*Continued from article Sizing of power cables for circuit breaker controlled feeders (part 2)*

## 3. Criteria Starting and running voltage drops in cable

This criterion is applied so that the cross sectional area of the cable is sufficient to keep the voltage drop (due to impedance of cable conductor) within the specified limit so that the equipment which is being supplied power through that cable gets at least the minimum required voltage at its power supply input terminal during starting and running condition both.

Cables shall be sized so that the maximum voltage drop between the supply source and the load when carrying the design current does not exceed that which will ensure safe and efficient operation of the associated equipment. It is a requirement that the voltage at the equipment is greater than the lowest operating voltage specified for the equipment in the relevant equipment standard.

So before starting with calculation for voltage drop let us first analyze that what is the permissible voltage drop as per relevant standards and guidelines and what is the possible logic behind selecting these values as the permissible values.

* Indian standard 1255- CODE OF PRACTICE FOR INSTALLATION AND MAINTENANCE OF POWER CABLES UP TO AND INCLUDING 33 kV RATING* in its clause 4.2.3.4 mentions the permissible value for different cross sectional sizes of Aluminium conductor in volts/kM/Ampere for cables from voltage grade of 1.1kV till 33kV. Since we calculate voltage drop in terms of percentage of source voltage, this clause is not very widely used in basic as well as detailed engineering fraternity.

Its complex unit requires to be multiplied by cable length and ampacity. However one can definitely check for any cable size and length, what value is obtained in terms of percentage?

* IEEE standard 525 – Guide for the Design and Installation of Cable Systems in Substations* in its annexure C, clause number C3 mentions that Voltage drop is commonly expressed as a percentage of the source voltage. An acceptable voltage drop is determined based on an overall knowledge of the system. Typical limits are 3% from source to load center, 3% from load center to load, and 5% total from source to load. These values are indicated diagrammatically below.

**dV1**is the drop from source (Transformer) to load center (PCC) which should be less than or equal to 3%. Feeder connecting source to load center is also known as primary feeder.

**dV2** is the drop from load center (PCC) to individual loads which should be less than 3%. Feeder connecting load center to individual loads is also known as secondary feeder.

**dV2** = dV1 + dV2 is the total drop from source (Transformer) to load which should be less than or equal to 5%

### So far we have understood:

1. What are primary and secondary feeders?

2. What are the permissible values of voltage drop in cables for different types of feeder?

3. What are the governing standards for permissible voltage drop values?

*Now before proceeding further some fundamental question that should be asked is:*

Even though all the electrical equipments are rated for negative tolerance of 10% in voltage, and system voltage variation allowed is also 10% on negative side than why do we design the cable from source to load for a voltage drop of 5% maximum, what is wrong if the cable is also designed for voltage drop of 10%?

Well answer to this lies in the fact that there is a rule of thumb that 2 percent of voltage is lost at terminations and other points like cable joints in a circuit between the power source and the load. Such voltage loss are not indicated and accounted for in cable sizing calculation. The cable sizing calculation only considers the voltage drop in cable conductor from source to load. It is prudent to make certain that the designed voltage drop does not exceed 5% to avoid problems after installation.

It is much more costly to remove and replace an existing cable or piece of equipment that is under rated versus the cost of equipment and cables designed with a degree of extra size and avoid problems due to inadequate voltage at the load.

The NEC recommends or requires a maximum voltage drop of 5%, but realistically connection impedances, deterioration of terminals due to heat and age, etc; add resistance to the total circuit.

## Difference between voltage drop and voltage dip?

A voltage dip is a decrease in the magnitude of a supply voltage having the duration of some cycles to seconds. A voltage dip is a power quality problem which occurs due to:

Sudden change in the load, such as suddenly switching ON the large inductive load or any temporary fault in the utility side of the system and impedance of source (Transformer)

Voltage dip is a sort of transient negative side fluctuation of bus voltage which is experienced by all other loads connected to that bus, however it is caused by switching ON of any one single load of large magnitude. It is mainly experienced as a decrease in bus voltage due to starting of large motor. Since bus voltage decreases so other loads connected to that bus experience a fluctuation of voltage. We often come across this phenomenon at our home also when due to sudden switching ON of refrigerator or an air condition the voltage fluctuates.

Even in case of utility the addition of a large load will normally be scheduled with the utility so they can project the time of day that a load, such as an office or industrial plant, is turned on. Whereas the voltage drop is the drop in supply voltage before it reaches to the load. It is totally because of impedance of the connecting cable. It is because of this reason that for checking the adequacy of transformer MVA capacity and suitability of its percentage impedance that we conduct voltage dip calculation after sizing of transformer. Same can also be done by motor starting studies.

Now let us come back to the original topic that is voltage drop and its calculation. As we already know about the permissible values of voltage drop so let us calculate and derive an expression for the same in terms of impedance of cable, cable length and source voltage.

Let us consider a reference phasor as V. Direction of V as X axis and perpendicular to V as Y axis. Approximation OC = OF which is almost equal to OE as EF can be neglected because EF << OF

*X component of voltage drop:*

= Vdx = AE = AD + DE = AD + BG

= IRCosф + IX Sinф (Equation-1)

*Y Component of voltage drop:*

= Vdy = CE = CG-EG

= CG-BD

= IXCosф – IRSinф (Equation-2)

** X component of VS:**VSx = OE = √ (OC2 –CE2)

**VSx = √ VS2 – Vdy2** (Equation-3)

**V = OE –AE = VSx – Vdx** (Equation-4)

** Now Voltage drop Vd is:**Vd = VS – V = VS – (VSx –Vdx) (Putting the value of V from equation-4)

Vd = VS + Vdx – VSx

Vd = VS + Vdx – VS2 – Vdy2 Equation -5 (Putting the value of VSx from equation-3)

**Now substituting the values of Vdy and Vdx from equation-2 and equation-1 respectively:**

**Vd = VS + (IRCosф + IX Sinф) – √ (VS2 – (IXCosф – IRSinф)2** (Equation -6)

**Equation-6 is the final expression for voltage drop where:**

**VS** = the supply voltage

**I** = the load current

**R** = the resistance of cable conductor in Ohms/kM

**X** = the reactance of cable conductor in Ohms/kM

The above equation for voltage drop is recommended for exact calculation as per IEEE-241, Recommended Practice for Electric Power Systems in Commercial Buildings, clause number 3.6.1 and IEEE-141, Recommended Practice for Electric Power Distribution for Industrial Plants, clause number 3.11.1

Many consultants recommend the use of above formula for exact calculation of voltage drop in cables meant for power plants. However as per IEEE-525, Guide for the Design and Installation of Cable Systems in Substations, equation number C.2b of Annexure C recommends the use of following formula:

**Vd = IRCosф + IXSinф** (Equation-7)

Since cable length is usually expressed in meters so before substituting in above expression proper unit conversion should be done.

Sometimes multiple runs of cable are used so number of runs should come as division factor in above expression for equivalent resistance. Multiplying factor of √3 is to be taken for 3 phase system.

**Vdy**is negligible as compared to supply voltage.

That is we are neglecting the vertical component of both the inductive drop and resistive drop. So approximating **VS-Vdy** almost equal to VS the formula in * equation-6* will be reduced to formula in equation-7.

## Resistance of cable conductor

Resistance of cable conductor is calculated from resistivity value of conductor material at 20 C, which is a standard temperature for testing adopted by all cable manufacturers. Resistivity is concerted into resistance by following formula:

**Rdc = ρ X L / A**

*Where:*

**ρ** = Resistivity at 20 C

**L**= 1 kM length

**A** = Cross sectional area of conductor.

This resistance is DC resistance at 20C. It is converted to DC resistance at 90 C by the following conversion formula:

**Rt = R _{20} (1 + αT)**

*Where:*

**R _{20}** = Resistance at 20 C

**α**= Coefficient of linier expansion of Aluminium

**T**= Temperature at which resistance is to be calculated

For sizing of cables for AC system the resistance of conductor to be selected should be AC resistance at 90 C and not DC resistance. DC resistance is selected for sizing of cables for DC system like battery, battery charger etc….

A conductor offers a greater resistance to a flow of alternating current than it does to direct current. When the term “ac resistance of a conductor” is used, it means the DC resistance of that conductor plus an increment that reflects the increased apparent resistance in the conductor. This increment is mainly caused by:

### Skin effect

This results in a decrease of current density toward the center of a conductor. A longitudinal element of the conductor near the center is surrounded by more magnetic lines of force than is an element near the rim.

Thus, the counter-emf is greater in the center of the element. The net driving emf at the center element is thus reduced with consequent reduction of current density. In simple terms the current tends to crowd toward the outer surface.

### Proximity Effect

In closely spaced ac conductors, there is a tendency for the current to shift to the portion of the conductor that is away from the other conductors of that cable. This is called proximity effect. The flux linking the conductor current in one conductor is distorted by the current in a nearby conductor which in turn causes a distortion of the cross-sectional current distribution.

The above mentioned two factors are for increased resistance is generally expressed as the AC/DC resistance ratio. There are other magnetic effects can also cause an additional increase in AC/DC resistance ratios. However we are not going to discuss them in this article. ac/dc ratio is determined by skin effect factor and proximity effect factor.

**Rac = (AC/DC) ratio x Rdc**

For frequencies higher than 60 hertz, a correction factor for the values of resistance is applied as follows:

**x = 0.027678 √ f/Rdc**

*Where:*

**f** = frequency in hertz

**Rdc** = conductor DC resistance at operating temperature, in ohms per 1000 feet. The inductance of a multi-conductor cable mainly depends on the thickness of the insulation over the conductor.

## Inductive reactance of cable conductor

The inductive reactance of an electrical circuit is based on Faraday’s law. That law states that the induced voltage appearing in a circuit is proportional to the rate of change of the magnetic flux that links it. The inductance of an electrical circuit consisting of parallel conductors, such as a single-phase concentric neutral cable may be calculated from the following equation:

**XL = 2π f (0.1404 log S/r + 0.153) x 10 ^{-3}**

*Where:*

**XL** = Ohms per 1000 feet

**S** = Distance from the center of the cable conductor to the center of the neutral

**r** = Radius of the center conductor

**S** and r must be expressed in the same unit, such as inches.

Now, in technical articles part-2 and part-1 we had considered the sizing of cable for DOL motor feeder rated at 160kW supplied by 415V. Minimum required area was calculated as 3CX240 Sq mm Al, XLPE, however due to continuous current requirement the cable cross section required was calculated as 3CX300 Sq mm.

Now let us check the running and starting voltage drop for the same using exact * equation-6* as well as approximated

*.*

**equation-7**- Resistance of conductor of 3CX300 mm Sq Al, XLPE cable = 0.128 Ohms/kM (From manufacturers catalog)
- Reactance of conductor of 3CX300 mm Sq Al, XLPE cable = 0.071 Ohms/kM (From manufacturers catalog)
- Cable length = 150Mtr (assumed for this calculation)
- Running power factor of motor = 0.85
- Starting power factor of Motor = 0.3
- Starting current of motor = 6 times rated current

Assuming a drop of 1.5% in the cable for incomer feeder, that is from (source) to load center (PCC) which we have not calculated here for sake of simplicity and space limitation.

Modifying * equation-6* for proper units:

**L** = length of cable = 150 Mtr

**N** = Number of parallel runs of cable = 1

Substituting the values all the values in the above equation:

**voltage drop = 2.52%**from load center (PCC) to Motor.

Total running voltage drop from source to load = dV1 + dV2 = 1.5% + 2.52% = 4.02% which is < 5%.

**voltage drop = 11.4%**from load center (PCC) to Motor.

Hence total starting voltage drop from source to load = dV1 + dV2 = 1.5% + 11.4% = 12.9% which is < 15%.

Now we can verify the above obtained result by the approximate formula so that we can analyze the amount of approximation involved in using that formula.

Modifying equation-7 for proper units:

**L** = length of cable = 150 Mtr

**N** = Number of parallel runs of cable = 1

Substituting the values all the values in the above equation

**voltage drop = 2.5%**from load center (PCC) to Motor.

Total running voltage drop from source to load = dV1 + dV2 = 1.5% + 2.5% = 4.0% which is < 5%.

**voltage drop = 11.05%**from load center (PCC) to Motor.

Hence total starting voltage drop from source to load = dV1 + dV2 = 1.5% + 11.05% = 12.55% which is < 15%.

Hence we can see that even the approximate formula does give accuracy till one place of decimal and can be used. We can do a small case study by varying the cable length from 50 Mtrs to 150 Mtrs in steps of 15 Mtrs and analyze the difference in voltage drop by the use of two formulas.

No. | Cable Length | Exact Formula | Approximate Formula | ||

Running | Starting | Running | Starting | ||

1 | 50 | 2.35% | 5.20% | 2.35% | 5.18% |

2 | 65 | 2.56% | 6.35% | 2.61% | 6.29% |

3 | 80 | 2.80% | 7.47% | 2.86% | 7.39% |

4 | 95 | 3.10% | 8.60% | 3.12% | 8.50% |

5 | 110 | 3.30% | 9.70% | 3.37% | 9.60% |

6 | 125 | 3.63% | 10.00% | 3.63% | 10.70% |

7 | 140 | 3.90% | 12.10% | 3.88% | 11.81% |

8 | 150 | 4.02% | 12.90% | 4.05% | 12.55% |

Hence we can observer that voltage drop only after one place of decimal as obtained by exact formula is on lesser side where as approximate formula till the route length of 100 Mtrs gives voltage drop on higher side. For route length above 100 Mtrs both the formulas almost converge to give same value of running voltage drop.

Hence it is advisable to go for exact formula as far as possible however the approximate formula also gives the fairly accurate result.

With the completion of third and final criteria of voltage drop we come to the end of sizing of power cables for breaker controlled motor feeders supplied by 415V supply. With this methodology readers can develop a formulated excel sheet for sizing of power cables for circuit breaker controlled feeders.

###### References:

1. Electrical power cable engineering, edited by William A Thue, Publishers: MARCELD EKKER INC. NEW YORK

2. IEEE Red book

3. IEEE Grey book

4. IEEE-525

5. IEEE-835

6. Indian standard-1255 (second revision)