Complex power in an unbalanced three phase circuit.

To start, you need to know what that all means. Theres a few different methods to measure power, I’m going to teach you complex power. Your normal 240V wall outlet that you plug your toaster, your hair dryer, television all works on real power. Real power is measured in Watts (named after the Scottish Engineer James Watt 1736-1819). Complex power has units of Volt-Amperes. To define complex power you must first understand what a complex number is. A complex number is defined as the sum of a real part and an imaginary part and is denoted by z=x+iy (where i is the imaginary unit and is equal to sqrt(-1)). A complex number is a point in the complex plane. It has a magnitude which is calculated by sqrt(x^2+y^2) and a phase angle calculated by theta=arctan y/x

Complex power is a complex number and thus has the properties I described above. It is denoted by S=P+iQ where P is the active (real) power and Q is the reactive (imaginary) power. P is defined by the voltage parallel to the impedance multiplied by the current through the impedance multiplied by cosine of the phase angle between the current and the voltage so, P=Vicos(theta). Q is defined as the same thing but instead of multiplying cosine of the phase angle, we multiply by sine of the phase angle so Q=Visin(theta). Putting these two results together, we get complex power S= Vicos(theta) + iVIsin(theta). A balanced three phase circuit consists of three voltage sources connected to three load impedances that are equal in magnitude and phase. The transmission power lines that travel down the street and across the country are powered by a three phase generator. When you plug your toaster into the power point, you’re connecting it to a single phase circuit, that is, the main switchboard in your house converts the power lines from three phase to single phase for use. In an unbalanced circuit, the three load impedances are all different.

To calculate the current through the load impedance, you must follow Ohms Law which states that the Voltage is equal to Resistance times Current V=IR. Rearranging the formula yeilds I=V/R Once you have calculated the current, you can multiply it with the voltage and cosine of the angle between the voltage and the current to get the active power. Do the same thing with the sin of the angle to get the reactive power. Add them together in the form of S=P+iQ to get the complex power across that particular impedance. Repeat the above two more times to get the complex power for the other two impedances in the three phase load. You should now have S1 S2 and S3. Add them altogether to get the total complex power in the three phase system. The units of measurement will be Volt-Amperes.

This is an extremely cut down version of events, but the main bits are all there.

Hope you enjoyed this engineering lesson in Power and Machines.