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RLC CIRCUITS AND TRANSFORMERS

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Inductors and Inductance

In a straight piece of wire, an alternating current (with its changing magnetic field) will cause a voltage to be induced that opposes the applied voltage. This voltage is extremely small, however, and not noticeable except when the wire is long, like a transmission line wire. If the wire is in the form of coil, however, the effect of the wires electromagnetic field is strong, and the induced AC voltage is large enough to be taken into account.

The ability of a coil to induce an opposition voltage into itself is called self inductance, or simply inductance (L).The inductance of a coil depends on such things as length, spacing of the turns, core material, etc. Inductance is measured in henries (symbol: H). A coil has an inductance of one henry if a current change of 1 ampere per second through it produces a counter-voltage of 1 volt.

However, what if we have a second coil whose conductors are parallel to the first, but with no electrical connection between them? The same kind of induction takes place; this time, however, it is called mutual induction. In fact, if all of the flux from the first coil links all of the turns in the second coil (a coefficient of coupling of 1), the same CEMF will be induced in both coils. Even more, the varying magnetic field of the second coil will induce a voltage back in the first one. Mutual inductance, or LM, is also measured in henries. Two coils have a mutual inductance of 1 henry when a current change of 1 ampere per second in either coil induces 1 volt in the other one.

The mutual inductance can be determined from the equation:

where K is the coefficient of coupling between coils L1 and L2. Coefficients of coupling generally range between 0.1 for air core coils to 1 for iron core coils. K also depends on the physical relationship of the coils. The closer they are together, the higher K will be.

The total inductance of coils connected in series (with no mutual inductance) is the sum of the individual inductors (in henries).

If connected in series, the total inductance depends not only on the amount of coupling, but also on whether the coils are connected series-aiding or series-opposing. Series-aiding means that the common current produces a magnetic field of the same direction for the two coils. The series-opposing connection results in opposite fields.

  • When two coils are connected series-aiding, the total inductance, or LT, is computed from the following equation:
  • When two coils are connected series-opposing, LT is computed from the following equation:

Transformers

Close to every large power-generating plant is a bank of large transformers. These step the generated voltage up to a higher level for transmission. Then, at various strategic locations, there are substations that step-down the transmission-level voltages to distribution level. Industrial plants and commercial buildings each have their own transformer to step the voltage down once again to the level required by lights, motors, etc. In residential areas, pole-mounted transformers do the same thing for a number of residential users. In addition, most homes have a small transformer to step-down the distributed 1 AC voltage. For example, 120 volts in the US might be stepped down to 12 volts for operating the doorbell.

All of these transformers have an iron core, which strengthens the magnetic field caused by the alternating current. There is a coil connected to the power source called the primary. Wound on the same core (magnetically linked with the primary) is a second coil, called the secondary. The expanding and collapsing magnetic field in the core induces a counter-voltage in the primary and a voltage in the secondary. Since it is the same field operating on both coils, it induces the same voltage in each turn of both coils. This is what allows transformers to step-up and step-down voltages.

The ratio between the number of turns on the primary, NP, to the number of turns on the secondary, NS, is the same as the ratio of the voltage applied to the primary, VP, to the voltage across the secondary, VS.

The term turns ratio usually is applied to the ratio of the high side (the larger number of turns) to the low side (the smaller number of turns). For example, a 2:1 step-up transformer has half as many turns on the primary as the secondary, while a 2:1 step-down transformer has twice as many turns on the primary as the secondary.

Now we come to the reason why you do not get large voltage drops on high-voltage AC transmission lines. In an idealized transformer (most large transformers are 97%, or better, efficient), there is no power loss. The power in (VPIP) equals the power out (VSIS). The higher the voltage, the less current is needed to supply the same power.

For example, assume an operational heater is drawing 10 amps from a 120-volt line. The electricity was transmitted at 600,000 volts. The voltage was stepped down by a factor of 5,000. At the same time, current was stepped up by a factor of 5,000. That 10 amps looked like only 2 milliamps to the transmission line. Since power line losses are I2R, the difference is considerable. The ratio between the current drawn by the secondary, IS, and the current supplied to the primary, IP, is the same as the ratio between the number of turns on the primary, NP, and the number of turns on the secondary, NS.

Inductive Reactance and Impedance

The inductance of a coil depends on four physical factors: (1) the number of turns, (2) the diameter of the form the coil is wound on, (3) the permeability of the core, and (4) the length of the coil.

Since the current is induced in the coil (in the opposite direction as the current that caused it) only when current is changing, we say that inductance opposes any change in current.

Remember that the induced voltage depends on the rate that the current is changing, not the value of the current itself.

At the instant DC is applied, current starts from zero, climbing rapidly toward its steady value. The expanding field around the coils conductors induces an opposition voltage into the conductors. This counter-electromotive force, or CEMF, tries to create a current flow of its own in the opposite direction as the current that caused it. Net current flow during this brief period is current caused by the applied voltage minus induced current. This is why we say that inductance opposes a changing current. The more rapidly the current changes, the more the inductance tries to hold it back.

When the DC reaches its final steady value and current is no longer changing, net current flow is simply the applied voltage divided by the resistance of the wire, or I = V/R. Note, however, that at this point, the current-carrying conductors do have a magnetic field around them. The coil has absorbed energy temporarily in this magnetic field. If the DC is suddenly removed so that the magnetic field starts collapsing rapidly, the induced voltage (CEMF) can become quite high and may be many times the voltage across the resistance of the coil. Care must always be taken when dealing with switches in DC circuits that contain coils. Unless some sort of protection is present to absorb the energy released by the collapsing field, the resulting "inductive kick" can cause considerable damage to equipment and even personnel.

When we look at the voltage across the coil and current through it, we find the voltage at its peak when the current crosses zero. (The current is changing at its fastest rate then.) In addition, voltage is at zero when the current peaks out in both directions. (Current is not changing at all at this instant.) The voltage reaches its maximum one-fourth cycle, or 90 electrical degrees, ahead of the current. In an inductive circuit, therefore, current lags voltage by 90.

If the current through the coil, or IL, is less, as we said it was, then the opposition to current must be greater. We call this extra opposition inductive reactance, which is measured in ohms and abbreviated as XL. As we might expect, the XL of a coil depends on the coils inductance and the frequency at which the current through it is changing. The equation is XL = 2fL, where XL is in ohms, frequency is in hertz, and inductance is in henries. Because of the out-of-phase relationship between voltage and current, we cannot just add inductive reactance and resistance together to find the total impedance, Z, of a circuit.

Power Factor

In a pure inductive circuit (with negligible resistance), current lags behind voltage one-fourth cycle, or 90 electrical degrees. Resistors do not cause this phase shift. Therefore, in circuits containing both resistance and inductance, current lags voltage by an angle greater than 0 but less than 90. There is only one current, but it can be thought of as being made up of two parts. One part is in phase with the applied voltage. The 90-out-of-phase part is often called the quadrature current.

The electrical power supplied by the in-phase part of the current is called active power (P) and is measured in watts. This power is converted to some other form of energy (heat, light, motion, etc.). The power supplied by the quadrature current does no work at all. It is merely the energy stored by an expanding magnetic field and is returned to the source when the field collapses. This is called reactive power (PQ) and is measured in VARs (for volt ampere reactives). Reactive power can be thought of as a "shuttle power" that shuttles back and forth between the source and the load.

Just as the in-phase and quadrature currents make up current flow, the active and reactive powers make up what is called apparent power (PS). Apparent power is the product of applied voltage and total current and is measured in volt-amperes. Apparent power is equal to the square root of the sum of the squares of active power and reactive power. The equation is:

The "power factor" of a circuit tells us how much of the apparent power is active power. The equation is: PF = P/PS. The highest possible power factor is 1 (also called unity). Unity power factor occurs when the load is purely resistive. The lowest possible power factor is 0, which is when the load is purely reactive. Most industrial circuits (motors, etc.) have a power factor of about 0.8.

Capacitors and Capacitance

Whenever a difference in potential exists between two conductors that are close to each other, an electrostatic field is created in the insulating material.

In capacitors, the conductors are called plates. The insulating material between the plates is called the dielectric. Since the dielectric is an insulator, electrons cannot freely move through it. However, the voltage difference across the plates causes the orbits of the electrons to become stretched out of shape. This is the meaning of an electrostatic field. Stretching the electrons orbits is like stretching a rubber band or a coiled spring. It represents stored energy that is released when the pressure is removed.

The ability to store energy in an electrostatic field is called capacitance. Capacitance can occur between the parallel wires of power line cables and in the circuits of high-frequency equipment, like radios and TV sets. This stray capacitance usually is unwanted, however. Factors that govern the amount of capacitance are size and spacing of the plates and the dielectric material. Air is the most frequently used dielectric for tuning capacitors. Other dielectric materials are paper, mica, and ceramic. Electrolytic capacitors use aluminum oxide as the dielectric and electrolyte as one of the plates.

As the voltage across a capacitor begins to build, current enters one plate and leaves the other. If you did not know better, you would think that current was passing through the capacitor. However, as the voltage approaches a maximum, the backpressure from the field increases too, making current decrease. At the instant the capacitor becomes fully charged (its back voltage equals the applied voltage), current is 0. Visualize pushing against a coil spring. At first, there is considerable motion. However, by the time your pressure against it is maximum, the motion stops.

In fact, the current reaches its peak 90 degrees ahead of the voltage. We can, therefore, say that (1) capacitors cause current to lead voltage by 90, and (2) capacitors oppose any change in voltage.

To help distinguish between the actions of coils and capacitors, remember "ELI and ICE man." ELI helps remind you that I (current) follows (lags) E (voltage) when the circuit is L (inductive), while ICE reminds you that I (current) is ahead of (leads) E (voltage) when the circuit is C (capacitive).

Capacitance is measured in farads (F). One farad is the amount of capacitance that produces one ampere of charging current when the voltage changes at a rate of one volt per second.

Capacitors in Series and Parallel

Capacitors cause current to lead voltage by 90 electrical degrees. A perfect capacitor uses no power. Energy is stored in the electrostatic field during the following quarter-cycle. A capacitor tends to produce what is called a "leading power factor."

The ratio of the voltage across a capacitor to its charging current (VC/IC) is called capacitive reactance (XC), which is measured in ohms.

The greater the capacitance, the greater the current. It follows, therefore, that the ohmic value of the capacitive reactance must decrease as the capacitance increases. The equation is:

where XC is in ohms and C is in farads.

Since most capacitors have values in microfarads, conversion to farads must be made. For 60 Hz AC, the factor 2f works out to be approximately 377.

Capacitance itself is a different story, however. Besides the dielectric material, the capacitance of a capacitor varies directly with plate area and inversely with dielectric thickness. The larger the plate size, the greater the capacitance. The thicker the dielectric, the smaller the capacitance.

Having capacitors in parallel is like increasing the plate size. The total capacitance of capacitors in parallel is equal to the sum of the individual capacitances.

Having capacitors in series is like increasing the thickness of the dielectric. The equivalent capacitance is less than the smallest one.

Capacitive Reactance and Impedance

Consider a capacitor and resistor in series. The current through the resistor is in phase with the voltage across it. In the capacitor, however, the current is leading the voltage by 90 electrical degrees. We can represent this relationship by a phasor diagram (see Figure 1).

Since the voltage drop across R is directly proportional to the resistance (VR = IR) and the voltage drop across C is directly proportional to the capacitive reactance (VC = IXC), the impedance diagram is exactly the same as the voltage diagram for a series circuit. The ratio between applied voltage and total current (VA/IT) is called impedance (Z). It is represented by the hypotenuse of the impedance right triangle and can be calculated from the following equation:


or

It may seem strange at first that one cannot simply add the resistance to the capacitive reactance directly. It may help to understand that the phasor impedance relationship is based on the voltage relationships.

Impedance of Series RLC Circuits

An inductance in a circuit causes current to lag voltage by 90. A capacitor in a circuit causes current to lead voltage 90. That is why adding resistance and reactance algebraically cannot be calculated accurately.

When there is both inductive reactance and capacitive reactance in a circuit, however, one can combine these two algebraically. One can always subtract the smaller from the larger. The reason is because the current through the inductor is 180 out of phase with the current in the capacitor. They are directly opposed at every point in their respective cycles. The result, then, is the difference between the two.

If the total reactance of a coil and capacitor in series is less than either one separately, the impedance must also be less. This is shown by the equation for RLC circuits:

Even if the capacitance reactance is larger (producing a negative reactance), the equation still holds true. The square of a negative number is still positive.

Impedance of Parallel RL and RC Circuits

In a parallel circuit, there is just one voltage. The same voltage appears across the resistor and capacitor in a parallel RC circuit. The current flowing through the resistor is in phase with this voltage. The charging current of the capacitor is 90 ahead of this voltage. To find the total current, one must find the "phasor sum" of these two currents. This can be done (1) graphically, (2) by the "sum of the squares" method, or (3) by trigonometry (see Figure 2).


Figure 2: Phasor Diagram

Once the total current has been identified, IT, one can calculate the impedance, Z, from the equation:

In a parallel circuit containing resistance and inductance, the current through the inductance lags the voltage by 90. Total current is the phasor sum of the individual currents (see Figure 3):


Figure 3: Phasor Diagram

Again, as far as the power source is concerned, there is only one current, IT, being supplied at a certain voltage, V.

Impedance of Parallel RLC Circuits

In solving parallel AC circuits, we cannot start with a voltage vector diagram because the voltage is the same across each leg. Neither can we start with an impedance diagram. We have to combine currents. Current, remember, leads voltage by 90 in a capacitor and lags voltage by 90 in an inductor.

Since we are still dealing with the relationships of a right triangle, all of the equations still apply:

Once we have determined the total current (the vector sum of the current in the individual legs), we can find the impedance from the equation:

Having calculated the phase angle (the angle whose tangent equals IX/IR), one can calculate the equivalent resistance and equivalent reactance from the equations:

One will know whether the reactance is inductive or capacitive by whether IX, and consequently the phase angle, or Z, is positive or negative.

L/R and RC Time Constants

At the instant that voltage is applied to a capacitor, the voltage across it is zero and the capacitor begins to charge. At full charge (zero current flow), the voltage across it is at its maximum. If a resistor is in series with the capacitor, the voltage drop across the resistor goes from the maximum to the minimum during the charging time (VT = VC + VR). The time it takes the capacitor to charge depends on both the capacitance of the capacitor and the resistance of the resistor. The time constant is defined as "the time it takes for the capacitor to charge to approximately 63% of the applied voltage." The equation is:

Time constant (in seconds) = R (in ohms) times C (in Farads)

Similarly, a capacitor does not discharge instantaneously. The voltage decreases (decays) at a rate that depends on the values of the resistor and capacitor. It takes one time constant for the voltage to decay approximately 63%.

An inductor tends to oppose a change in current. In a circuit having a resistor in series with an inductor, the time constant is defined as the time it takes current through the inductor to reach approximately 63% of its final value when there has been a step change in applied voltage. This definition applies equally to increasing or decaying current. The equation for computing time constant is:

Time constant (in seconds) = L (in henries) / R (in Ohms)

Measurement of RC Time Constants

Refer to the Universal Time Constant Chart, shown in Figure 4 below. It shows the rate of rise and decay in RL circuits and the rate of charge and discharge in RC circuits. From these curves, one can see the discharge of RC circuits. From these curves, one can see that the rate of change starts out directly proportionate to time. In one-tenth of a time constant, the change is 10%; in two tenths, it is 20%. The curves start to level off at about that point, however, because the more energy L and C give up, the less they have left to oppose the change.


Figure 4: Universal Time Constant Chart

It is known that at the end of one time constant period, 63% of the change has taken place. At the end of two constants, that value is 86%; after three time constants, it is 96%; and after four time constants, 98%. At the end of five time constants, effectively 100% of the change has taken place.

The ability of capacitors to delay voltage rise and fall, coupled with the fact that this delay is adjustable by varying R and/or C, makes the RC circuit useful for many timing and time-delay applications.

Transformers

Transformers are used extensively for AC power transmissions and for various control and indication circuits. Knowledge of the basic theory of how these components operate is necessary to understand the role transformers play in todays various utilities.

Theory of Transformer Operation

If flux lines from the expanding and contracting magnetic field of one coil cut the windings of another nearby coil, a voltage will be induced in that coil. The inducing of an EMF in a coil by magnetic flux lines generated in another coil is called mutual induction. The amount of electromotive force (EMF) that is induced depends on the relative positions of the two coils.

A transformer works on the principle that energy can be transferred by magnetic induction from one set of coils to another set by means of a varying magnetic flux. The magnetic flux is produced by an AC source. AC sources must be used to provide a changing (expanding and contracting) magnetic field. If a DC source were used, there would be a constant magnetic field and no voltage would be induced.

The coil of a transformer that is energized from an AC source is called the primary winding; the coil that delivers AC to the load is called the secondary winding (see Figure 5).


Figure 5: Core-Type Transformer

When alternating voltage is applied to the primary winding, an alternating current will flow that will magnetize the magnetic core, first in one direction and then in the other direction. This alternating flux flowing around the entire length of the magnetic circuit induces a voltage in the secondary windings. The voltage induced will have the same frequency as the source.

A voltage will also be induced into the primary windings, since they are also in close proximity to the changing magnetic field. This voltage opposes the voltage applied to the primary winding and is called counter-electromotive force (CEMF).

Transformer Functions

According to Faraday's law, the voltage that is induced into a conductor is dependent upon both the rate of change of magnetic flux and the number of turns in the conductor. Transformers use this law to change the magnitude of the voltage from the primary to the secondary windings.

  • If the secondary winding of a transformer contains more turns of wire than the primary winding, more voltage will be induced into the secondary winding and the transformer will act to increase the voltage. These types of transformers are known as step-up transformers.
  • If the secondary winding of a transformer contains fewer turns of wire than the primary winding, less voltage will be induced in the secondary winding than exists in the primary winding and the transformer will act to decrease the voltage. These types of transformers are known as step-down transformers.

This feature provides transformers with the ability to change the magnitude of AC voltages. In some applications, changing the voltage of an AC source may be beneficial for system performance.

For example, electrical transmission systems that carry voltage over long distances operate more efficiently at higher voltages. If this higher voltage was present at the generators that produce the electricity and the loads that consume the electricity, more insulation and safety equipment would be required for operation, resulting in a higher cost. By using step-up and step-down transformers though, the electrical power can be generated at a low voltage, stepped up to a high voltage that is more efficient for transmission, and then stepped down to a voltage suitable for consumption.

Another function of transformers is to provide isolation capabilities. Recall that there are no physical connections between the primary and secondary windings, only a magnetic connection. This feature ensures that the power source and the load are not electrically connected; therefore, an electrical anomaly on one side of the transformer will not affect the equipment on the other side of the transformer.

Types of Transformers

Transformers are constructed so that their characteristics match the application for which they are intended. The differences in construction may involve the size of the windings or the relationship between the primary and secondary windings. Transformer types are also designated by the function the transformer serves in a circuit, such as an isolation transformer.

Distribution Transformer

Distribution transformers generally are used in electrical power distribution and transmission systems. This class of transformer has the highest power, or volt-ampere, ratings and the highest continuous voltage rating. The power rating normally is determined by the type of cooling methods the transformer may use. One commonly-used method of cooling is by using oil or some other heat-conducting material. Ampere rating is increased in a distribution transformer by increasing the size of the primary and secondary windings; voltage ratings are increased by increasing the voltage rating of the insulation used in making the transformer.

Power Transformer

Power transformers are used in electronic circuits and come in many different types and applications. Power transformers are sometimes considered to be those with ratings of 300 volt-amperes and below. These transformers normally provide power to the power supply of an electronic device, such as in power amplifiers in audio receivers.

Control Transformer

Control transformers generally are used in electronic circuits that require constant voltage or constant current with a low power or volt-amp rating. Various filtering devices, such as capacitors, are used to minimize the variations in the output. This results in a more constant voltage or current.

Auto Transformer

The auto transformer generally is used in low-power applications where a variable voltage is required. The auto transformer is a special type of power transformer. It consists of only one winding. By tapping or connecting at certain points along the winding, different voltages can be obtained (Figure 6).


Figure 6: Auto Transformer Schematic

Isolation Transformer

Isolation transformers normally are low-power transformers used to isolate noise from or to ground electronic circuits. Since a transformer cannot pass DC voltage from the primary to the secondary windings, any DC voltage (such as noise) cannot be passed, which is why the transformer is able to isolate this noise.

Instrument Potential Transformer

The instrument potential transformer (PT) steps down voltage of a circuit to a low value that can be effectively and safely used for the operation of instruments such as ammeters, voltmeters, wattmeters, and relays used for various protective purposes.

Instrument Current Transformer

The instrument current transformer (CT) steps down the current of a circuit to a lower value and is used in the same types of equipment as a potential transformer. This is done by constructing the secondary coil, which consists of many turns of wire, around the primary coil, which contains only a few turns of wire. In this manner, measurements of high values of current can be obtained.

A current transformer must always be short-circuited when not connected to an external load. Because the magnetic circuit of a current transformer is designed for low magnetizing current when under load, this large increase in magnetizing current will build up a large flux in the magnetic circuit and cause the transformer to act as a step-up transformer, inducing an excessively high voltage in the secondary winding when energized under no load. Shorting the secondary of a current transformer prevents this very high voltage.