When a load is driven simultaneously by more than one source with each source being of its own frequency, the individual load power deliveries from those sources are independent of each other. Whatever power any one of those sources would provide to the load all by itself, that power delivery will not be affected by the presence of the absence of the other sources.

Imagine a stack of voltage sources connected in series and feeding into some load resistance, R. It could look something like **Figure 1**.

**Figure 1** A stack of voltage sources connected in series and feeding into some load resistance, R.

Of course, we could have more sources, say four, five or more, but three is a nice and convenient number. For the sake of discussion, we can meaningfully call the voltage from this stack of three a “triplet”. We further say of our triplet that each source is delivering its voltage at a different frequency. The frequency of the DC source is of course, zero.

The instantaneous power delivered to R is the instantaneous voltage at the top of the stack squared and then divided by R. The value of R is not of concern for now, so we will just look at that stack-top voltage which is our triplet.

When we square the triplet expression, we get several components per the following algebra in **Figure 2**.

**Figure 2** Squaring the triplet expression to obtain the instantaneous power delivered to R.

Just as a double check of this algebra to demonstrate equality, by choosing deliberately different frequencies W1 and W2, we can graphically plot the triplet squared and then plot the sum of the derived terms as shown in **Figure 3**. We see that they are indeed identical.

**Figure 3** Graphical check of squaring the triplet where, by choosing different frequencies (W1 and W2), we can graphically plot the triplet squared and plot the sum of the derived terms. From this, we can visually confirm that they are identical.

Getting back to the algebra, the results of squaring the triplet are shown above. The value of the first line is never negative, only positive, but the values of the second and third lines swing back and forth from positive to negative, to positive to negative, and so on and so on.

The energy delivered to R is the integral of the power over time. The integral for the first line is positive which means that R does indeed receive energy from the terms of that first line, __but__ the integrals of the second and third lines each come to zero. As time goes on, the positive swings of the second and third lines giveth while the negative swings of the second and third lines taketh away. Therefore, the integrals of those two lines come to zero which means that those two lines deliver __no energy__ to the load and no energy delivery means no power delivery.

Only the terms of the first line deliver power to R where that power is shown in **Figure 4**.

**Figure 4** The power delivery to R. As shown in the image, only the terms of the first line deliver power (to R).

The upshot of all this is that each voltage source of our triplet delivers as much power to R as it would deliver if it were connected to R all by itself. The power delivered by each source is independent of the presence or absence of each of the other sources.

If we’d had four sources or five sources or more, it wouldn’t matter. As long as their frequencies are not equal, the power deliveries of each source would still be independent of all of the others.

With more sources, the algebra would be more complex, but their independence of each other would remain the case.

*John Dunn** is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).*

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