Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer. To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be (or should be) made as close to zero as possible. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine. The efficiency is the ratio of power dissipated by the load resistance to total power dissipated by circuit (which includes the voltage source's resistance of as well as ): Consider three particular cases: If , then Efficiency approaches 0% if the load resistance approaches zero (a short circuit), since all power is consumed in the source and no power is consumed in the short. Note, voltage sources must have some resistance. If , then Efficiency is only 50% if the load resistance equals the source resistance (which is the condition of maximum power transfer). If , then Efficiency approaches 100% if the load resistance approaches infinity (though the total power level tends towards zero) or if the source resistance approaches zero. Using a large ratio is called impedance bridging. Impedance matching A related concept is reflectionless impedance matching. In radio frequency transmission lines, and other electronics, there is often a requirement to match the source impedance (at the transmitter) to the load impedance (such as an antenna) to avoid reflections in the transmission line that could overload or damage the transmitter. Calculus-based proof for purely resistive circuits In the simplified model of powering a load with resistance by a source with voltage and source resistance , then by Ohm's law the resulting current is simply the source voltage divided by the total circuit resistance: The power dissipated in the load is the square of the current multiplied by the resistance: The value of for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of for which the denominator is a minimum. The result will be the same in either case. Differentiating the denominator with respect to : For a maximum or minimum, the first derivative is zero, so or In practical resistive circuits, and are both positive, so the positive sign in the above is the correct solution. To find out whether this solution is a minimum or a maximum, the denominator