Patent-Invent: Alternating Current (AC) Calculations

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Alternating Current (AC) Calculations

Charles Proteus Steinmetz solved many of the problems associated with electricity generation and transmission using alternating current. Among others was a practical method for making calculations concerning alternating current circuits. This method was an example of using mathematical aids for engineering the design of machinery and power lines, so that the performance of the electrical system could be predicted in advance without the necessity of going through the expensive and uncertain process of building the system first and then testing it for its efficiency. Steinmetz developed a symbolic method of calculating alternating-current phenomena and in so doing simplified an extremely complicated and barely understood field so that the average engineer could work with alternating current. This accomplishment was largely responsible for the rapid progress made in the commercial introduction of alternating-current apparatus.

A few simple alternating current (AC) calculations follow:

Alternating currents are usually associated with alternating voltages. An AC voltage v can be described mathematically as a function of time by the following equation:

where

angular frequency

radians

Since angular frequency is of more interest to mathematicians than to engineers, this is commonly rewritten as:

where

f is the frequency in hertz.

The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak. Since the maximum value of sin(x) is +1 and the minimum value is -1, an AC voltage swings between +A and -A. The peak-to-peak voltage, written as V_P-P, is therefore (+A) - (-A) = 2 × A.

The size of an AC voltage can also be stated as a root mean square (RMS) value, written V_rms. For a sinusoidal voltage:

V_rms is useful in calculating the power consumed by a load. If a DC voltage of V_DC delivers a certain power P into a given load, then an AC voltage of V_rms will deliver the same average power P into the same load if V_rms = V_DC. Because of this fact rms is the normal means of measuring voltage in mains (power) systems.

To illustrate these concepts, consider the 240 V AC mains used in the UK (it should be noted that the UK is now officially 230 V +10% -6% but in reality voltages are still closer to 240 V than 230 V in most cases). It is so called because its rms value is (at least nominally) 240 V. This means that it has the same heating effect as 240 V DC. To work out its peak voltage (amplitude), we can modify the above equation to:

For our 240 V AC, the peak voltage V_P or A is therefore 240 V × √2 = 339 V (approx.). The peak-to-peak value V_P-P of the 240 V AC mains is even higher: 2 × 240 V × √2 = 679 V (approx.)

Note that non-sinusoidal waveforms have a different relationship between their peak magnitude and effective (RMS) value. This is of practical significance when working with non-linear circuit elements that produce harmonic currents.

The European Union (including the UK) have now officially harmonized on a supply of 230 V 50 Hz. However they made the tolerance bands very wide at ±10%. Some countries actually specify stricter standards than this for example the UK specifies 230 V +10% -6%. Most supplies to the old standards therefore conform to the new one and do not need to be changed.

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This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia Encyclopedia article "Alternating Current"