# IEEE 141 Fig 3-11 : Voltage difference between source and load ends of a cable

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One of the frequent electrical issues we have to deal with from time-to-time is the effect cable sizing has on loads, or in reverse on source voltages.

Naturally there is a reduction in voltage from the source end of a cable to the load end.  Overall we need to ensure that the voltages at either end are within acceptable regulatory or operating limits. These limits are in terms of the voltage magnitude, which for the purpose of these calculations are the phase-to-neutral voltages, i.e. the magnitude of Vsource and Vload in this diagram.

IEEE 141 is quite a useful document worth buying as it discusses many issues including this issue as shown in Figure 3-11

One of the other problems we face in the industry is sometimes being a little "free" with our terminology and hence leading to some confusion when others refer to what ever we are trying to describe e.g. my other posts regarding to the fact that there is no such thing as "negative Power Factor"!! But getting back to voltage drop ...
IEEE 141 is actually trying to estimate the relative magnitude of the phase-to-neutral voltage at source relative to the phase-to-neutral voltage at the load - this relativity they refer to as the "voltage drop" ... personally I would call this a "voltage difference" because I would argue that "voltage drop" is the voltage seen across the ends of the cable ... but who am I to argue the semantics of the Standard, as long as you understand what they really mean!!

The IEEE 141 diagram shows the relationship between the source voltage and the load voltage, i.e. for a given load voltage and a given load current at a certain angle (i.e. power factor), we can determine the voltage appearing across the cable.

The cable voltage can then be added to the load voltage vector to  determine the source voltage vector and hence its magnitude.

In general we are usually more interested in what is the resultant voltage at the load end of the cable is for a given source voltage, load current and cable impedance.
This is important to ensure the cable is sized appropriately to maintain the voltage at the load within regulatory and operational limits.
But that is "just" a matter of "reverse engineering" the maths

I am guessing that the IEEE 141 authors felt that the maths associated with this could be a little tricky to work out the relative magnitudes of the phase-to-neutral voltages at each end ... so they developed a formula as an estimate of the difference in magnitudes, i.e. the line above formula does say it is APPROXIMATE:

Approximate difference in magnitude of Vp-n souce c.f. Vp-n load = I x R x cos(Φ)   +  I x X x sin(Φ)

Obviously this implies you know the magnitude of the total load current, the impedance of the cable and the phase angle of the load.
Of course you can determine the phase angle if you know the Power Factor as measured at the load end. (Note this is not the angle or Power Factor measured at the source!)

As can be seen in the diagram this an approximation which is shown as a seemingly minor error.

The idea is that they have used this highly simplified formula to approximate the voltage appearing across the end-to-ends of the cable, but only so far as the estimate is restricted to the component that is in phase with the load voltage.
They then propose artificially rotating the actual source voltage back to be in phase with the load voltage where it would seem there is a small error, hence justifying the approximation.

However we can model the maths in a simple Excel spreadsheet and determine the exact difference in the magnitude of the source voltage and the load voltage as well as the voltage appearing end-to-end of the cable - you can download my version below

The default scenario (Vload = 1000 @0, Iload = 500 @-30, Rc = 0.001 ohms/m, Xc = 0.0015 ohms/m, 1000 m) yields these results
Vsource magnitude1851.6 V
The difference in magnitude of Vsource compared to Vload_ph-n is |Vsource| - |Vload|851.6 V
The voltage magnitude end-to-end across the cable is |Vcable|901.4 V
IEEE 141 Formula approximation of |Vsource| - |Vload| 808.0 Vi.e. 0.95 times actual Vdiff and therefore underestimating the voltage drop

There are a some things to note from this analysis

• The magnitude of the voltage across the two ends of the cable (i.e. if you connected a voltmeter to each end of the cable), is independent of the phase angle Φ of the current relative to the load voltage - try just changing the phase angle from -30, to 0 to +30 degrees, the voltage magnitude end-to-end is exactly the same.  You will see that this voltage does change its angle relative to the load voltage according to the angle of the current, but the V=IZ triangle is the same shape and magnitude.
• Depending on your load current, load angle and cable impedances, this IEEE 141 formula could be highly inaccurate and misrepresent the difference in voltage magnitudes measured at each end - again try just changing the phase angle from -30, to 0 to +30 degrees. i.e. if we consider the source voltage as the "reference", IEEE 141 voltage drop estimate of 808.0 V would suggest the load voltage will be 1043.6 V whereas it will actually be only 1000 V

My overall take on this is if you are going to use an approximation, you should also know the accuracy boundaries of that approximation based on the influencing factors.

... However it may just be easier to do the full calculation!!!

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