fault unit measures angle between the quantities fed to it only in case of L-g and L-L-g faults. In case of L-g faults, the residual voltage is equal to depression of voltage in faulted phase for effectively earthed systems. This voltage causes the earth-fault relay to operate if residual current is more than plug-setting of a relay and the phase angle between residual voltage and residual current is within the range &ndash;90<sup>0</sup> to + 90<sup>0</sup> (i.e., direction is operative) as discussed. In case of R-g fault immediately after relaying point, V<sub>Rg</sub> = 0. But voltage fed to relay (V<sub>R</sub> + V<sub>Y</sub> + V<sub>B</sub> = V<sub>Y</sub> + V<sub>B</sub>) is not equal to zero. Similar argument holds good for Y-g and B-g faults too). Hence there is no dead zone in case of L-g faults too. Dead zone is found only for L-L-L-g faults (0.2 % probability) in which case one has to depend on remote back-up relay only. The detailed protective scheme of directional overcurrent protection of transmission line is shown in figure 7.27.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16253873411625387341.jpg" style="height:726px; width:800px" /></span></p>