TRANSFORMERS
Mutual inductance and basic operation
Suppose we were to wrap a coil of insulated wire around a loop of ferromagnetic material and energize this coil with an AC voltage source:
As an inductor, we would expect this iron-core coil to oppose the applied voltage with its inductive reactance, limiting current through the coil as predicted by the equations XL = 2πfL and I=E/X (or I=E/Z). For the purposes of this example, though, we need to take a more detailed look at the interactions of voltage, current, and magnetic flux in the device.
Kirchhoff's voltage law describes how the algebraic sum of all voltages in a loop must equal zero. In this example, we could apply this fundamental law of electricity to describe the respective voltages of the source and of the inductor coil. Here, as in any one-source, one-load circuit, the voltage dropped across the load must equal the voltage supplied by the source, assuming zero voltage dropped along the resistance of any connecting wires. In other words, the load (inductor coil) must produce an opposing voltage equal in magnitude to the source, in order that it may balance against the source voltage and produce an algebraic loop voltage sum of zero. From where does this opposing voltage arise? If the load were a resistor, the opposing voltage would originate from the "friction" of electrons flowing through the resistance of the resistor. With a perfect inductor (no resistance in the coil wire), the opposing voltage comes from another mechanism: the reaction to a changing magnetic flux in the iron core.
Michael Faraday discovered the mathematical relationship between magnetic flux (Φ) and induced voltage with this equation:
The instantaneous voltage (voltage dropped at any instant in time) across a wire coil is equal to the number of turns of that coil around the core (N) multiplied by the instantaneous rate-of-change in magnetic flux (dΦ/dt) linking with the coil. Graphed, this shows itself as a set of sine waves (assuming a sinusoidal voltage source), the flux wave 90o lagging behind the voltage wave:
Magnetic flux through a ferromagnetic material is analogous to current through a conductor: it must be motivated by some force in order to occur. In electric circuits, this motivating force is voltage (a.k.a. electromotive force, or EMF). In magnetic "circuits," this motivating force is magnetomotive force, or mmf. Magnetomotive force (mmf) and magnetic flux (Φ) are related to each other by a property of magnetic materials known as reluctance (the latter quantity symbolized by a strange-looking letter "R"):
In our example, the mmf required to produce this changing magnetic flux (Φ) must be supplied by a changing current through the coil. Magnetomotive force generated by an electromagnet coil is equal to the amount of current through that coil (in amps) multiplied by the number of turns of that coil around the core (the SI unit for mmf is the amp-turn). Because the mathematical relationship between magnetic flux and mmf is directly proportional, and because the mathematical relationship between mmf and current is also directly proportional (no rates-of-change present in either equation), the current through the coil will be in-phase with the flux wave:
This is why alternating current through an inductor lags the applied voltage waveform by 90o: because that is what is required to produce a changing magnetic flux whose rate-of-change produces an opposing voltage in-phase with the applied voltage. Due to its function in providing magnetizing force (mmf) for the core, this current is sometimes referred to as the magnetizing current.
It should be mentioned that the current through an iron-core inductor is not perfectly sinusoidal (sine-wave shaped), due to the nonlinear B/H magnetization curve of iron. In fact, if the inductor is cheaply built, using as little iron as possible, the magnetic flux density might reach high levels (approaching saturation), resulting in a magnetizing current waveform that looks something like this:
When a ferromagnetic material approaches magnetic flux saturation, disproportionately greater levels of magnetic field force (mmf) are required to deliver equal increases in magnetic field flux (Φ). Because mmf is proportional to current through the magnetizing coil (mmf = NI, where "N" is the number of turns of wire in the coil and "I" is the current through it), the large increases of mmf required to supply the needed increases in flux results in large increases in coil current. Thus, coil current increases dramatically at the peaks in order to maintain a flux waveform that isn't distorted, accounting for the bell-shaped half-cycles of the current waveform in the above plot.
The situation is further complicated by energy losses within the iron core. The effects of hysteresis and eddy currents conspire to further distort and complicate the current waveform, making it even less sinusoidal and altering its phase to be lagging slightly less than 90o behind the applied voltage waveform. This coil current resulting from the sum total of all magnetic effects in the core (dΦ/dt magnetization plus hysteresis losses, eddy current losses, etc.) is called the exciting current. The distortion of an iron-core inductor's exciting current may be minimized if it is designed for and operated at very low flux densities. Generally speaking, this requires a core with large cross-sectional area, which tends to make the inductor bulky and expensive. For the sake of simplicity, though, we'll assume that our example core is far from saturation and free from all losses, resulting in a perfectly sinusoidal exciting current.
As we've seen already in the inductors chapter, having a current waveform 90o out of phase with the voltage waveform creates a condition where power is alternately absorbed and returned to the circuit by the inductor. If the inductor is perfect (no wire resistance, no magnetic core losses, etc.), it will dissipate zero power.
Let us now consider the same inductor device, except this time with a second coil wrapped around the same iron core. The first coil will be labeled the primary coil, while the second will be labeled the secondary:
If this secondary coil experiences the same magnetic flux change as the primary (which it should, assuming perfect containment of the magnetic flux through the common core), and has the same number of turns around the core, a voltage of equal magnitude and phase to the applied voltage will be induced along its length. In the following graph, the induced voltage waveform is drawn slightly smaller than the source voltage waveform simply to distinguish one from the other:
This effect is called mutual inductance: the induction of a voltage in one coil in response to a change in current in the other coil. Like normal (self-) inductance, it is measured in the unit of Henrys, but unlike normal inductance it is symbolized by the capital letter "M" rather than the letter "L":
No current will exist in the secondary coil, since it is open-circuited. However, if we connect a load resistor to it, an alternating current will go through the coil, in phase with the induced voltage (because the voltage across a resistor and the current through it are always in phase with each other).
At first, one might expect this secondary coil current to cause additional magnetic flux in the core. In fact, it does not. If more flux were induced in the core, it would cause more voltage to be induced voltage in the primary coil (remember that e = dΦ/dt). This cannot happen, because the primary coil's induced voltage must remain at the same magnitude and phase in order to balance with the applied voltage, in accordance with Kirchhoff's voltage law. Consequently, the magnetic flux in the core cannot be affected by secondary coil current. However, what does change is the amount of mmf in the magnetic circuit.
Magnetomotive force is produced any time electrons move through a wire. Usually, this mmf is accompanied by magnetic flux, in accordance with the mmf=ΦR "magnetic Ohm's Law" equation. In this case, though, additional flux is not permitted, so the only way the secondary coil's mmf may exist is if a counteracting mmf is generated by the primary coil, of equal magnitude and opposite phase. Indeed, this is what happens, an alternating current forming in the primary coil -- 180o out of phase with the secondary coil's current -- to generate this counteracting mmf and prevent additional core flux. Polarity marks and current direction arrows have been added to the illustration to clarify phase relations:
If you find this process a bit confusing, do not worry. Transformer dynamics is a complex subject. What is important to understand is this: when an AC voltage is applied to the primary coil, it creates a magnetic flux in the core, which induces AC voltage in the secondary coil in-phase with the source voltage. Any current drawn through the secondary coil to power a load induces a corresponding current in the primary coil, drawing current from the source.
Notice how the primary coil is behaving as a load with respect to the AC voltage source, and how the secondary coil is behaving as a source with respect to the resistor. Rather than energy merely being alternately absorbed and returned the primary coil circuit, energy is now being coupled to the secondary coil where it is delivered to a dissipative (energy-consuming) load. As far as the source "knows," it's directly powering the resistor. Of course, there is also an additional primary coil current lagging the applied voltage by 90o, just enough to magnetize the core to create the necessary voltage for balancing against the source (the exciting current).
We call this type of device a transformer, because it transforms electrical energy into magnetic energy, then back into electrical energy again. Because its operation depends on electromagnetic induction between two stationary coils and a magnetic flux of changing magnitude and "polarity," transformers are necessarily AC devices. Its schematic symbol looks like two inductors (coils) sharing the same magnetic core:
The two inductor coils are easily distinguished in the above symbol. The pair of vertical lines represent an iron core common to both inductors. While many transformers have ferromagnetic core materials, there are some that do not, their constituent inductors being magnetically linked together through the air.
The following photograph shows a power transformer of the type used in gas-discharge lighting. Here, the two inductor coils can be clearly seen, wound around an iron core. While most transformer designs enclose the coils and core in a metal frame for protection, this particular transformer is open for viewing and so serves its illustrative purpose well:
Both coils of wire can be seen here with copper-colored varnish insulation. The top coil is larger than the bottom coil, having a greater number of "turns" around the core. In transformers, the inductor coils are often referred to as windings, in reference to the manufacturing process where wire is wound around the core material. As modeled in our initial example, the powered inductor of a transformer is called the primary winding, while the unpowered coil is called the secondary winding.
In the next photograph, a transformer is shown cut in half, exposing the cross-section of the iron core as well as both windings. Like the transformer shown previously, this unit also utilizes primary and secondary windings of differing turn counts. The wire gauge can also be seen to differ between primary and secondary windings. The reason for this disparity in wire gauge will be made clear in the next section of this chapter. Additionally, the iron core can be seen in this photograph to be made of many thin sheets (laminations) rather than a solid piece. The reason for this will also be explained in a later section of this chapter.
It is easy to demonstrate simple transformer action using SPICE, setting up the primary and secondary windings of the simulated transformer as a pair of "mutual" inductors. The coefficient of magnetic field coupling is given at the end of the "k" line in the SPICE circuit description, this example being set very nearly at perfection (1.000). This coefficient describes how closely "linked" the two inductors are, magnetically. The better these two inductors are magnetically coupled, the more efficient the energy transfer between them should be.
transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 100 l2 3 5 100 ** This line tells SPICE that the two inductors ** l1 and l2 are magnetically "linked" together k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 1k .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end
Note: the Rbogus resistors are required to satisfy certain quirks of SPICE. The first breaks the otherwise continuous loop between the voltage source and L1 which would not be permitted by SPICE. The second provides a path to ground (node 0) from the secondary circuit, necessary because SPICE cannot function with any ungrounded circuits.
freq v(2) i(v1) 6.000E+01 1.000E+01 9.975E-03 Primary winding
freq v(3,5) i(vi1) 6.000E+01 9.962E+00 9.962E-03 Secondary winding
Note that with equal inductances for both windings (100 Henrys each), the AC voltages and currents are nearly equal for the two. The difference between primary and secondary currents is the magnetizing current spoken of earlier: the 90o lagging current necessary to magnetize the core. As is seen here, it is usually very small compared to primary current induced by the load, and so the primary and secondary currents are almost equal. What you are seeing here is quite typical of transformer efficiency. Anything less than 95% efficiency is considered poor for modern power transformer designs, and this transfer of power occurs with no moving parts or other components subject to wear.
If we decrease the load resistance so as to draw more current with the same amount of voltage, we see that the current through the primary winding increases in response. Even though the AC power source is not directly connected to the load resistance (rather, it is electromagnetically "coupled"), the amount of current drawn from the source will be almost the same as the amount of current that would be drawn if the load were directly connected to the source. Take a close look at the next two SPICE simulations, showing what happens with different values of load resistors:
transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 100 l2 3 5 100 k l1 l2 0.999 vi1 3 4 ac 0 ** Note load resistance value of 200 ohms rload 4 5 200 .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end
freq v(2) i(v1) 6.000E+01 1.000E+01 4.679E-02
freq v(3,5) i(vi1) 6.000E+01 9.348E+00 4.674E-02
Notice how the primary current closely follows the secondary current. In our first simulation, both currents were approximately 10 mA, but now they are both around 47 mA. In this second simulation, the two currents are closer to equality, because the magnetizing current remains the same as before while the load current has increased. Note also how the secondary voltage has decreased some with the heavier (greater current) load. Let's try another simulation with an even lower value of load resistance (15 Ω):
transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 100 l2 3 5 100 k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 15 .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end
freq v(2) i(v1) 6.000E+01 1.000E+01 1.301E-01
freq v(3,5) i(vi1) 6.000E+01 1.950E+00 1.300E-01
Our load current is now 0.13 amps, or 130 mA, which is substantially higher than the last time. The primary current is very close to being the same, but notice how the secondary voltage has fallen well below the primary voltage (1.95 volts versus 10 volts at the primary). The reason for this is an imperfection in our transformer design: because the primary and secondary inductances aren't perfectly linked (a k factor of 0.999 instead of 1.000) there is "stray" or "leakage" inductance. In other words, some of the magnetic field isn't linking with the secondary coil, and thus cannot couple energy to it:
Consequently, this "leakage" flux merely stores and returns energy to the source circuit via self-inductance, effectively acting as a series impedance in both primary and secondary circuits. Voltage gets dropped across this series impedance, resulting in a reduced load voltage: voltage across the load "sags" as load current increases.
If we change the transformer design to have better magnetic coupling between the primary and secondary coils, the figures for voltage between primary and secondary windings will be much closer to equality again:
transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 100 l2 3 5 100 ** Coupling factor = 0.99999 instead of 0.999 k l1 l2 0.99999 vi1 3 4 ac 0 rload 4 5 15 .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end
freq v(2) i(v1) 6.000E+01 1.000E+01 6.658E-01
freq v(3,5) i(vi1) 6.000E+01 9.987E+00 6.658E-01
Here we see that our secondary voltage is back to being equal with the primary, and the secondary current is equal to the primary current as well. Unfortunately, building a real transformer with coupling this complete is very difficult. A compromise solution is to design both primary and secondary coils with less inductance, the strategy being that less inductance overall leads to less "leakage" inductance to cause trouble, for any given degree of magnetic coupling inefficiency. This results in a load voltage that is closer to ideal with the same (heavy) load and the same coupling factor:
transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 ** inductance = 1 henry instead of 100 henrys l1 2 0 1 l2 3 5 1 k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 15 .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end
freq v(2) i(v1) 6.000E+01 1.000E+01 6.664E-01
freq v(3,5) i(vi1) 6.000E+01 9.977E+00 6.652E-01
Simply by using primary and secondary coils of less inductance, the load voltage for this heavy load has been brought back up to nearly ideal levels (9.977 volts). At this point, one might ask, "If less inductance is all that's needed to achieve near-ideal performance under heavy load, then why worry about coupling efficiency at all? If it's impossible to build a transformer with perfect coupling, but easy to design coils with low inductance, then why not just build all transformers with low-inductance coils and have excellent efficiency even with poor magnetic coupling?"
The answer to this question is found in another simulation: the same low-inductance transformer, but this time with a lighter load (1 kΩ instead of 15 Ω):
transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 1 l2 3 5 1 k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 1k .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end
freq v(2) i(v1) 6.000E+01 1.000E+01 2.835E-02
freq v(3,5) i(vi1) 6.000E+01 9.990E+00 9.990E-03
With lower winding inductances, the primary and secondary voltages are closer to being equal, but the primary and secondary currents are not. In this particular case, the primary current is 28.35 mA while the secondary current is only 9.990 mA: almost three times as much current in the primary as the secondary. Why is this? With less inductance in the primary winding, there is less inductive reactance, and consequently a much larger magnetizing current. A substantial amount of the current through the primary winding merely works to magnetize the core rather than transfer useful energy to the secondary winding and load.
An ideal transformer with identical primary and secondary windings would manifest equal voltage and current in both sets of windings for any load condition. In a perfect world, transformers would transfer electrical power from primary to secondary as smoothly as though the load were directly connected to the primary power source, with no transformer there at all. However, you can see this ideal goal can only be met if there is perfect coupling of magnetic flux between primary and secondary windings. Being that this is impossible to achieve, transformers must be designed to operate within certain expected ranges of voltages and loads in order to perform as close to ideal as possible. For now, the most important thing to keep in mind is a transformer's basic operating principle: the transfer of power from the primary to the secondary circuit via electromagnetic coupling.
- REVIEW:
- Mutual inductance is where the magnetic flux of two or more inductors are "linked" so that voltage is induced in one coil proportional to the rate-of-change of current in another.
- A transformer is a device made of two or more inductors, one of which is powered by AC, inducing an AC voltage across the second inductor. If the second inductor is connected to a load, power will be electromagnetically coupled from the first inductor's power source to that load.
- The powered inductor in a transformer is called the primary winding. The unpowered inductor in a transformer is called the secondary winding.
- Magnetic flux in the core (Φ) lags 90o behind the source voltage waveform. The current drawn by the primary coil from the source to produce this flux is called the magnetizing current, and it also lags the supply voltage by 90o.
- Total primary current in an unloaded transformer is called the exciting current, and is comprised of magnetizing current plus any additional current necessary to overcome core losses. It is never perfectly sinusoidal in a real transformer, but may be made more so if the transformer is designed and operated so that magnetic flux density is kept to a minimum.
- Core flux induces a voltage in any coil wrapped around the core. The induces voltage(s) are ideally in phase with the primary winding source voltage and share the same waveshape.
- Any current drawn through the secondary winding by a load will be "reflected" to the primary winding and drawn from the voltage source, as if the source were directly powering a similar load.
Step-up and step-down transformers
So far, we've observed simulations of transformers where the primary and secondary windings were of identical inductance, giving approximately equal voltage and current levels in both circuits. Equality of voltage and current between the primary and secondary sides of a transformer, however, is not the norm for all transformers. If the inductances of the two windings are not equal, something interesting happens:transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 10000 l2 3 5 100 k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 1k .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end
freq v(2) i(v1) 6.000E+01 1.000E+01 9.975E-05 Primary winding
freq v(3,5) i(vi1) 6.000E+01 9.962E-01 9.962E-04 Secondary winding
Notice how the secondary voltage is approximately ten times less than the primary voltage (0.9962 volts compared to 10 volts), while the secondary current is approximately ten times greater (0.9962 mA compared to 0.09975 mA). What we have here is a device that steps voltage down by a factor of ten and current up by a factor of ten:
This is a very useful device, indeed. With it, we can easily multiply or divide voltage and current in AC circuits. Indeed, the transformer has made long-distance transmission of electric power a practical reality, as AC voltage can be "stepped up" and current "stepped down" for reduced wire resistance power losses along power lines connecting generating stations with loads. At either end (both the generator and at the loads), voltage levels are reduced by transformers for safer operation and less expensive equipment. A transformer that increases voltage from primary to secondary (more secondary winding turns than primary winding turns) is called a step-up transformer. Conversely, a transformer designed to do just the opposite is called a step-down transformer.
Let's re-examine a photograph shown in the previous section:
This is a step-down transformer, as evidenced by the high turn count of the primary winding and the low turn count of the secondary. As a step-down unit, this transformer converts high-voltage, low-current power into low-voltage, high-current power. The larger-gauge wire used in the secondary winding is necessary due to the increase in current. The primary winding, which doesn't have to conduct as much current, may be made of smaller-gauge wire.
In case you were wondering, it is possible to operate either of these transformer types backwards (powering the secondary winding with an AC source and letting the primary winding power a load) to perform the opposite function: a step-up can function as a step-down and visa-versa. However, as we saw in the first section of this chapter, efficient operation of a transformer requires that the individual winding inductances be engineered for specific operating ranges of voltage and current, so if a transformer is to be used "backwards" like this it must be employed within the original design parameters of voltage and current for each winding, lest it prove to be inefficient (or lest it be damaged by excessive voltage or current!).
Transformers are often constructed in such a way that it is not obvious which wires lead to the primary winding and which lead to the secondary. One convention used in the electric power industry to help alleviate confusion is the use of "H" designations for the higher-voltage winding (the primary winding in a step-down unit; the secondary winding in a step-up) and "X" designations for the lower-voltage winding. Therefore, a simple power transformer will have wires labeled "H1", "H2", "X1", and "X2". There is usually significance to the numbering of the wires (H1 versus H2, etc.), which we'll explore a little later in this chapter.
The fact that voltage and current get "stepped" in opposite directions (one up, the other down) makes perfect sense when you recall that power is equal to voltage times current, and realize that transformers cannot produce power, only convert it. Any device that could output more power than it took in would violate the Law of Energy Conservation in physics, namely that energy cannot be created or destroyed, only converted. As with the first transformer example we looked at, power transfer efficiency is very good from the primary to the secondary sides of the device.
The practical significance of this is made more apparent when an alternative is considered: before the advent of efficient transformers, voltage/current level conversion could only be achieved through the use of motor/generator sets. A drawing of a motor/generator set reveals the basic principle involved:
In such a machine, a motor is mechanically coupled to a generator, the generator designed to produce the desired levels of voltage and current at the rotating speed of the motor. While both motors and generators are fairly efficient devices, the use of both in this fashion compounds their inefficiencies so that the overall efficiency is in the range of 90% or less. Furthermore, because motor/generator sets obviously require moving parts, mechanical wear and balance are factors influencing both service life and performance. Transformers, on the other hand, are able to convert levels of AC voltage and current at very high efficiencies with no moving parts, making possible the widespread distribution and use of electric power we take for granted.
In all fairness it should be noted that motor/generator sets have not necessarily been obsoleted by transformers for all applications. While transformers are clearly superior over motor/generator sets for AC voltage and current level conversion, they cannot convert one frequency of AC power to another, or (by themselves) convert DC to AC or visa-versa. Motor/generator sets can do all these things with relative simplicity, albeit with the limitations of efficiency and mechanical factors already described. Motor/generator sets also have the unique property of kinetic energy storage: that is, if the motor's power supply is momentarily interrupted for any reason, its angular momentum (the inertia of that rotating mass) will maintain rotation of the generator for a short duration, thus isolating any loads powered by the generator from "glitches" in the main power system.
Looking closely at the numbers in the SPICE analysis, we should see a correspondence between the transformer's ratio and the two inductances. Notice how the primary inductor (l1) has 100 times more inductance than the secondary inductor (10000 H versus 100 H), and that the measured voltage step-down ratio was 10 to 1. The winding with more inductance will have higher voltage and less current than the other. Since the two inductors are wound around the same core material in the transformer (for the most efficient magnetic coupling between the two), the parameters affecting inductance for the two coils are equal except for the number of turns in each coil. If we take another look at our inductance formula, we see that inductance is proportional to the square of the number of coil turns:
So, it should be apparent that our two inductors in the last SPICE transformer example circuit -- with inductance ratios of 100:1 -- should have coil turn ratios of 10:1, because 10 squared equals 100. This works out to be the same ratio we found between primary and secondary voltages and currents (10:1), so we can say as a rule that the voltage and current transformation ratio is equal to the ratio of winding turns between primary and secondary.
The step-up/step-down effect of coil turn ratios in a transformer is analogous to gear tooth ratios in mechanical gear systems, transforming values of speed and torque in much the same way:
Step-up and step-down transformers for power distribution purposes can be gigantic in proportion to the power transformers previously shown, some units standing as tall as a home. The following photograph shows a substation transformer standing about twelve feet tall:
- REVIEW:
- Transformers "step up" or "step down" voltage according to the ratios of primary to secondary wire turns.
-
- A transformer designed to increase voltage from primary to secondary is called a step-up transformer. A transformer designed to reduce voltage from primary to secondary is called a step-down transformer.
- The transformation ratio of a transformer will be equal to the square root of its primary to secondary inductance (L) ratio.
-
Electrical isolation
Aside from the ability to easily convert between different levels of voltage and current in AC and DC circuits, transformers also provide an extremely useful feature called isolation, which is the ability to couple one circuit to another without the use of direct wire connections. We can demonstrate an application of this effect with another SPICE simulation: this time showing "ground" connections for the two circuits, imposing a high DC voltage between one circuit and ground through the use of an additional voltage source:
v1 1 0 ac 10 sin rbogus1 1 2 1e-12 v2 5 0 dc 250 l1 2 0 10000 l2 3 5 100 k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 1k .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end
DC voltages referenced to ground (node 0): (1) 0.0000 (2) 0.0000 (3) 250.0000 (4) 250.0000 (5) 250.0000
AC voltages: freq v(2) i(v1) 6.000E+01 1.000E+01 9.975E-05 Primary winding
freq v(3,5) i(vi1) 6.000E+01 9.962E-01 9.962E-04 Secondary winding
SPICE shows the 250 volts DC being impressed upon the secondary circuit elements with respect to ground, but as you can see there is no effect on the primary circuit (zero DC voltage) at nodes 1 and 2, and the transformation of AC power from primary to secondary circuits remains the same as before. The impressed voltage in this example is often called a common-mode voltage because it is seen at more than one point in the circuit with reference to the common point of ground. The transformer isolates the common-mode voltage so that it is not impressed upon the primary circuit at all, but rather isolated to the secondary side. For the record, it does not matter that the common-mode voltage is DC, either. It could be AC, even at a different frequency, and the transformer would isolate it from the primary circuit all the same.
There are applications where electrical isolation is needed between two AC circuit without any transformation of voltage or current levels. In these instances, transformers called isolation transformers having 1:1 transformation ratios are used. A benchtop isolation transformer is shown in the following photograph:
- REVIEW:
- By being able to transfer power from one circuit to another without the use of interconnecting conductors between the two circuits, transformers provide the useful feature of electrical isolation.
- Transformers designed to provide electrical isolation without stepping voltage and current either up or down are called isolation transformers.
Phasing
Since transformers are essentially AC devices, we need to be aware of the phase relationships between the primary and secondary circuits. Using our SPICE example from before, we can plot the waveshapes for the primary and secondary circuits and see the phase relations for ourselves:legend: *: v(2) Primary voltage +: v(3,5) Secondary voltage time v(2) (*)----------- -10 -5 0 5 10 (+)----------- -10 -5 0 5 10 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0.000E+00 0.000E+00 . . x . . 1.000E-03 3.675E+00 . . . + * . . 2.000E-03 6.803E+00 . . . . + * . 3.000E-03 9.008E+00 . . . . +* . 4.000E-03 9.955E+00 . . . . x 5.000E-03 9.450E+00 . . . . *+. 6.000E-03 7.672E+00 . . . . * + . 7.000E-03 4.804E+00 . . . *.+ . 8.000E-03 1.245E+00 . . . * + . . 9.000E-03 -2.474E+00 . . * + . . . 1.000E-02 -5.864E+00 . *+ . . . 1.100E-02 -8.390E+00 . *+ . . . . 1.200E-02 -9.779E+00 .x . . . . 1.300E-02 -9.798E+00 +* . . . . 1.400E-02 -8.390E+00 . +* . . . . 1.500E-02 -5.854E+00 . + *. . . . 1.600E-02 -2.479E+00 . . + * . . . 1.700E-02 1.246E+00 . . .+ * . . 1.800E-02 4.795E+00 . . . + *. . 1.900E-02 7.686E+00 . . . . + * . 2.000E-02 9.451E+00 . . . . x . 2.100E-02 9.937E+00 . . . . x 2.200E-02 9.025E+00 . . . . *+ . 2.300E-02 6.802E+00 . . . . *+ . 2.400E-02 3.667E+00 . . . * + . . 2.500E-02 -1.487E-03 . . * + . . 2.600E-02 -3.658E+00 . . * + . . . 2.700E-02 -6.814E+00 . * + . . . . 2.800E-02 -9.026E+00 . *+ . . . . 2.900E-02 -9.917E+00 *+ . . . . 3.000E-02 -9.511E+00 .x . . . . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
legend: *: i(v1) Primary current +: i(vi1) Secondary current time i(v1) (*)---------- -2.000E-04 -1.000E-04 0 1.000E-04 2.000E-04 (+)---------- -1.000E-03 -5.000E-04 0 5.000E-04 1.000E-03 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0.000E+00 0.000E+00 . . x . . 1.000E-03 -2.973E-05 . . + *. . . 2.000E-03 -6.279E-05 . + . * . . . 3.000E-03 -8.772E-05 . + . * . . . 4.000E-03 -1.008E-04 + * . . . 5.000E-03 -9.954E-05 .+ * . . . 6.000E-03 -8.522E-05 . + . * . . . 7.000E-03 -5.919E-05 . +. * . . . 8.000E-03 -2.500E-05 . . + *. . . 9.000E-03 1.212E-05 . . . * + . . 1.000E-02 4.736E-05 . . . * .+ . 1.100E-02 7.521E-05 . . . * . + . 1.200E-02 9.250E-05 . . . *. +. 1.300E-02 9.648E-05 . . . *. + 1.400E-02 8.602E-05 . . . * . + . 1.500E-02 6.362E-05 . . . * . + . 1.600E-02 3.177E-05 . . . * + . . 1.700E-02 -4.998E-06 . . x . . 1.800E-02 -4.136E-05 . . + * . . . 1.900E-02 -7.246E-05 . + . * . . . 2.000E-02 -9.331E-05 . + .* . . . 2.100E-02 -1.019E-04 + * . . . 2.200E-02 -9.651E-05 . + * . . . 2.300E-02 -7.749E-05 . + . * . . . 2.400E-02 -4.842E-05 . . + * . . . 2.500E-02 -1.275E-05 . . x. . . 2.600E-02 2.428E-05 . . . * + . . 2.700E-02 5.761E-05 . . . * .+ . 2.800E-02 8.261E-05 . . . * . + . 2.900E-02 9.514E-05 . . . *. +. 3.000E-02 9.487E-05 . . . *. +. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
It would appear that both voltage and current for the two transformer windings are in phase with each other, at least for our resistive load. This is simple enough, but it would be nice to know which way we should connect a transformer in order to ensure the proper phase relationships be kept. After all, a transformer is nothing more than a set of magnetically-linked inductors, and inductors don't usually come with polarity markings of any kind. If we were to look at an unmarked transformer, we would have no way of knowing which way to hook it up to a circuit to get in-phase (or 180o out-of-phase) voltage and current:
Since this is a practical concern, transformer manufacturers have come up with a sort of polarity marking standard to denote phase relationships. It is called the dot convention, and is nothing more than a dot placed next to each corresponding leg of a transformer winding:
Typically, the transformer will come with some kind of schematic diagram labeling the wire leads for primary and secondary windings. On the diagram will be a pair of dots similar to what is seen above. Sometimes dots will be omitted, but when "H" and "X" labels are used to label transformer winding wires, the subscript numbers are supposed to represent winding polarity. The "1" wires (H1 and X1) represent where the polarity-marking dots would normally be placed.
The similar placement of these dots next to the top ends of the primary and secondary windings tells us that whatever instantaneous voltage polarity seen across the primary winding will be the same as that across the secondary winding. In other words, the phase shift from primary to secondary will be zero degrees.
On the other hand, if the dots on each winding of the transformer do not match up, the phase shift will be 180o between primary and secondary, like this:
Of course, the dot convention only tells you which end of each winding is which, relative to the other winding(s). If you want to reverse the phase relationship yourself, all you have to do is swap the winding connections like this:
- REVIEW:
- The phase relationships for voltage and current between primary and secondary circuits of a transformer are direct: ideally, zero phase shift.
- The dot convention is a type of polarity marking for transformer windings showing which end of the winding is which, relative to the other windings. Remaining Info..
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