How much must efficiency improve to provide twice the maximum power output from a new supply that has the same dimensions as the old supply? The demand for increasing power while decreasing the costs and volume of power systems and their components leads to the need for a constantly increasing power density.

In many practical cases, a bigger output power is required for a given ambient temperature range and mechanical and thermal system design, including package type, cooling conditions, and other factors. Obviously, the maximum allowable power dissipations (Pd) for original and modified systems must be the same in both cases.

In general, system size, geometry, thermal design, and ambient temperature range all limit Pd. Assuming Pd is proportional to the system volume, we can write:

Pd1/V1 = Pd2/V2 (1)

where Pd1and Pd2 along with V1 and V2 are the maximum allowable power dissipations and volumes for power systems 1 and 2, respectively. Substituting into Equation 1 the expressions for Pd1 and Pd2 as functions of their corresponding efficiencies (η1 and η2) and output power levels (Po1 and Po2), we have:

(Po1/V1)*(1 – η1)/η1 = (Po2/V2)*(1 – η2)/η2 (2)

The terms p1 = (Po1/V1) and p2 = (Po2/V2) in Equation 2 represent the power densities of the systems. Solving Equation 2 with respect to η2, we obtain:

η2 = η1*p/\\[1 + η1*(p – 1)\\] (3)

Equation 3 makes it possible to calculate efficiency η2 needed to increase power density by a factor of p = (p2/p1) for a given initial efficiency η1. If, for example, power density needs to be doubled, i.e., p = 2, and the efficiency of the original system is 90%, i.e., η1 = 0.9, the new system efficiency η2 should be:

η2 = 0.9*2/\\[1 + 0.9*(2 – 1)\\] = 0.947 (4)

or 94.7%. In other words, an efficiency boost of 4.7% is needed. In general, based on Equation 3, the following expression for delta efficiency Δη = (η2 – η1) can be obtained:

Δη = η1*(1 – η1)*(p – 1)/\\[1 + η1*(p – 1)\\] (5)

According to Equation 5, delta efficiency Δη (the efficiency boost) needed to increase the power density of a power system by a factor of p is defined by its initial efficiency η1.

A set of curves can be constructed based on Equation 5 to simplify estimations of delta efficiency Δη for an initial efficiency range η1 from 80% to 100% and power density boost factors from unity to 2 (see the figure).