1. Resistors and Alternating Current
1.1. AC Resistor Circuits
Pure resistive AC circuit: resistor voltage and current are in phase.
If we were to plot the current and voltage for a very simple AC circuit consisting of a source and a resistor (Figure above), it would look something like this: (Figure below)
Voltage and current "in phase" for resistive circuit.
Because the resistor simply and directly resists the flow of electrons at all periods of time, the waveform for the voltage drop across the resistor is exactly in phase with the waveform for the current through it. We can look at any point in time along the horizontal axis of the plot and compare those values of current and voltage with each other (any "snapshot" look at the values of a wave are referred to as instantaneous values, meaning the values at that instant in time). When the instantaneous value for current is zero, the instantaneous voltage across the resistor is also zero. Likewise, at the moment in time where the current through the resistor is at its positive peak, the voltage across the resistor is also at its positive peak, and so on. At any given point in time along the waves, Ohm’s Law holds true for the instantaneous values of voltage and current.
We can also calculate the power dissipated by this resistor, and plot those values on the same graph: (Figure below)
Instantaneous AC power in a pure resistive circuit is always positive.
Note that the power is never a negative value. When the current is positive (above the line), the voltage is also positive, resulting in a power (p=ie) of a positive value. Conversely, when the current is negative (below the line), the voltage is also negative, which results in a positive value for power (a negative number multiplied by a negative number equals a positive number). This consistent "polarity" of power tells us that the resistor is always dissipating power, taking it from the source and releasing it in the form of heat energy. Whether the current is positive or negative, a resistor still dissipates energy.
2. Inductors and Alternating Current
This might be a good place to recall what you learned about phase in chapter 1. When two things are in step, going through a cycle together, falling together and rising together, they are in phase. When they are out of phase, the angle of lead or lagthe number of electrical degrees by which one of the values leads or lags the otheris a measure of the amount they are out of step. The time it takes the current in an inductor to build up to maximum and to fall to zero is important for another reason. It helps illustrate a very useful characteristic of inductive circuitsthe current through the inductor always lags the voltage across the inductor.
A circuit having pure resistance (if such a thing were possible) would have the alternating current through it and the voltage across it rising and failing together. This is illustrated in Figure 1, which shows the sine waves for current and voltage in a purely resistive circuit having an ac source. The current and voltage do not have the same amplitude, but they are in phase.
In the case of a circuit having inductance, the opposing force of the counter emf would be enough to keep the current from remaining in phase with the applied voltage. You learned that in a de circuit containing pure inductance the current took time to rise to maximum even though the full applied voltage was immediately at maximum. Figure 2 shows the wave forms for a purely inductive ac circuit in steps of quartercycles.
With an ac voltage, in the first quartercycle (0° to 90° ) the applied ac voltage is continually increasing. If there was no inductance in the circuit, the current would also increase during this first quartercycle. You know this circuit does have inductance. Since inductance opposes any change in current flow, no current flows during the first quartercycle. In the next quartercycle (90° to 180° ) the voltage decreases back to zero; current begins to flow in the circuit and reaches a maximum value at the same instant the voltage reaches zero. The applied voltage now begins to build up to maximum in the other direction, to be followed by the resulting current. When the voltage again reaches its maximum at the end of the third quartercycle (270°) all values are exactly opposite to what they were during the first halfcycle. The applied voltage leads the resulting current by one quartercycle or 90 degrees. To complete the full 360° cycle of the voltage, the voltage again decreases to zero and the current builds to a maximum value.
You must not get the idea that any of these values stops cold at a particular instant. Until the applied voltage is removed, both current and voltage are always changing in amplitude and direction.
As you know the sine wave can be compared to a circle. Just as you mark off a circle into 360 degrees, you can mark off the time of one cycle of a sine wave into 360 electrical degrees. This relationship is shown in Figure 3. By referring to this figure you can see why the current is said to lag the voltage, in a purely inductive circuit, by 90 degrees. Furthermore, by referring to Figure 3 and Figure 1 you can see why the current and voltage are said to be in phase in a purely resistive circuit. In a circuit having both resistance and inductance then, as you would expect, the current lags the voltage by an amount somewhere between 0 and 90 degrees.
A simple memory aid to help you remember the relationship of voltage and current in an inductive circuit is the word ELI. Since E is the symbol for voltage, L is the symbol for inductance, and I is the symbol for current; the word ELI demonstrates that current comes after (Lags) voltage in an inductor.
2.1. Inductive Reactance
When the current flowing through an inductor continuously reverses itself, as in the case of an ac source, the inertia effect of the cemf is greater than with dc. The greater the amount of inductance (L), the greater the opposition from this inertia effect. Also, the faster the reversal of current, the greater this inertial opposition. This opposing force which an inductor presents to the FLOW of alternating current cannot be called resistance, since it is not the result of friction within a conductor. The name given to it is INDUCTIVE REACTANCE because it is the "reaction" of the inductor to the changing value of alternating current. Inductive reactance is measured in ohms and its symbol is X_{L}.
As you know, the induced voltage in a conductor is proportional to the rate at which magnetic lines of force cut the conductor. The greater the rate (the higher the frequency), the greater the cemf. Also, the induced voltage increases with an increase in inductance; the more ampereturns, the greater the cemf. Reactance, then, increases with an increase of frequency and with an increase of inductance. The formula for inductive reactance is as follows:
The following example problem illustrates the computation of X ,.
O4. What is the formula used to compute the value of this opposition?
2.2. AC Inductor Circuits
Inductors do not behave the same as resistors. Whereas resistors simply oppose the flow of electrons through them (by dropping a voltage directly proportional to the current), inductors oppose changes in current through them, by dropping a voltage directly proportional to the rate of change of current. In accordance with Lenz’s Law, this induced voltage is always of such a polarity as to try to maintain current at its present value. That is, if current is increasing in magnitude, the induced voltage will "push against" the electron flow; if current is decreasing, the polarity will reverse and "push with" the electron flow to oppose the decrease. This opposition to current change is called reactance, rather than resistance.
Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such:
The expression di/dt is one from calculus, meaning the rate of change of instantaneous current (i) over time, in amps per second. The inductance (L) is in Henrys, and the instantaneous voltage (e), of course, is in volts. Sometimes you will find the rate of instantaneous voltage expressed as "v" instead of "e" (v = L di/dt), but it means the exact same thing. To show what happens with alternating current, let’s analyze a simple inductor circuit: (Figure below)
Pure inductive circuit: Inductor current lags inductor voltage by 90^{o}.
If we were to plot the current and voltage for this very simple circuit, it would look something like this: (Figure below)
Pure inductive circuit, waveforms.
Remember, the voltage dropped across an inductor is a reaction against the change in current through it. Therefore, the instantaneous voltage is zero whenever the instantaneous current is at a peak (zero change, or level slope, on the current sine wave), and the instantaneous voltage is at a peak wherever the instantaneous current is at maximum change (the points of steepest slope on the current wave, where it crosses the zero line). This results in a voltage wave that is 90^{o} out of phase with the current wave. Looking at the graph, the voltage wave seems to have a "head start" on the current wave; the voltage "leads" the current, and the current "lags" behind the voltage. (Figure below)
Current lags voltage by 90^{o} in a pure inductive circuit.
Things get even more interesting when we plot the power for this circuit: (Figure below)
In a pure inductive circuit, instantaneous power may be positive or negative
Because instantaneous power is the product of the instantaneous voltage and the instantaneous current (p=ie), the power equals zero whenever the instantaneous current or voltage is zero. Whenever the instantaneous current and voltage are both positive (above the line), the power is positive. As with the resistor example, the power is also positive when the instantaneous current and voltage are both negative (below the line). However, because the current and voltage waves are 90^{o} out of phase, there are times when one is positive while the other is negative, resulting in equally frequent occurrences of negative instantaneous power.
But what does negative power mean? It means that the inductor is releasing power back to the circuit, while a positive power means that it is absorbing power from the circuit. Since the positive and negative power cycles are equal in magnitude and duration over time, the inductor releases just as much power back to the circuit as it absorbs over the span of a complete cycle. What this means in a practical sense is that the reactance of an inductor dissipates a net energy of zero, quite unlike the resistance of a resistor, which dissipates energy in the form of heat. Mind you, this is for perfect inductors only, which have no wire resistance.
An inductor’s opposition to change in current translates to an opposition to alternating current in general, which is by definition always changing in instantaneous magnitude and direction. This opposition to alternating current is similar to resistance, but different in that it always results in a phase shift between current and voltage, and it dissipates zero power. Because of the differences, it has a different name: reactance. Reactance to AC is expressed in ohms, just like resistance is, except that its mathematical symbol is X instead of R. To be specific, reactance associated with an inductor is usually symbolized by the capital letter X with a letter L as a subscript, like this: X_{L}.
Since inductors drop voltage in proportion to the rate of current change, they will drop more voltage for fasterchanging currents, and less voltage for slowerchanging currents. What this means is that reactance in ohms for any inductor is directly proportional to the frequency of the alternating current. The exact formula for determining reactance is as follows:
If we expose a 10 mH inductor to frequencies of 60, 120, and 2500 Hz, it will manifest the reactances in Table Figure below.
Reactance of a 10 mH inductor:
Frequency (Hertz)  Reactance (Ohms) 

60 
3.7699 
120 
7.5398 
2500 
157.0796 
In the reactance equation, the term "2πf" (everything on the righthand side except the L) has a special meaning unto itself. It is the number of radians per second that the alternating current is "rotating" at, if you imagine one cycle of AC to represent a full circle’s rotation. A radian is a unit of angular measurement: there are 2π radians in one full circle, just as there are 360^{o} in a full circle. If the alternator producing the AC is a doublepole unit, it will produce one cycle for every full turn of shaft rotation, which is every 2π radians, or 360^{o}. If this constant of 2π is multiplied by frequency in Hertz (cycles per second), the result will be a figure in radians per second, known as the angular velocity of the AC system.
Angular velocity may be represented by the expression 2πf, or it may be represented by its own symbol, the lowercase Greek letter Omega, which appears similar to our Roman lowercase "w": ω. Thus, the reactance formula X_{L} = 2πfL could also be written as X_{L} = ωL.
It must be understood that this "angular velocity" is an expression of how rapidly the AC waveforms are cycling, a full cycle being equal to 2π radians. It is not necessarily representative of the actual shaft speed of the alternator producing the AC. If the alternator has more than two poles, the angular velocity will be a multiple of the shaft speed. For this reason, ω is sometimes expressed in units of electrical radians per second rather than (plain) radians per second, so as to distinguish it from mechanical motion.
Any way we express the angular velocity of the system, it is apparent that it is directly proportional to reactance in an inductor. As the frequency (or alternator shaft speed) is increased in an AC system, an inductor will offer greater opposition to the passage of current, and vice versa. Alternating current in a simple inductive circuit is equal to the voltage (in volts) divided by the inductive reactance (in ohms), just as either alternating or direct current in a simple resistive circuit is equal to the voltage (in volts) divided by the resistance (in ohms). An example circuit is shown here: (Figure below)
Inductive reactance
+
However, we need to keep in mind that voltage and current are not in phase here. As was shown earlier, the voltage has a phase shift of +90^{o} with respect to the current. (Figure below) If we represent these phase angles of voltage and current mathematically in the form of complex numbers, we find that an inductor’s opposition to current has a phase angle, too:
+ (((02061.png)))
Current lags voltage by 90^{o} in an inductor.
Mathematically, we say that the phase angle of an inductor’s opposition to current is 90^{o}, meaning that an inductor’s opposition to current is a positive imaginary quantity. This phase angle of reactive opposition to current becomes critically important in circuit analysis, especially for complex AC circuits where reactance and resistance interact. It will prove beneficial to represent any component’s opposition to current in terms of complex numbers rather than scalar quantities of resistance and reactance.

REVIEW:

Inductive reactance is the opposition that an inductor offers to alternating current due to its phaseshifted storage and release of energy in its magnetic field. Reactance is symbolized by the capital letter "X" and is measured in ohms just like resistance ®.

Inductive reactance can be calculated using this formula: X_{L} = 2πfL

The angular velocity of an AC circuit is another way of expressing its frequency, in units of electrical radians per second instead of cycles per second. It is symbolized by the lowercase Greek letter "omega," or ω.

Inductive reactance increases with increasing frequency. In other words, the higher the frequency, the more it opposes the AC flow of electrons.
3. Capacitors and Alternating Current
Figure 4, Figure 5, Figure 6, and Figure 7 show the variation of the alternating voltage and current in a capacitive circuit, for each quarter of one cycle. The solid line represents the voltage across the capacitor, and the dotted line represents the current. The line running through the center is the zero, or reference point, for both the voltage and the current. The bottom line marks off the time of the cycle in terms of electrical degrees. Assume that the ac voltage has been acting on the capacitor for some time before the time represented by the starting point of the sine wave in the figure.
At the beginning of the first quartercycle (0° to 90°) the voltage has just passed through zero and is increasing in the positive direction. Since the zero point is the steepest part of the sine wave, the voltage is changing at its greatest rate. The charge on a capacitor varies directly with the voltage, and therefore the charge on the capacitor is also changing at its greatest rate at the beginning of the first quartercycle. In other words, the greatest number of electrons are moving off one plate and onto the other plate. Thus the capacitor current is at its maximum value, as Figure 4 shows.
As the voltage proceeds toward maximum at 90 degrees, its rate of change becomes less and less, hence the current must decrease toward zero. At 90 degrees the voltage across the capacitor is maximum, the capacitor is fully charged, and there is no further movement of electrons from plate to plate. That is why the current at 90 degrees is zero.
At the end of this first quartercycle the alternating voltage stops increasing in the positive direction and starts to decrease. It is still a positive voltage, but to the capacitor the decrease in voltage means that the plate which has just accumulated an excess of electrons must lose some electrons. The current flow, therefore, must reverse its direction. Figure 5 shows the current curve to be below the zero line (negative current direction) during the second quartercycle (90° to 180°).
At 180 degrees the voltage has dropped to zero. This means that for a brief instant the electrons are equally distributed between the two plates; the current is maximum because the rate of change of voltage is maximum. Just after 180 degrees the voltage has reversed polarity and starts building up its maximum negative peak which is reached at the end of the third quartercycle (180° to 270°). During this third quartercycle the rate of voltage change gradually decreases as the charge builds to a maximum at 270 degrees. At this point the capacitor is fully charged and it carries the full impressed voltage. Because the capacitor is fully charged there is no further exchange of electrons; therefore, the current flow is zero at this point. The conditions are exactly the same as at the end of the first quartercycle (90°) but the polarity is reversed.
Just after 270 degrees the impressed voltage once again starts to decrease, and the capacitor must lose electrons from the negative plate. It must discharge, starting at a minimum rate of flow and rising to a maximum. This discharging action continues through the last quartercycle (270° to 360°) until the impressedvoltage has reached zero. At 360 degrees you are back at the beginning of the entire cycle, and everything starts over again.
If you examine the complete voltage and current curves Figure 7, you will see that the current always arrives at a certain point in the cycle 90 degrees ahead of the voltage, because of the charging and discharging action. You know that this time and place relationship between the current and voltage is called the phase relationship. The voltagecurrent phase relationship in a capacitive circuit is exactly opposite to that in an inductive circuit. The current of a capacitor leads the voltage across the capacitor by 90 degrees.
You realize that the current and voltage are both going through their individual cycles at the same time during the period the ac voltage is impressed. The current does not go through part of its cycle (charging or discharging), stop, and wait for the voltage to catch up. The amplitude and polarity of the voltage and the amplitude and direction of the current are continually changing. Their positions with respect to each other and to the zero line at any electrical instantany degree between zero and 360 degreescan be seen by reading upwards from the timedegree line. The current swing from the positive peak at zero degrees to the negative peak at 180 degrees is NOT a measure of the number of electrons, or the charge on the plates. It is a picture of the direction and strength of the current in relation to the polarity and strength of the voltage appearing across the plates.
At times it is convenient to use the word "ICE" to recall to mind the phase relationship of the current and voltage in capacitive circuits. [is the symbol for current, and in the word ICE it leads, or comes before, the symbol for voltage, E. C, of course, stands for capacitor. This memory aid is similar to the "ELI" used to remember the current and voltage relationship in an inductor. The phrase "ELI the ICE man" is helpful in remembering the phase relationship in both the inductor and capacitor.
Since the plates of the capacitor are changing polarity at the same rate as the ac voltage, the capacitor seems to pass an alternating current. Actually, the electrons do not pass through the dielectric, but their rushing back and forth from plate to plate causes a current flow in the circuit. It is convenient, however, to say that the alternating current flows "through" the capacitor. You know this is not true, but the expression avoids a lot of trouble when speaking of current flow in a circuit containing a capacitor. By the same short cut, you may say that the capacitor does not pass a direct current (if both plates are connected to a de source, current will flow only long enough to charge the capacitor). With a capacitor type of hookup in a circuit containing both ac and dc, only the ac will be "passed" on to another circuit.
You have now learned two things to remember about a capacitor: A capacitor will appear to conduct an alternating current and a capacitor will not conduct a direct current.
O8. What is the phase relationship between current and voltage in a capacitor?
3.1. Capacitive Reactance
So far you have been dealing with the capacitor as a device which passes ac and in which the only opposition to the alternating current has been the normal circuit resistance present in any conductor. However, capacitors themselves offer a very real opposition to current flow. This opposition arises from the fact that, at a given voltage and frequency, the number of electrons which go back and forth from plate to plate is limited by the storage abilitythat is, the capacitanceof the capacitor. As the capacitance is increased, a greater number of electrons change plates every cycle, and (since current is a measure of the number of electrons passing a given point in a given time) the current is increased.
Increasing the frequency will also decrease the opposition offered by a capacitor. This occurs because the number of electrons which the capacitor is capable of handling at a given voltage will change plates more often. As a result, more electrons will pass a given point in a given time (greater current flow). The opposition which a capacitor offers to ac is therefore inversely proportional to frequency and to capacitance. This opposition is called CAPACITIVE REACTANCE. You may say that capacitive reactance decreases with increasing frequency or, for a given frequency, the capacitive reactance decreases with increasing capacitance. The symbol for capacitive reactance is X_{C}.
Now you can understand why it is said that the X_{C} varies inversely with the product of the frequency and capacitance. The formula is:
O10. What is the formula used to compute this opposition? QI1. What happens to the value of X_{C} as frequency decreases?
3.2. AC Capacitor Circuits
Capacitors do not behave the same as resistors. Whereas resistors allow a flow of electrons through them directly proportional to the voltage drop, capacitors oppose changes in voltage by drawing or supplying current as they charge or discharge to the new voltage level. The flow of electrons "through" a capacitor is directly proportional to the rate of change of voltage across the capacitor. This opposition to voltage change is another form of reactance, but one that is precisely opposite to the kind exhibited by inductors.
Expressed mathematically, the relationship between the current "through" the capacitor and rate of voltage change across the capacitor is as such:
The expression de/dt is one from calculus, meaning the rate of change of instantaneous voltage (e) over time, in volts per second. The capacitance © is in Farads, and the instantaneous current (i), of course, is in amps. Sometimes you will find the rate of instantaneous voltage change over time expressed as dv/dt instead of de/dt: using the lowercase letter "v" instead or "e" to represent voltage, but it means the exact same thing. To show what happens with alternating current, let’s analyze a simple capacitor circuit: (Figure below)
Pure capacitive circuit: capacitor voltage lags capacitor current by 90^{o}
If we were to plot the current and voltage for this very simple circuit, it would look something like this: (Figure below)
Pure capacitive circuit waveforms.
Remember, the current through a capacitor is a reaction against the change in voltage across it. Therefore, the instantaneous current is zero whenever the instantaneous voltage is at a peak (zero change, or level slope, on the voltage sine wave), and the instantaneous current is at a peak wherever the instantaneous voltage is at maximum change (the points of steepest slope on the voltage wave, where it crosses the zero line). This results in a voltage wave that is 90^{o} out of phase with the current wave. Looking at the graph, the current wave seems to have a "head start" on the voltage wave; the current "leads" the voltage, and the voltage "lags" behind the current. (Figure below)
Voltage lags current by 90^{o} in a pure capacitive circuit.
As you might have guessed, the same unusual power wave that we saw with the simple inductor circuit is present in the simple capacitor circuit, too: (Figure below)
In a pure capacitive circuit, the instantaneous power may be positive or negative.
As with the simple inductor circuit, the 90 degree phase shift between voltage and current results in a power wave that alternates equally between positive and negative. This means that a capacitor does not dissipate power as it reacts against changes in voltage; it merely absorbs and releases power, alternately.
A capacitor’s opposition to change in voltage translates to an opposition to alternating voltage in general, which is by definition always changing in instantaneous magnitude and direction. For any given magnitude of AC voltage at a given frequency, a capacitor of given size will "conduct" a certain magnitude of AC current. Just as the current through a resistor is a function of the voltage across the resistor and the resistance offered by the resistor, the AC current through a capacitor is a function of the AC voltage across it, and the reactance offered by the capacitor. As with inductors, the reactance of a capacitor is expressed in ohms and symbolized by the letter X (or X_{C} to be more specific).
Since capacitors "conduct" current in proportion to the rate of voltage change, they will pass more current for fasterchanging voltages (as they charge and discharge to the same voltage peaks in less time), and less current for slowerchanging voltages. What this means is that reactance in ohms for any capacitor is inversely proportional to the frequency of the alternating current. (Table below)
Reactance of a 100 uF capacitor:
Frequency (Hertz)  Reactance (Ohms) 

60 
26.5258 
120 
13.2629 
2500 
0.6366 
Please note that the relationship of capacitive reactance to frequency is exactly opposite from that of inductive reactance. Capacitive reactance (in ohms) decreases with increasing AC frequency. Conversely, inductive reactance (in ohms) increases with increasing AC frequency. Inductors oppose faster changing currents by producing greater voltage drops; capacitors oppose faster changing voltage drops by allowing greater currents.
As with inductors, the reactance equation’s 2πf term may be replaced by the lowercase Greek letter Omega (ω), which is referred to as the angular velocity of the AC circuit. Thus, the equation X_{C} = 1/(2πfC) could also be written as X_{C} = 1/(ωC), with ω cast in units of radians per second.
Alternating current in a simple capacitive circuit is equal to the voltage (in volts) divided by the capacitive reactance (in ohms), just as either alternating or direct current in a simple resistive circuit is equal to the voltage (in volts) divided by the resistance (in ohms). The following circuit illustrates this mathematical relationship by example: (Figure below)
Capacitive reactance.
+
However, we need to keep in mind that voltage and current are not in phase here. As was shown earlier, the current has a phase shift of +90^{o} with respect to the voltage. If we represent these phase angles of voltage and current mathematically, we can calculate the phase angle of the capacitor’s reactive opposition to current.
+ (((02073.png)))
Voltage lags current by 90^{o} in a capacitor.
Mathematically, we say that the phase angle of a capacitor’s opposition to current is 90^{o}, meaning that a capacitor’s opposition to current is a negative imaginary quantity. (Figure above) This phase angle of reactive opposition to current becomes critically important in circuit analysis, especially for complex AC circuits where reactance and resistance interact. It will prove beneficial to represent any component’s opposition to current in terms of complex numbers, and not just scalar quantities of resistance and reactance.

REVIEW:

Capacitive reactance is the opposition that a capacitor offers to alternating current due to its phaseshifted storage and release of energy in its electric field. Reactance is symbolized by the capital letter "X" and is measured in ohms just like resistance ®.

Capacitive reactance can be calculated using this formula: X_{C} = 1/(2πfC)

Capacitive reactance decreases with increasing frequency. In other words, the higher the frequency, the less it opposes (the more it "conducts") the AC flow of electrons.