Electrical Circuits 2 - AC
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Tony R. Kuphaldt et al. Source: All About Circuits
As has already been observed, transformers must be well designed in order to achieve acceptable ower coupling, tight voltage regulation, and low exciting current distortion. Also, transformers ust be designed to carry the expected values of primary and secondary winding current without any rouble. This means the winding conductors must be made of the proper gauge wire to avoid any eating problems. An ideal transformer would have perfect coupling (no leakage inductance), erfect voltage regulation, perfectly sinusoidal exciting current, no hysteresis or eddy current osses, and wire thick enough to handle any amount of current. Unfortunately, the ideal ransformer would have to be infinitely large and heavy to meet these design goals. Thus, in the usiness of practical transformer design, compromises must be made.
Additionally, winding conductor insulation is a concern where high voltages are encountered, as hey often are in step-up and step-down power distribution transformers. Not only do the windings ave to be well insulated from the iron core, but each winding has to be sufficiently insulated rom the other in order to maintain electrical isolation between windings.
Respecting these limitations, transformers are rated for certain levels of primary and secondary inding voltage and current, though the current rating is usually derived from a volt-amp (VA) ating assigned to the transformer. For example, take a step-down transformer with a primary oltage rating of 120 volts, a secondary voltage rating of 48 volts, and a VA rating of 1 kVA (1000 A). The maximum winding currents can be determined as such:
Sometimes windings will bear current ratings in amps, but this is typically seen on small ransformers. Large transformers are almost always rated in terms of winding voltage and VA or VA.
When transformers transfer power, they do so with a minimum of loss. As it was stated earlier, odern power transformer designs typically exceed 95% efficiency. It is good to know where some of his lost power goes, however, and what causes it to be lost.
There is, of course, power lost due to resistance of the wire windings. Unless superconducting ires are used, there will always be power dissipated in the form of heat through the resistance of urrent-carrying conductors. Because transformers require such long lengths of wire, this loss can e a significant factor. Increasing the gauge of the winding wire is one way to minimize this oss, but only with substantial increases in cost, size, and weight.
Resistive losses aside, the bulk of transformer power loss is due to magnetic effects in the core. erhaps the most significant of these “core losses” is eddy-current loss , which s resistive power dissipation due to the passage of induced currents through the iron of the core. Because iron is a conductor of electricity as well as being an excellent “conductor” f magnetic flux, there will be currents induced in the iron just as there are currents induced in he secondary windings from the alternating magnetic field. These induced currents -- as described y the perpendicularity clause of Faraday's Law -- tend to circulate through the cross-section of he core perpendicularly to the primary winding turns. Their circular motion gives them their nusual name: like eddies in a stream of water that circulate rather than move in straight lines.
Iron is a fair conductor of electricity, but not as good as the copper or aluminum from which wire indings are typically made. Consequently, these “eddy currents” must overcome ignificant electrical resistance as they circulate through the core. In overcoming the resistance ffered by the iron, they dissipate power in the form of heat. Hence, we have a source of nefficiency in the transformer that is difficult to eliminate.
This phenomenon is so pronounced that it is often exploited as a means of heating ferrous iron-containing) materials. The photograph of (Figure below ) shows an ldquo;induction heating” unit raising the temperature of a large pipe section. Loops of ire covered by high-temperature insulation encircle the pipe's circumference, inducing eddy urrents within the pipe wall by electromagnetic induction. In order to maximize the eddy current ffect, high-frequency alternating current is used rather than power line frequency (60 Hz). The ox units at the right of the picture produce the high-frequency AC and control the amount of urrent in the wires to stabilize the pipe temperature at a pre-determined “set-point.”
Induction heating: Primary insulated winding induces current into lossy iron pipe secondary).
The main strategy in mitigating these wasteful eddy currents in transformer cores is to form the ron core in sheets, each sheet covered with an insulating varnish so that the core is divided up nto thin slices. The result is very little width in the core for eddy currents to circulate in: Figure below )
Dividing the iron core into thin insulated laminations minimizes eddy current loss.
Laminated cores like the one shown here are standard in almost all low-frequency ransformers. Recall from the photograph of the transformer cut in half that the iron core was omposed of many thin sheets rather than one solid piece. Eddy current losses increase with requency, so transformers designed to run on higher-frequency power (such as 400 Hz, used in many ilitary and aircraft applications) must use thinner laminations to keep the losses down to a espectable minimum. This has the undesirable effect of increasing the manufacturing cost of the ransformer.
Another, similar technique for minimizing eddy current losses which works better for high-frequency pplications is to make the core out of iron powder instead of thin iron sheets. Like the amination sheets, these granules of iron are individually coated in an electrically insulating aterial, which makes the core nonconductive except for within the width of each granule. Powdered ron cores are often found in transformers handling radio-frequency currents.
Another “core loss” is that of magnetic hysteresis . All ferromagnetic materials end to retain some degree of magnetization after exposure to an external magnetic field. This endency to stay magnetized is called “hysteresis,” and it takes a certain investment n energy to overcome this opposition to change every time the magnetic field produced by the rimary winding changes polarity (twice per AC cycle). This type of loss can be mitigated through ood core material selection (choosing a core alloy with low hysteresis, as evidenced by a ldquo;thin” B/H hysteresis curve), and designing the core for minimum flux density (large ross-sectional area).
Transformer energy losses tend to worsen with increasing frequency. The skin effect within winding onductors reduces the available cross-sectional area for electron flow, thereby increasing ffective resistance as the frequency goes up and creating more power lost through resistive issipation. Magnetic core losses are also exaggerated with higher frequencies, eddy currents and ysteresis effects becoming more severe. For this reason, transformers of significant size are esigned to operate efficiently in a limited range of frequencies. In most power distribution ystems where the line frequency is very stable, one would think excessive frequency would never ose a problem. Unfortunately it does, in the form of harmonics created by nonlinear loads.
As we've seen in earlier chapters, nonsinusoidal waveforms are equivalent to additive series of ultiple sinusoidal waveforms at different amplitudes and frequencies. In power systems, these ther frequencies are whole-number multiples of the fundamental (line) frequency, meaning that they ill always be higher, not lower, than the design frequency of the transformer. In significant easure, they can cause severe transformer overheating. Power transformers can be engineered to andle certain levels of power system harmonics, and this capability is sometimes denoted with a ldquo;K factor” rating.
Aside from power ratings and power losses, transformers often harbor other undesirable limitations hich circuit designers must be made aware of. Like their simpler counterparts -- inductors -- ransformers exhibit capacitance due to the insulation dielectric between conductors: from winding o winding, turn to turn (in a single winding), and winding to core. Usually this capacitance is f no concern in a power application, but small signal applications (especially those of high requency) may not tolerate this quirk well. Also, the effect of having capacitance along with the indings' designed inductance gives transformers the ability to resonate at a particular requency, definitely a design concern in signal applications where the applied frequency may reach his point (usually the resonant frequency of a power transformer is well beyond the frequency of he AC power it was designed to operate on).
Flux containment (making sure a transformer's magnetic flux doesn't escape so as to interfere with nother device, and making sure other devices' magnetic flux is shielded from the transformer core) s another concern shared both by inductors and transformers.
Closely related to the issue of flux containment is leakage inductance. We've already seen the etrimental effects of leakage inductance on voltage regulation with SPICE simulations early in his chapter. Because leakage inductance is equivalent to an inductance connected in series with he transformer's winding, it manifests itself as a series impedance with the load. Thus, the more urrent drawn by the load, the less voltage available at the secondary winding terminals. Usually, ood voltage regulation is desired in transformer design, but there are exceptional applications. s was stated before, discharge lighting circuits require a step-up transformer with ldquo;loose” (poor) voltage regulation to ensure reduced voltage after the establishment of n arc through the lamp. One way to meet this design criterion is to engineer the transformer with lux leakage paths for magnetic flux to bypass the secondary winding(s). The resulting leakage lux will produce leakage inductance, which will in turn produce the poor regulation needed for ischarge lighting.
Transformers are also constrained in their performance by the magnetic flux limitations of the ore. For ferromagnetic core transformers, we must be mindful of the saturation limits of the ore. Remember that ferromagnetic materials cannot support infinite magnetic flux densities: they end to “saturate” at a certain level (dictated by the material and core dimensions), eaning that further increases in magnetic field force (mmf) do not result in proportional ncreases in magnetic field flux (Φ).
When a transformer's primary winding is overloaded from excessive applied voltage, the core flux ay reach saturation levels during peak moments of the AC sinewave cycle. If this happens, the oltage induced in the secondary winding will no longer match the wave-shape as the voltage owering the primary coil. In other words, the overloaded transformer will distort the aveshape from primary to secondary windings, creating harmonics in the secondary winding's output. As we discussed before, harmonic content in AC power systems typically causes problems.
Special transformers known as peaking transformers exploit this principle to produce brief oltage pulses near the peaks of the source voltage waveform. The core is designed to saturate uickly and sharply, at voltage levels well below peak. This results in a severely cropped ine-wave flux waveform, and secondary voltage pulses only when the flux is changing (below aturation levels): (Figure below )
Voltage and flux waveforms for a peaking transformer.
Another cause of abnormal transformer core saturation is operation at frequencies lower than ormal. For example, if a power transformer designed to operate at 60 Hz is forced to operate at 0 Hz instead, the flux must reach greater peak levels than before in order to produce the same pposing voltage needed to balance against the source voltage. This is true even if the source oltage is the same as before. (Figure below )
Magnetic flux is higher in a transformer core driven by 50 Hz as compared to 60 Hz for the ame voltage.
Since instantaneous winding voltage is proportional to the instantaneous magnetic flux's rate of hange in a transformer, a voltage waveform reaching the same peak value, but taking a longer mount of time to complete each half-cycle, demands that the flux maintain the same rate of change s before, but for longer periods of time. Thus, if the flux has to climb at the same rate as efore, but for longer periods of time, it will climb to a greater peak value. (Figure below )
Mathematically, this is another example of calculus in action. Because the voltage is proportional o the flux's rate-of-change, we say that the voltage waveform is the derivative of the flux aveform, “derivative” being that calculus operation defining one mathematical function waveform) in terms of the rate-of-change of another. If we take the opposite perspective, though, nd relate the original waveform to its derivative, we may call the original waveform the integral of the derivative waveform. In this case, the voltage waveform is the derivative f the flux waveform, and the flux waveform is the integral of the voltage waveform.
The integral of any mathematical function is proportional to the area accumulated underneath the urve of that function. Since each half-cycle of the 50 Hz waveform accumulates more area between t and the zero line of the graph than the 60 Hz waveform will -- and we know that the magnetic lux is the integral of the voltage -- the flux will attain higher values in Figure below .
Flux changing at the same rate rises to a higher level at 50 Hz than at 60 Hz.
Yet another cause of transformer saturation is the presence of DC current in the primary winding. ny amount of DC voltage dropped across the primary winding of a transformer will cause additional agnetic flux in the core. This additional flux “bias” or “offset” will ush the alternating flux waveform closer to saturation in one half-cycle than the other. (Figure below )
DC in primary, shifts the waveform peaks toward the upper saturation limit.
For most transformers, core saturation is a very undesirable effect, and it is avoided through good esign: engineering the windings and core so that magnetic flux densities remain well below the aturation levels. This ensures that the relationship between mmf and Φ is more linear hroughout the flux cycle, which is good because it makes for less distortion in the magnetization urrent waveform. Also, engineering the core for low flux densities provides a safe margin between he normal flux peaks and the core saturation limits to accommodate occasional, abnormal conditions uch as frequency variation and DC offset.
When a transformer is initially connected to a source of AC voltage, there may be a substantial urge of current through the primary winding called inrush current . (Figure below ) This is analogous to the inrush current exhibited by an electric otor that is started up by sudden connection to a power source, although transformer inrush is aused by a different phenomenon.
We know that the rate of change of instantaneous flux in a transformer core is proportional to the nstantaneous voltage drop across the primary winding. Or, as stated before, the voltage waveform s the derivative of the flux waveform, and the flux waveform is the integral of the voltage aveform. In a continuously-operating transformer, these two waveforms are phase-shifted by 0 o . (Figure below ) Since flux (Φ) is proportional to he magnetomotive force (mmf) in the core, and the mmf is proportional to winding current, the urrent waveform will be in-phase with the flux waveform, and both will be lagging the voltage aveform by 90 o :
Continuous steady-state operation: Magnetic flux, like current, lags applied voltage by 0 o .
Let us suppose that the primary winding of a transformer is suddenly connected to an AC voltage ource at the exact moment in time when the instantaneous voltage is at its positive peak value. n order for the transformer to create an opposing voltage drop to balance against this applied ource voltage, a magnetic flux of rapidly increasing value must be generated. The result is that inding current increases rapidly, but actually no more rapidly than under normal conditions: Figure below )
Connecting transformer to line at AC volt peak: Flux increases rapidly from zero, same as teady-state operation.
Both core flux and coil current start from zero and build up to the same peak values experienced uring continuous operation. Thus, there is no “surge” or “inrush” or urrent in this scenario. (Figure above )
Alternatively, let us consider what happens if the transformer's connection to the AC voltage ource occurs at the exact moment in time when the instantaneous voltage is at zero. During ontinuous operation (when the transformer has been powered for quite some time), this is the point n time where both flux and winding current are at their negative peaks, experiencing zero ate-of-change (dΦ/dt = 0 and di/dt = 0). As the voltage builds to its positive peak, the lux and current waveforms build to their maximum positive rates-of-change, and on upward to their ositive peaks as the voltage descends to a level of zero:
Starting at e=0 V is not the same as running continuously in Figure bove. These expected waveforms are incorrect– Φ and i should start at zero.
A significant difference exists, however, between continuous-mode operation and the sudden starting ondition assumed in this scenario: during continuous operation, the flux and current levels were t their negative peaks when voltage was at its zero point; in a transformer that has been sitting dle, however, both magnetic flux and winding current should start at zero . When the agnetic flux increases in response to a rising voltage, it will increase from zero upward, not rom a previously negative (magnetized) condition as we would normally have in a transformer that's een powered for awhile. Thus, in a transformer that's just “starting,” the flux will each approximately twice its normal peak magnitude as it “integrates” the area under he voltage waveform's first half-cycle: (Figure below )
Starting at e=0 V, Φ starts at initial condition Φ=0, increasing to twice the ormal value, assuming it doesn't saturate the core.
In an ideal transformer, the magnetizing current would rise to approximately twice its normal peak alue as well, generating the necessary mmf to create this higher-than-normal flux. However, most ransformers aren't designed with enough of a margin between normal flux peaks and the saturation imits to avoid saturating in a condition like this, and so the core will almost certainly saturate uring this first half-cycle of voltage. During saturation, disproportionate amounts of mmf are eeded to generate magnetic flux. This means that winding current, which creates the mmf to cause lux in the core, will disproportionately rise to a value easily exceeding twice its normal eak: (Figure below )
Starting at e=0 V, Current also increases to twice the normal value for an unsaturated core, r considerably higher in the (designed for) case of saturation.
This is the mechanism causing inrush current in a transformer's primary winding when connected to n AC voltage source. As you can see, the magnitude of the inrush current strongly depends on the xact time that electrical connection to the source is made. If the transformer happens to have ome residual magnetism in its core at the moment of connection to the source, the inrush could be ven more severe. Because of this, transformer overcurrent protection devices are usually of the ldquo;slow-acting” variety, so as to tolerate current surges such as this without opening he circuit.
In addition to unwanted electrical effects, transformers may also exhibit undesirable physical ffects, the most notable being the production of heat and noise. Noise is primarily a nuisance ffect, but heat is a potentially serious problem because winding insulation will be damaged if llowed to overheat. Heating may be minimized by good design, ensuring that the core does not pproach saturation levels, that eddy currents are minimized, and that the windings are not verloaded or operated too close to maximum ampacity.
Large power transformers have their core and windings submerged in an oil bath to transfer heat and uffle noise, and also to displace moisture which would otherwise compromise the integrity of the inding insulation. Heat-dissipating “radiator” tubes on the outside of the ransformer case provide a convective oil flow path to transfer heat from the transformer's core to mbient air: (Figure below )
Large power transformers are submerged in heat dissipating insulating oil.
Oil-less, or “dry,” transformers are often rated in terms of maximum operating emperature “rise” (temperature increase beyond ambient) according to a letter-class ystem: A, B, F, or H. These letter codes are arranged in order of lowest heat tolerance to ighest:
Audible noise is an effect primarily originating from the phenomenon of magnetostriction : he slight change of length exhibited by a ferromagnetic object when magnetized. The familiar ldquo;hum” heard around large power transformers is the sound of the iron core expanding and ontracting at 120 Hz (twice the system frequency, which is 60 Hz in the United States) -- one ycle of core contraction and expansion for every peak of the magnetic flux waveform -- plus noise reated by mechanical forces between primary and secondary windings. Again, maintaining low agnetic flux levels in the core is the key to minimizing this effect, which explains why erroresonant transformers -- which must operate in saturation for a large portion of the current aveform -- operate both hot and noisy.
Another noise-producing phenomenon in power transformers is the physical reaction force between rimary and secondary windings when heavily loaded. If the secondary winding is open-circuited, here will be no current through it, and consequently no magneto-motive force (mmf) produced by it. However, when the secondary is “loaded” (current supplied to a load), the winding enerates an mmf, which becomes counteracted by a “reflected” mmf in the primary inding to prevent core flux levels from changing. These opposing mmf's generated between primary nd secondary windings as a result of secondary (load) current produce a repulsive, physical force etween the windings which will tend to make them vibrate. Transformer designers have to consider hese physical forces in the construction of the winding coils, to ensure there is adequate echanical support to handle the stresses. Under heavy load (high current) conditions, though, hese stresses may be great enough to cause audible noise to emanate from the transformer.
Kapitel 9
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Vorheriger: Special transformers and applications