so that the peak current is for all phases and the instantaneous currents are Now the instantaneous powers in the phases are Using angle subtraction formulae: which add up for a total instantaneous power Since the three terms enclosed in square brackets are a three-phase system, they add up to zero and the total power becomes or showing the above contention. Again, using the root mean square voltage , can be written in the usual form . No neutral current For the case of equal loads on each of three phases, no net current flows in the neutral. The neutral current is the inverted vector sum of the line currents. See Kirchhoff's circuit laws. We define a non-dimensionalized current, : Since we have shown that the neutral current is zero we can see that removing the neutral core will have no effect on the circuit, provided the system is balanced. Such connections are generally used only when the load on the three phases is part of the same piece of equipment (for example a three-phase motor), as otherwise switching loads and slight imbalances would cause large voltage fluctuations. Unbalanced systems In practice, systems rarely have perfectly balanced loads, currents, voltages and impedances in all three phases. The analysis of unbalanced cases is greatly simplified by the use of the techniques of symmetrical components. An unbalanced system is analysed as the superposition of three balanced systems, each with the positive, negative or zero sequence of balanced voltages. When specifying wiring sizes in a three-phase system, we only need to know the magnitude of the phase and neutral currents. The neutral current can be determined by adding the three phase currents together as complex numbers and then converting from rectangular to polar co-ordinates. If the three-phase root mean square (RMS) currents are , , and , the neutral RMS current is: which resolves to The polar magnitude of this is the square root of the sum of the squares of the real and imaginary parts, which reduces to Non-linear loads With linear loads, the neutral only carries the current due to imbalance between the phases. Devices that utilize rectifier-capacitor front ends (such as switch-mode power supplies for computers, office equipment and the like) introduce third order harmonics. Third harmonic currents are in-phase on each of the supply phases and therefore will add together in the neutral which can cause the neutral current