ESP Motor and Cable Sizing - Horsepower, Amperage, and Voltage Drop

Overview

The motor and cable are the power delivery system for an ESP installation. The motor must provide sufficient horsepower to drive the pump at the required brake horsepower, and the cable must deliver adequate voltage and current from the surface to the motor without excessive losses.

Undersized motors lead to overloading, overheating, and premature failure. Undersized cables cause excessive voltage drop, reduced motor performance, and wasted energy. Both components must be sized together as part of an integrated electrical system design.

---

Motor Horsepower Requirements

Required Motor HP

The motor must deliver at least the brake horsepower demanded by the pump:

$$HP_{required} = \frac{BHP_{pump}}{\eta_{motor}} + HP_{protector}$$

Where:

• $BHP_{pump}$ = brake horsepower required by the pump
• $\eta_{motor}$ = motor efficiency (typically 0.80 to 0.92)
• $HP_{protector}$ = power consumed by the seal/protector section (typically 2-10 HP)

In most practical designs, the protector losses are small relative to the pump BHP and are accounted for in the motor selection margin.

Brake Horsepower from Pump

The pump BHP depends on the number of stages, flow rate, and fluid properties:

$$BHP_{pump} = N_{stages} \times BHP_{stage}(Q) \times SG_f$$

Where $BHP_{stage}(Q)$ is the power consumption per stage at the operating rate (from the manufacturer's curve), and $SG_f$ is the fluid specific gravity.

Alternatively, from hydraulic horsepower and efficiency:

$$BHP_{pump} = \frac{Q \times TDH \times SG_f}{3960 \times \eta_{pump}}$$

Motor Selection

Motors are available in discrete nameplate HP ratings. The selected motor must satisfy:

$$HP_{nameplate} \geq HP_{required}$$

Standard ESP motor sizes (HP): 15, 20, 25, 30, 40, 50, 60, 75, 100, 125, 150, 200, 250, 300, 400, 500, 600, 750, 1000.

---

Motor Load Factor

Definition

The load factor is the ratio of actual shaft load to the motor's nameplate rating:

$$LF = \frac{BHP_{pump}}{HP_{nameplate}}$$

Load Factor Guidelines

Most ESP motors have a service factor of 1.0 to 1.15, meaning they can sustain 100-115% of nameplate HP under certain conditions. However, operating above nameplate reduces motor life due to increased winding temperature.

Temperature and Load

Motor winding temperature rise is approximately proportional to the square of the load:

$$\Delta T \propto LF^2$$

A motor at 110% load generates approximately 21% more heat than at 100% load, significantly reducing insulation life.

---

Motor Amperage

Nameplate Current

ESP motors are rated at a specific voltage and amperage at full load. The relationship between power, voltage, and current for a three-phase motor:

$$HP = \frac{\sqrt{3} \times V_{motor} \times I_{motor} \times PF \times \eta_{motor}}{746}$$

Where:

• $V_{motor}$ = motor terminal voltage (V)
• $I_{motor}$ = motor current (A)
• $PF$ = power factor (typically 0.80 to 0.85 for ESP motors)
• $\eta_{motor}$ = motor efficiency (decimal)
• 746 = watts per horsepower

Rearranging for motor current at any load:

$$I_{motor} = \frac{746 \times HP_{actual}}{\sqrt{3} \times V_{motor} \times PF \times \eta_{motor}}$$

Current at Partial Load

At partial load, motor current does not decrease linearly with HP because magnetizing current remains roughly constant:

$$I_{partial} \approx I_{no\text{-}load} + (I_{full\text{-}load} - I_{no\text{-}load}) \times LF$$

Where $I_{no\text{-}load}$ is typically 30-50% of $I_{full\text{-}load}$ for ESP motors.

Typical Motor Parameters

Note: These are representative values. Actual parameters vary by manufacturer and motor series.

---

Motor Temperature Considerations

Heat Dissipation

The ESP motor is cooled by the well fluid flowing past the motor housing. The minimum fluid velocity past the motor for adequate cooling:

$$v_{fluid} \geq 1.0 \text{ ft/s (typical minimum)}$$

The annular velocity between the motor and casing:

$$v_{annular} = \frac{Q_{total}}{A_{annular}} = \frac{Q_{total}}{\frac{\pi}{4}(D_{casing}^2 - D_{motor}^2)}$$

Where all dimensions are in consistent units.

Temperature Rating

The motor winding temperature equals the ambient well temperature plus the temperature rise from internal losses:

$$T_{winding} = T_{ambient} + \Delta T_{rise}$$$$T_{winding} \leq T_{rating}$$

---

Cable Voltage Drop

Importance

The cable delivers three-phase power from the surface to the downhole motor. Due to the cable's electrical resistance, voltage is lost along the cable length. If the voltage reaching the motor terminals is too low, the motor cannot develop rated torque and will overheat.

Voltage Drop Calculation

For a three-phase cable, the voltage drop per 1,000 ft of cable:

$$\Delta V_{1000} = \sqrt{3} \times I_{motor} \times R_{cable}$$

Where:

• $\Delta V_{1000}$ = voltage drop per 1,000 ft (V/1000 ft)
• $I_{motor}$ = motor current (A)
• $R_{cable}$ = cable resistance per 1,000 ft per conductor (ohm/1000 ft)

The total voltage drop along the full cable length:

$$\Delta V_{total} = \Delta V_{1000} \times \frac{L_{cable}}{1000}$$

Where $L_{cable}$ is the cable length in feet (typically equal to or slightly greater than the pump setting depth).

Temperature Effect on Resistance

Cable resistance increases with temperature. The resistance at downhole temperature:

$$R_T = R_{77} \times [1 + \alpha (T - 77)]$$

Where:

• $R_{77}$ = resistance at 77 F (published catalog value, ohm/1000 ft)
• $\alpha$ = temperature coefficient of resistance (0.00214 per F for copper)
• $T$ = average cable temperature (F)

The average cable temperature is typically estimated as:

$$T_{avg} = \frac{T_{surface} + T_{bottomhole}}{2}$$

Cable Resistance by Size

Note: Resistance values are per conductor. Ampacity ratings are approximate and depend on cable construction, insulation type, and installation conditions.

---

Cable Size Selection

Selection Criteria

Cable size must satisfy two independent requirements:

1. Ampacity: The cable must carry the motor current without exceeding its thermal rating
2. Voltage drop: The voltage drop must be within acceptable limits

Ampacity Requirement

I_{cable\text{-}rating} \geq I_{motor} \times 1.10 \quad \text{(10% margin)}

Voltage Drop Requirement

The voltage drop should not exceed a percentage of the motor nameplate voltage:

$$\Delta V_{total} \leq \Delta V_{max}$$

Common criteria:

Minimum Cable Size

The minimum cable size is the larger (lower AWG number) of:

1. The size that satisfies the ampacity requirement
2. The size that satisfies the voltage drop requirement

$$AWG_{min} = \min(AWG_{ampacity}, AWG_{vdrop})$$

Where lower AWG numbers indicate larger cables.

Selection Workflow

---

Power Loss in Cable

Cable Power Loss

The electrical power dissipated as heat in the cable:

$$P_{loss} = 3 \times I_{motor}^2 \times R_T \times \frac{L_{cable}}{1000}$$

Where:

• $P_{loss}$ = power loss in cable (watts)
• $I_{motor}$ = motor current per phase (A)
• $R_T$ = resistance at temperature per conductor per 1,000 ft (ohm/1000 ft)
• $L_{cable}$ = cable length (ft)
• Factor of 3 accounts for three conductors

Converting to horsepower:

$$HP_{loss} = \frac{P_{loss}}{746}$$

Cable Loss as Percentage of Total Power

$$\%_{loss} = \frac{P_{loss}}{P_{total}} \times 100 = \frac{3 I^2 R_T L / 1000}{\sqrt{3} \times V_{surface} \times I \times PF} \times 100$$

Typical Cable Losses

Note: Values calculated at 77 F. At higher temperatures, losses increase due to higher resistance.

---

Surface Electrical System

Required Surface Voltage

The surface transformer or VSD must supply sufficient voltage to overcome the cable drop and deliver rated voltage to the motor:

$$V_{surface} = V_{motor} + \Delta V_{total}$$

VSD Considerations

When a Variable Speed Drive (VSD) is used:

• Voltage scales with frequency: $V_{out} \propto f / f_{base}$
• At reduced speed, motor voltage and current change
• Cable voltage drop changes with current

The VSD output voltage at any frequency:

$$V_{VSD} = V_{base} \times \frac{f}{f_{base}} + \Delta V_{cable}(I_f)$$

Where $f$ is the operating frequency and $f_{base}$ is the base frequency (typically 60 Hz).

System Efficiency

The overall electrical efficiency from surface to pump shaft:

$$\eta_{system} = \eta_{VSD} \times \eta_{cable} \times \eta_{motor}$$

Where:

$$\eta_{cable} = 1 - \frac{P_{cable\text{-}loss}}{P_{input}}$$

---

Design Example

Given Data

Calculations

Load Factor:

$$LF = \frac{150}{200} = 0.75 \quad \text{(optimal range)}$$

Average Cable Temperature:

$$T_{avg} = \frac{80 + 220}{2} = 150 \text{ F}$$

Cable Resistance at Temperature:

$$R_{150} = 0.175 \times [1 + 0.00214 \times (150 - 77)] = 0.175 \times 1.156 = 0.202 \text{ ohm/1000 ft}$$

Voltage Drop:

$$\Delta V = \sqrt{3} \times 83 \times 0.202 \times \frac{8{,}000}{1{,}000} = 232 \text{ V}$$

Voltage Drop Percentage:

$$\% = \frac{232}{1{,}500} \times 100 = 15.5\%$$

This exceeds the 5% criterion. The designer should either:

• Increase cable size to #1 AWG (reduces resistance)
• Select a higher-voltage motor (reduces current for same HP)

With #1 AWG Cable:

$$R_{150} = 0.139 \times 1.156 = 0.161 \text{ ohm/1000 ft}$$$$\Delta V = \sqrt{3} \times 83 \times 0.161 \times 8 = 185 \text{ V} \quad (12.3\%)$$

Still high. A higher voltage motor (e.g., 2,400 V, 52 A) would yield:

$$\Delta V = \sqrt{3} \times 52 \times 0.161 \times 8 = 116 \text{ V} \quad (4.8\%)$$

This meets the 5% criterion with #1 AWG cable.

Cable Power Loss (final design, 2,400 V motor, #1 AWG):

$$P_{loss} = 3 \times 52^2 \times 0.161 \times 8 = 10{,}461 \text{ W} = 14.0 \text{ HP}$$

---

Related Topics

• ESP System Design Overview -- Complete design workflow with motor and cable steps
• ESP Pump Performance -- Determining BHP requirements for motor sizing
• ESP Gas Handling -- Gas effects on pump load and motor sizing
• ESP Viscosity Corrections -- Viscous BHP for motor sizing
• Well Flow Overview -- Production rate targets driving ESP sizing

---

References

1. Takacs, G. (2009). Electrical Submersible Pumps Manual: Design, Operations, and Maintenance. Gulf Professional Publishing.

2. Baker Hughes. (2021). ESP Cable Engineering Handbook. Baker Hughes Technical Publications.

3. IEEE Standard 1580. (2010). Recommended Practice for Marine Cable for Use on Shipboard and Fixed or Floating Platforms. IEEE.

4. Brown, K.E. (1984). The Technology of Artificial Lift Methods, Vol. 2b. PennWell Books.

5. Neely, A.B. and Patterson, M.M. (1984). "Soft Starting the Submersible Pump." SPE-12484-MS, Formation Damage Control Symposium, Bakersfield, California.