One of our AC source customers recently asked me to justify the reading on the front panel of one of our AC sources set to produce a sine wave with a DC offset. He had our 6812B AC Power Source/Analyzer set to a sine wave of 100 Vac (60 Hz) and added a DC offset of 50 Vdc. These AC sources can produce output voltages of up to 300 Vrms and DC voltages up to +/- 425 Vdc. With his settings of 100 Vac and 50 Vdc, the front panel meter was reading 111.79 V with the meter set to measure AC+DC. At first this seemed like an odd result to me, but then I realized that we are simply measuring the rms (root-mean-square) of the total waveform (AC plus DC) and that should be the square-root of the sum-of-the-squares of the individual rms values. This can be mathematically proven fairly easily. Since the AC source Vac is set in rms volts and the rms of DC is simply the DC voltage:
This works even if the DC value is set to -50 Vdc instead of +50 Vdc since the value is squared. And sure enough, when I set the AC source output to 100 Vac and -50 Vdc, the front panel measurement shows 111.82 as expected. The small variation in the measured value compared to the exact calculated value is due to the slight inaccuracies in both the output setting and measurement system.
So in summary, measurement of an rms waveform that consists of an AC signal plus a DC signal is the square-root of the sum-of-the-squares of the individual two values. It’s as simple as that!
