Select load power, load resistance, source resistance, voltage magnitude or maximum matched-load power for an ideal resistive circuit.

Ideal resistive Thรฉvenin circuit. Maximum power requires matched resistance; this is not maximum efficiency.

Related Calculators

Maximum Power Transfer Formula

The calculator uses the load power equation for a voltage source with internal or source resistance in series with a load resistance.

PL = (VSยฒ ร— RL) / (RS + RL)ยฒ
  • PL = power delivered to the load, in watts (W)
  • VS = source voltage, in volts (V)
  • RL = load resistance, in ohms (ฮฉ)
  • RS = source resistance, in ohms (ฮฉ)

Maximum power transfer occurs when the load resistance equals the source resistance.

RL = RS
Pโ‚˜ax = VSยฒ / (4 ร— RS)
  • Pmax = maximum possible power delivered to the load, in watts (W)
  • RS = source resistance, in ohms (ฮฉ)
  • VS = source voltage, in volts (V)

Select the quantity to solve and enter its displayed known values. The missing value is found by rearranging the load power formula:

VS = sqrt(PL ร— (RS + RL)ยฒ / RL)
PL ร— RLยฒ + (2 ร— PL ร— RS - VSยฒ) ร— RL + PL ร— RSยฒ = 0
PL ร— RSยฒ + 2 ร— PL ร— RL ร— RS + (PL ร— RLยฒ - VSยฒ ร— RL) = 0
  • Power delivered to load: uses the direct power formula with source voltage, source resistance, and load resistance.
  • Source voltage: rearranges the power formula and takes the positive square root of the voltage magnitude.
  • Load resistance: solves a quadratic equation because the load resistance appears in both the numerator and denominator.
  • Source resistance: uses RS = |VS|โˆš(RL/PL) โˆ’ RL and requires a nonnegative result.

Common Unit Conversions for Resistance, Voltage, and Power

Quantity Unit Base unit equivalent
Resistance 1 kฮฉ 1,000 ฮฉ
Resistance 1 Mฮฉ 1,000,000 ฮฉ
Voltage 1 mV 0.001 V
Voltage 1 kV 1,000 V
Power 1 mW 0.001 W
Power 1 kW 1,000 W
Load condition Meaning Power transfer result
RL < RS Load resistance is too low Less than maximum
RL = RS Matched resistance Maximum power transfer
RL > RS Load resistance is too high Less than maximum

Example Problems

Example 1: Calculate power delivered to the load

Suppose the source voltage is 12 V, the source resistance is 6 ฮฉ, and the load resistance is 6 ฮฉ.

PL = (12ยฒ ร— 6) / (6 + 6)ยฒ
PL = 864 / 144 = 6 W

The power delivered to the load is 6 W. Since the load resistance equals the source resistance, this is the maximum power transfer condition.

Example 2: Calculate source voltage

Suppose the load resistance is 10 ฮฉ, the source resistance is 10 ฮฉ, and the power delivered to the load is 2.5 W.

VS = sqrt(2.5 ร— (10 + 10)ยฒ / 10)
VS = sqrt(100) = 10 V

The source voltage is 10 V.

FAQ

What is the maximum power transfer condition?

Maximum power transfer occurs when the load resistance equals the source resistance. In symbols, RL = RS. At that point, the load receives the greatest possible power from the source for that source resistance and voltage.

Is maximum power transfer the same as maximum efficiency?

No. At maximum power transfer, the load resistance equals the source resistance, so the load and source resistance dissipate equal power. That means the efficiency is 50% in the ideal resistive case. Maximum power transfer is useful when getting the most load power matters more than minimizing losses.

Why can solving for resistance give more than one answer?

The power formula contains resistance in a squared denominator, so solving backward for a missing resistance creates a quadratic equation. With source voltage and source resistance fixed, a positive load power below the maximum corresponds to two positive load resistances: one below the matched resistance and one above it. The calculator reports both. Inverting source resistance with load resistance fixed instead gives at most one nonnegative solution. The maximum occurs at the single point where RL = RS.