Voltage Drop - Line Losses and Calculations

Overview

When several resistors are connected in series in a circuit, the total voltage divides among them. Voltage drop also occurs in the connecting cables between the voltage source and the consumer. This is because cables have resistance too.

Practical Impact:

The voltage drop on connecting cables represents energy lost as heat. This loss reduces the effective voltage available to the load and must be considered in circuit design.

Voltage Drop Calculation

The voltage drop across a cable is proportional to the current flowing through it and the cable's resistance.

Definition:

Loss voltage (voltage drop) is the voltage lost in the connecting cable due to its resistance.

Basic Voltage Drop Formula

\(\displaystyle U_V = I \cdot R_L\)

Formula symbols

\(U_V\) = voltage drop (loss voltage) in volts
\(I\) = current in amperes
\(R_L\) = line resistance in ohms

Effective Voltage at Consumer

The effective voltage available to the consumer is the supply voltage minus the voltage drop in the cable.

\(\displaystyle U_N = U - U_V = U - I \cdot R_L\)

Formula symbols

\(U_N\) = effective voltage at consumer (terminal voltage)
\(U\) = supply voltage from source
\(U_V\) = voltage drop in cables

Important:

If the voltage drop is too large, the consumer may not operate properly or efficiently. This is why proper cable sizing is critical in electrical installations.

Factors Affecting Voltage Drop

The voltage drop depends on the current and the line resistance. The line resistance, in turn, depends on the cable's material properties and geometry.

Line Resistance Dependence

The line resistance increases with:

Cable length: Longer cables have higher resistance
Resistivity of material: Different materials conduct differently

The line resistance decreases with:

Cable cross-section: Thicker cables have lower resistance

Line Resistance Formulas

Single Wire

\(\displaystyle R_L = \frac{\rho \cdot l}{A}\)

Outgoing and Return Line

\(\displaystyle R_L = \frac{\rho \cdot 2l}{A}\)

Formula symbols

\(\rho\) = specific resistance (resistivity) of the cable material
\(l\) = cable length in meters
\(A\) = cable cross-section in mm²

Combined Formula for Voltage Drop

Substituting the line resistance formula into the voltage drop formula gives the complete expression that directly relates voltage drop to current, cable length, and cross-section:

Comprehensive Voltage Drop Formula:

\(\displaystyle U_V = \rho \cdot \frac{2 \cdot l \cdot I}{A}\)

Interpretation:

This formula shows that voltage drop:

Increases with cable length \(l\)
Increases with current \(I\)
Decreases with larger cross-section \(A\)
Is proportional to material resistivity \(\rho\)

Worked Example

Calculate Voltage Drop for a 50m Cable

Given:

Cable length: \(l = 50\,\text{m}\)
Cable cross-section: \(A = 2.5\,\text{mm}^2\)
Current: \(I = 15\,\text{A}\)
Supply voltage: \(U = 230\,\text{V}\)
Cable material: Copper, \(\rho = 0.018\,\Omega\text{·mm}^2/\text{m}\)

Calculate voltage drop:

\(\displaystyle U_V = \rho \cdot \frac{2 \cdot l \cdot I}{A} = 0.018 \cdot \frac{2 \cdot 50 \cdot 15}{2.5} = 0.018 \cdot \frac{1500}{2.5} = 0.018 \cdot 600 = 10.8\,\text{V}\)

Calculate effective voltage at consumer:

\(\displaystyle U_N = U - U_V = 230 - 10.8 = 219.2\,\text{V}\)

Result: The voltage drop is \(10.8\,\text{V}\) (approximately \(4.7\%\) loss), leaving \(219.2\,\text{V}\) for the consumer. This is generally acceptable but approaches the typical limit of \(5\%\) voltage drop.

Minimizing Voltage Drop

Use Thicker Cables

Increase cross-section to reduce resistance and voltage drop

Shorten Cable Runs

Place power source closer to load when possible

Choose Good Materials

Use materials with low resistivity (copper better than aluminum)

Reduce Current

Use higher voltage for the same power (\(P = U \cdot I\))

Key Points

Voltage drop is caused by resistance in connecting cables
Basic formula: \(U_V = I \cdot R_L\)
Effective voltage at consumer: \(U_N = U - U_V\)
Voltage drop increases with cable length and current
Voltage drop decreases with larger cable cross-section
Typical acceptable voltage drop limit is \(3\%\) to \(5\%\)
Proper cable sizing prevents excessive losses and ensures equipment operates correctly

Quick Calculation

Use the voltage drop calculator to quickly determine cable losses for your installation:

Voltage Drop Calculator →