The Flat Earth by Jefferson Bronfeld: In Search of Ferroresonance
By Jefferson Bronfeld, Cadick Corporation
Draw a line from Fort Lauderdale to Puerto Rico, and on to Bermuda, and you have traced a section of Atlantic known as the Bermuda Triangle, or the Devil's Triangle. Many unexplained and bizarre happenings have been attributed to the Bermuda Triangle. Ships throughout the centuries have disappeared as they sailed through the area, Christopher Columbus was supposed to have seen odd lights and flames on the water and unexplainable compass malfunctioning in 1492, and five Navy bombers mysteriously vanished "without a trace" in 1945, as did the rescue plane sent to search for them.
Luckily there is a huge difference between the unexplained, and the unexplainable.
Take for instance ferroresonance. Ferroresonance is a power system phenomenon characterized by the sudden onset of very high and sustained overvoltages (sometimes in excess of 4 per unit), with high levels of harmonic distortion. Transformers heat up and hum excessively, insulation breaks down, and protective relays mis-operate.
The phenomenon does not seem to occur regularly, or in response to a specific stimulus. The power system just seems to take a sudden non-linear jump from its normal state to one of sever harmonic distortion and several per-unit overvoltage. Laboratory modeling confirms ferroresonant circuits can respond in more than one way to the same initial response: some responses benign, and some extremely dangerous.
Real Vincent Price in a white lab coat nightmare stuff.
In order to understand ferroresonance, an extremely non-linear phenomenon, it is necessary to review regular old fashioned and well understood resonance in electrical circuits. Typically, the impedance of a circuit changes with frequency. Circuits can be intentionally designed to go have a maximum or minimum value of impedance at a certain frequency. Sometimes this happens unintentionally. This is resonance, and the frequency at which this occurs is known, oddly enough, as the resonant frequency.
Capacitive impedances drop with increased frequency, and inductive impedances rise. (This latter you should have remembered from my last column.) Resonance occurs when the inductive and capacitive reactances of a circuit exactly balance, and cancel each other out. At the resonant frequency, the impedance of a series combination of capacitance and inductance goes to a minimum, (theoretically goes to zero) and the current therefore goes through a maximum. In a parallel resonant circuit, the total impedance is a maximum at the resonant frequency, and for a given current flowing through the circuit, the voltage can be a maximum. In both series and parallel resonance, infinite voltage or infinite current are avoided by the fact that all real circuits also contain some pure resistance. In fact the introduction of resistance is used to mitigate the extremes of voltage or current at resonance. Why? Because the value of a pure resistance is the same at all frequencies.
Plain old resonant circuits exhibit regular and predictable responses to the applied voltage. A steady sinusoidal voltage will result in sinusoidal voltages and currents everywhere in the circuit. Applying twice, or half, the voltage will result in predictable and repeatable changes in the voltages and currents everywhere. Changing the frequency of the voltage will also change the magnitude of voltages and currents everywhere, in a very predictable manner, as described above. Even the circuit's response to transients, though somewhat more complicated, is at least theoretically predictable and repeatable.
Ferroresonance is like something else, something out of the Twilight Zone. There exist several stable responses to any given change in parameters, and the responses are often unpredictable. The resonant frequency can be different for different stable responses, so unlike regular old resonance, it's hard to know how the circuit will respond to frequency changes, or even if an experienced response is repeatable.
As you may have guessed, the key to ferroresonance is that the inductance in the circuit is ferromagnetic, meaning it has a ferromagnetic core.
Remember in the last column we talked about self-inductance? A current through a piece of wire causes a magnetic field. The ratio of magnetic field to current is known as inductance, which is measured in Henries, though the symbol is "L", and the phenomenon was discovered by Ampere. Go figure. Anyway the value of inductance is entirely determined by the geometry of the wire. Whether the wire is tightly coiled, the number of turns, the diameter of the turns, things like that.
It turns out another way to change the inductance of a coil, besides changing its geometry, is to introduce a ferromagnetic material, such as iron, into the core of the coil. You see, ferromagnetic materials have the property of causing an increase in the magnetic flux density, and therefore the magnetic induction. The amount of magnetic induction attributable to the iron in the center of a solenoid can be much larger than the induction associated with the current in the coil by itself. The ratio of magnetic field to current is changed. The value of inductance is changed.
The reasons for this go deep into the crystal structure of such materials, a subject that fascinates me. I cannot say I understand sufficiently. It has to do with the orientation of electron spin and the establishing of domains of similar spin with the material.
As the current in an iron core solenoid increases, the magnetic flux density increases. The ratio of magnetic flux density to current is a constant, a constant known as the inductance as I said earlier. The inductance is a higher value than that which would occur if there were no iron in the core. There are limits however. As the current is increased, a point is reached where further increases yield smaller and smaller increases in flux density. Its as if the iron cannot take it anymore. It has become saturated. Large increases in current no longer create large increases in flux density. At these extreme values of current, the ratio of flux density to current changes, it gets smaller: the inductance is smaller.
Figure 1 shows the magnetic flux density - current characteristic. The bottom curve indicates the flux, which occurs when current is increasing from a negative maximum value. The top curve indicates the flux, which occurs when the current is decreasing from a positive maximum. The two curves are not the same, that much is obvious. But the implication here is interesting: The magnitude of current that causes the iron to go into saturation is not the same as the magnitude of current at which the iron comes out of saturation. This is behavior is called hysteresis, and is due to residual flux density stored in the iron, which must be overcome when the current changes direction. This also explains why the iron may or may not saturate for the same value of applied current: it depends on the previous value of current. The response to a particular value of current is not always the same, it depends on the history of the circuit.
We now begin to see how this all fits together. Resonance is resonance. The capacitive reactance cancels the inductive reactance and extreme values of voltage and or current can occur. But with ferroresonance, the value of inductance is not fixed. It depends on more than just the geometry of the inductive element, it depends on the degree to which the iron is saturated and it depends on the prior current history as well. Is the present value of current the result of rising from a lower current or dropping from a higher current? Is the iron going into saturation or coming out of saturation. And don't forget, a change in inductance means a change in the frequency at which resonance will occur.
In a power system, a transformer is an excellent ferromagnetic inductance. A transformer can and does saturate at high voltages. When it does so, its inductance changes. With the change in inductance there is a change in the frequency at which the power circuit may resonate. Previously benign harmonic frequencies suddenly can be suddenly amplified and cause excess voltage. This excess voltage drives the transformer further into saturation, thus making the resonance and the overvoltages a sustained, stable phenomenon. Magnetic hysteresis further explains how more than one stable response is possible from the same values of voltage and current. The system is very sensitive to initial condition of the iron, as determined by its recent history.
If this occurred randomly without warning it would be quite disconcerting, and probably the subject of late night radio. However, with our understanding of ferroresonance we can identify, at least, the conditions under which it is likely to occur. These are:
- A sinusoidal voltage source - A power system generator for example.
- Ferromagnetic inductances - Power system transformers or instrument transformers.
- Capacitance - Installed power system capacitors or line to ground capacitance of transmission lines and underground cable.
- Low resistance - this can be lightly loaded power system equipment, unloaded transformers, low source impedances, or low circuit losses.
- The existence of at least one point in the system whose potential is not fixed - an isolated ungrounded neutral, a blown fuse on one phase, for example.
At the very least we don't have to leave it unexplained, or resort to mystical pseudoscientific half way explanations, (like that the earth is flat).
Until next week, stay away from the edge, oh and ummm... perhaps the Bermuda Triangle as well.
Questions? Comments? Send them to editor@electricnet.com.
Jefferson Bronfeld is the Chief Engineer for Cadick Corporation, an electric power engineering consulting and training company. A licensed professional engineer with two patents, Jeff has over 19 years experience in the power industry. Jeff makes his home in Binghamton, NY, where, when not working, Jeff is active in the IEEE, and tutors mathematics to local junior high and high school students. When not being a nerd Jeff can be found playing fiddle tunes on the mandolin, fly fishing, or asking friends to take him sailing on their boat. Jeff's column appears every Monday on ElectricNet. (Back to top)