As I've mentioned in the past, Kirchhoff’s Laws hold the key to determining the source of power quality phenomena. They apply to nearly every PQ investigation I encountered. The few that didn’t were because something else was askew, such as a malfunction of the recording instrument or someone playing a hoax. It is worth having a working understanding of his current and voltage laws that apply to electrical circuits for many situations, including power quality ones.
German physicist Gustav Kirchhoff first published his voltage and current circuit laws in 1845. (Kirchoff also has well-known laws in thermodynamics, fluid dynamics and thermal emissions.) They followed the work of Georg Ohm and his law relating voltage, current and impedance. Kirchhoff’s Laws preceded works by James Clerk Maxwell with his four infamous, but not generally understood, Maxwell’s Equations. As someone once stated, “The microscopic Maxwell equations have universal applicability but are unwieldy for common calculations.”
Not so with Kirchhoff’s Laws, which can be applied individually or simultaneously to circuits. At a high level, they are merely an extension of the conservation of energy principle, stating that energy cannot be created or destroyed, just altered in its form. In the Kirchhoff’s Current Law (KCL), current at a node cannot be created or destroyed because all of the currents flowing into a node must sum to zero. As shown in Figure 1a, this would be that I1+I2+I3=0. This equality can also be expressed as all of the current entering the node must equal the current leaving the node, regardless of how many connections there are to the node. In Figure 1b, this would be described as I1 = –I2 + –I3. However, exiting and entering are merely a sign convention referring to the direction of current flow. If we initially assume it is flowing in and actually flowing out, we can use the negative sign to reflect this and all the math works out.
Kirchhoff’s Voltage Law (KVL) follows the conservation principle that the sum of all voltages developed by generators or consumed by loads in a closed loop should equal zero. The sign conventions used in the voltages are labeled positive where the current enters the load (Z for impedances are loads) and negative where it exits the load. For generators, the convention is that the current enters the negative terminal of the source or generator (G) and exits the positive terminal as it is supplying the power to the circuit. If we apply KVL on the simple circuit in Figure 2, Vg1 = Vz1 + Vz2. To get it in the zero-equality form initially used in the KCL example, it would be –Vg1 + Vz1 +Vz2 = 0, as we “walk” around the closed-loop circuit, adding up the voltage. Of course, in this simple example, all the currents are the same, Ig1=Iz1=Iz2=Ix, provided there are no other paths for the current, such as leakage.
In a typical rms voltage-variation event, such as a sag or swell, it is considered an ideal source with no internal impedance that would change the voltage output when the current changes. Those “real world” internal impedances are summed up with all of the impedances from the wires, transformers, breakers and any other equipment or component (Z1) between the voltage source (G1) and the load (Z2). If the load is simple resistive with a single operating mode (either on or off), then we can calculate what the voltage will be at the load by algebraically rearranging the previous equation to be Vz2 = Vg1 – Vz1.
Since we said that Vg1 is an ideal source, the voltage is always the same. The voltage drop across the source impedance, Z1, varies as the current Ix varies. With no current flowing, there is no voltage drop across Z1 so Vg1=Vz2. As the current gets larger, more voltage develops across Z1 and less is left for the load. When an electric motor starts up, its impedance is 6 to 10 times lower than when running steading state. Remember, Ohm’s Law can be stated as I = V/Z. A lower impedance will result in a much larger current flow. More current, more Vz1, which results in less Vz2, the typical voltage sag scenario. The motor actually starves itself for voltage when it starts up, which requires even more current to get things rolling.
Next month, I will look at applying the special K’s to other PQ phenomena such as harmonics, interharmonics, transients and even noise, though it gets a bit more challenging.