# Kirchhoff’s Voltage and Current Laws

A hypothetical AC-DC converter which violates Kirchoff’s Voltage Law.

Granted, a switch circuit can perform power conversion, and granted, it can be described as a switch matrix. How do we make the choices of switch operation? Consider first the simple circuit shown to the right – something we might try for AC-DC conversion. This circuit has problems. Kirchoff’s Voltage Law (KVL) tells us:

The sum of voltage drops around a closed loop is zero.

Imagine what will happen here if the switch is closed: the sum of voltages around the loop is not zero! in reality, a very large current will flow, and cause a large I X R (current times resistance) voltage drop in the wires. KVL will be satisfied by the wire voltage drop, but a fire might result! We would hope that fuses will blow or circuit breakers will trip instead. KVL should serve as a warning: Dont connect unequal voltage sources directly.

You are familiar with KVL as an abstract rule of circuit analysis, In power electronics, KVL takes on a much more concrete meaning. There is nothing to prevent someone from building the circuit above, but it will cause problems as soon as the switch is closed. This is the reality of KVL: We can try to violate KVL, but will not succeed. As you should realize, this means certain things about the switch matrix. We must avoid switching operations that connect unequal voltage sources. Notice that a wire, or a dead short, can be thought of as a voltage source with V=0, so the warning is a generalization for avoiding shorts across an individual voltage source.

A similar constraint holds for Kirchoff’s Current Law (KCL). KCL states that:

Currents into a node must sum to zero.

A hypothetical current source converter which violates Kirchoff’s Current Law.

When current sources are present in a converter, we must avoid any attempts to violate KCL. Consider the simple circuit shown to the right. If the current sources are different and if the switch is opened, the sum of the currents into the node will not be zero. In a real circuit, high voltages will build up and cause an arc to create another current path. This situation has real potential for damage, and a fuse will not help. Again, this is the reality of KCL: We can try to violate KCL, but will not succeed. We must avoid operating switches so that unequal current sources are connected in series. An open circuit can be thought of as a current source with I=0, so the warning applies to the problem of opening an individual current source.

Catastrophic results of attempting to violate Kirchoff’s laws.

In contrast to conventional circuits, in which KVL and KCL are automatically satisfied, switches do not ‘know’ KVL or KCL. If a designer forgets to check, and accidentally shorts two voltages of breaks a current source connection, some problem will result. The photograph to the right shows a power handling board which was destroyed when an error connected the input voltage source to a reversed voltage. The circuit breakers did not have time to react.

KVL and KCL problems when simple energy storage elements are added to a circuit.

There are some interesting implications of the current law restrictions when energy storage is included. Look at the circuit diagrams to the right. Both represent ‘circuit law problems’. The voltage source will cause the inductor current to ramp up indefinitely since $V = L \frac{di}{dt}$. We micht consider this to be a ‘KVL problem’, since the long-term effect is similar to shorting the source. In the next example, the current source will cause the capacitor voltage to ramp up toward infinity. This causes a ‘KCL problem’; eventually an arc will form to create an additional current path, just as if the current source had been opened. Of course, these connections are not a problem if they are only temporary. However, it should be evident that an inductor will not support a DC voltage and a capacitor will not support an DC current.

Circuit laws are not exclusively linked to problems. Both KVL and KCL provide very positive guidance about how to design a switch matrix. They provide excellent working rules for design. Often, all but one or two possibilities for switches and their operation can be identified simply through careful application of basic circuit laws. Let us return to the diode bridge example to illustrate. Some of the possible connections are shown below. It is easy to see that any diode combination that connects both of the left devices or both of the right devices in the same direction will attempt to violate KVL. Other combinations will not create and KVL or KCL problems, but the diodes will never allow any current to flow through the resistor load. Just a little more thought quickly reduces the switch arrangements to one combination: there is only one connection of the diode bridge that permits energy to flow without violating KVL.

Many of the possibilities for diode bridge directions are not useful.

KVL determines many effects of current interconnections. We have compared a half-wave rectifier with resistive and R-L loads previously. Consider a half-wave rectifier with a current source load, as shown below. KCL requires that we provide a current path for the output source. Since there is only one loop in the circuit, this path requires the diode to be on at all times. No switching action will ever occur, and the combination will not function as a power converter. The result is quite different when a diode bridge supplies the current source, since several loops can be formed.

Power electronics is perhaps the only major subject in which a designer must think carefully about obeying circuit laws!