An understanding of how an AC generator develops an AC output will help the student analyze the AC power generation process.
- EO 1.1DESCRIBE the construction and operation of a simple AC generator.
- EO 1.2EXPLAIN the development of a sine-wave output in an AC generator.
The elementary AC generator (Figure 1) consists of a conductor, or loop of wire in a magnetic field that is produced by an electromagnet. The two ends of the loop are connected to slip rings, and they are in contact with two brushes. When the loop rotates it cuts magnetic lines of force, first in one direction and then the other.
Figure 1 Simple AC Generator
Development of a Sine-Wave Output
At the instant the loop is in the vertical position (Figure 2, 0°), the coil sides are moving parallel to the field and do not cut magnetic lines of force. In this instant, there is no voltage induced in the loop. As the coil rotates in a counter-clockwise direction, the coil sides will cut the magnetic lines of force in opposite directions. The direction of the induced voltages depends on the direction of movement of the coil.
The induced voltages add in series, making slip ring X (Figure 1) positive (+) and slip ring Y (Figure 1) negative (-). The potential across resistor R will cause a current to flow from Y to X through the resistor. This current will increase until it reaches a maximum value when the coil is horizontal to the magnetic lines of force (Figure 2, 90°). The horizontal coil is moving perpendicular to the field and is cutting the greatest number of magnetic lines of force. As the coil continues to turn, the voltage and current induced decrease until they reach zero, where the coil is again in the vertical position (Figure 2, 180°). In the other half revolution, an equal voltage is produced except that the polarity is reversed (Figure 2, 270°, 360°). The current flow through R is now from X to Y (Figure 1).
Figure 2 Developing a Sine-Wave Voltage
The periodic reversal of polarity results in the generation of a voltage, as shown in Figure 2. The rotation of the coil through 360° results in an AC sine wave output.
Summary
AC generation is summarized below.
AC Generation Summary
A simple generator consists of a conductor loop turning in a magnetic field, cutting across the magnetic lines of force.
The sine wave output is the result of one side of the generator loop cutting lines of force. In the first half turn of rotation this produces a positive current and in the second half of rotation produces a negative current. This completes one cycle of AC generation.
Generation Analysis.ocess.
Analysis of the AC power generation process and of the alternating current we use in almost every aspect of our lives is necessary to better understand how AC power is used in today’s technology.
EO1.3 DEFINE the following terms in relation to AC generation:
a. Radians/second
b. Hertz
c. Period
EO 1.4DEFINE effective value of an AC current relative to DC current.
EO 1.5Given a maximum value, CALCULATE the effective (RMS) and average values of AC voltage.
EO 1.6Given a diagram of two sine waves, DESCRIBE the phase relationship between the two waves.
Effective Values
The output voltage of an AC generator can be expressed in two ways. One is graphically by use of a sine wave (Figure 3). The second way is algebraically by the equation
e = Emax,sin cot, which will be covered later in the text.
Figure 3 Voltage Sine Wave
When a voltage is produced by an AC generator, the resulting current varies in step with the voltage. As the generator coil rotates 360°, the output voltage goes through one complete cycle. In one cycle, the voltage increases from zero to Emax in one direction, decreases to zero, increases to Emax in the opposite direction (negative E., ), and then decreases to zero again. The value of Emax occurs at 90° and is referred to as peak voltage. The time it takes for the generator to complete one cycle is called the period, and the number of cycles per second is called the frequency (measured in hertz).
One way to refer to AC voltage or current is by peak voltage (EP) or peak current (Ip). This is the maximum voltage or current for an AC sine wave.
Another value, the peak-to-peak value (Ep-por Ip-p), is the magnitude of voltage, or current range, spanned by the sine wave. However, the value most commonly used for AC is effective value. Effective value of AC is the amount of AC that produces the same heating effect as an equal amount of DC. In simpler terms, one ampere effective value of AC will produce the same amount of heat in a conductor, in a given time, as one ampere of DC. The heating effect of a given AC current is proportional to the square of the current. Effective value of AC can be calculated by squaring all the amplitudes of the sine wave over one period, taking the average of these values, and then taking the square root. The effective value, being the root of the mean (average) square of the currents, is known as the root-mean-square, or RMS value. In order to understand the meaning of effective current applied to a sine wave, refer to Figure 4.
The values of I are plotted on the upper curve, and the corresponding values of I2 are plotted on the lower curve. The I2 curve has twice the frequency of I and varies above and below a new axis. The new axis is the average of the I2 values, and the square root of that value is the RMS, or effective value, of current. The average value is 1/2 Imax2. The RMS value is then
There are six basic equations that are used to convert a value of AC voltage or current to another value, as listed below.
The values of current (I) and voltage (E) that are normally encountered are assumed to be RMS values; therefore, no subscript is used.
Figure 4 Effective Value of Current
Another useful value is the average value of the amplitude during the positive half of the cycle. Equation (7-7) is the mathematical relationship between Iav, Imax , and I.
Equation (7-8) is the mathematical relationship between Eav ,Emax , and E.
Example 1: The peak value of voltage in an AC circuit is 200 V. What is the RMS value of the voltage?
Example 2: The peak current in an AC circuit is 10 amps. What is the average value of current in the circuit?
Calculations
Equation (7-9) is a mathematical representation of the voltage associated with any particular orientation of a coil (inductor).
Example 1: What is the induced EMF in a coil producing a maximum EMF of 120 V when the angle from reference is 45°?
The maximum induced voltage can also be called peak voltage Ep. If (t) is the time in which the coil turns through the angle (0), then the angular velocity (co) of the coil is equal to 0/t and is expressed in units of radians/sec. Equation (7-10) is the mathematical representation of the angular velocity.
Using substitution laws, a relationship between the voltage induced, the maximum induced voltage, and the angular velocity can be expressed. Equation (7-11) is the mathematical representation of the relationship between the voltage induced, the maximum voltage, and the angular velocity, and is equal to the output of an AC Generator.
maximum induced EMF (volts) angular velocity (radians/sec) time to turn through the angle from reference (sec)
Current Calculations
Maximum induced current is calculated in a similar fashion. Equation (7-12) is a mathematical representation of the relationship between the maximum induced cu
The total power delivered by the source is the apparent power.Part of this apparent power, called true power, is dissipated by the circuit resistance in the form of heat. The rest of the apparent power is returned to the source by the circuit inductance and capacitance.
Power Factor
Power factor (pf) is the ratio between true power and apparent power. True power is the power consumed by an AC circuit, and reactive power is the power that is stored in an AC circuit. Cos is called the power factor (pf) of an AC circuit. It is the ratio of true power to apparent power, where is the phase angle between the applied voltage and current sine waves and also between P and S on a power triangle (Figure2). Equation (9-4) is a mathematical representation of power factor.
Power factor also determines what part of the apparent power is real power. It can vary from 1, when the phase angle is 0°, to 0, when the phase angle is 90°. In an inductive circuit, the current lags the voltage and is said to have a lagging power factor, as shown in Figure 2.
In a capacitive circuit, the current leads the voltage and is said to have a leading power factor, as shown in Figure 3.
A mnemonic memory device, “ELI the ICE man,” can be used to remember the voltage/current relationship in AC circuits. ELI refers to an inductive circuit (L) where current (I) lags voltage (E). ICE refers to a capacitive circuit (C) where current (I) leads voltage (E).
There are many natural causes of capacitance in AC power circuits, such as transmission lines, fluorescent lighting, and computer monitors. Normally, these are counteracted by the inductors previously discussed. However, where capacitors greatly outnumber inductive devices, we must calculate the amount of capacitance to add or subtract from an AC circuit by artificial means.
EO 1.5 DEFINE capacitive reactance (Xc).
EO 1.6 Given the operating frequency (f) and the value of capacitance (C), CALCULATE the capacitive reactance (Xc) of a simple AC circuit.
EO 1.7 DESCRIBE the effect on phase relationship between current (I) and voltage (E) in a capacitive circuit.
EO 1.8 DRAW a simple phasor diagram representing AC current (I) and voltage (E) in a capacitive circuit.
Capacitors
The variation of an alternating voltage applied to a capacitor, the charge on the capacitor, and the current flowing through the capacitor are represented by Figure 3.
The current flow in a circuit containing capacitance depends on the rate at which the voltage changes. The current flow in Figure 3 is greatest at points a, c, and e. At these points, the voltage is changing at its maximum rate (i.e., passing through zero). Between points a and b, the voltage and charge are increasing, and the current flow is into the capacitor, but decreasing in value. At point b, the capacitor is fully charged, and the current is zero. From points b to c, the voltage and charge are decreasing as the capacitor discharges, and its current flows in a direction opposite to the voltage. From points c to d, the capacitor begins to charge in the opposite direction, and the voltage and current are again in the same direction.
Figure 3 Voltage, Charge, and Current in a Capacitor
At point d, the capacitor is fully charged, and the current flow is again zero. From points d to e, the capacitor discharges, and the flow of current is opposite to the voltage. Figure 3 shows the current leading the applied voltage by 90°. In any purely capacitive circuit, current leads applied voltage by 90°
Calculations
The frequency of an alternating voltage or current can be related directly to the angular velocity of a rotating coil. The units of angular velocity are radians per second, and 2 radians is a full revolution. A radian is an angle that subtends an arc equal to the radius of a circle. One radian equals 57.3 degrees. One cycle of the sine wave is generated when the coil rotates 2 radians. Equation (7-13) is the mathematical relationship between frequency (f) and the angular velocity ( ) in an AC circuit.
Example 1: The frequency of a 120 V AC circuit is 60 Hz. Find the following:
1. Angular velocity
2. Angle from reference at 1 msec
3. Induced EMF at that point
Solution:
Summary
- AC generation analysis is summarized below.
- Voltage, Current, and Frequency Summary
- The following terms relate to the AC cycle: radians/second, the velocity the loop turns; hertz, the number of cycles in one second; period, the time to complete one cycle.
- Effective value of AC equals effective value of DC.
- Root mean square (RMS) values equate AC to DC equivalents:
Phase angle is used to compare two wave forms. It references the start, or zero point, of each wave. It compares differences by degrees of rotation. Wave forms with the same start point are “in-phase” while wave forms “out-of-phase” either lead or lag.
Phase Angle
Phase angle is the fraction of a cycle, in degrees, that has gone by since a voltage or current has passed through a given value. The given value is normally zero. Referring back to Figure 3, take point 1 as the starting point or zero phase. The phase at Point 2 is 30°, Point 3 is 60°, Point 4 is 90°, and so on, until Point 13 where the phase is 360°, or zero. A term more commonly used is phase difference. The phase difference can be used to describe two different voltages that have the same frequency, which pass through zero values in the same direction at different times. In Figure 5, the angles along the axis indicate the phases of voltages el and e2 at any point in time. At 120°, el passes through the zero value, which is 60° ahead of e2 (e2 equals zero at 180°). The voltage el is said to lead e2 by 60 electrical degrees, or it can be said that e2 lags el by 60 electrical degrees.
Figure 5 Phase Relationship
Phase difference is also used to compare two different currents or a current and a voltage. If the phase difference between two currents, two voltages, or a voltage and a current is zero degrees, they are said to be “in-phase.” If the phase difference is an amount other than zero, they are said to be “out-of-phase.”
Whenever inductive and capacitive components are used in an AC circuit, the calculation of their effects on the flow of current is important.
EO 1.9 DEFINE impedance (Z).
EO1.10 Given the values for resistance (R) and inductance (L) and a simple R-L series AC circuit, CALCULATE the impedance (Z) for that circuit.
EO1.11 Given the values for resistance (R) and capacitance (C) and a simple R-C series AC circuit, CALCULATE the impedance (Z) for that circuit.
EO1.12 Given a simple R-C-L series AC circuit and the values for resistance (R), inductive reactance (XL), and capacitive reactance (Xc), CALCULATE the impedance (Z) for that circuit.
EO1.13 STATE the formula for calculating total current (IT) in a simple parallel R-C-L AC circuit.
EO1.14 Given a simple R-C-L parallel AC circuit and the values for voltage (VT), resistance (R), inductive reactance (XL), and capacitive reactance (Xc), CALCULATE the impedance (Z) for that circuit.
Impedance
No circuit is without some resistance, whether desired or not. Resistive and reactive components in an AC circuit oppose current flow. The total opposition to current flow in a circuit depends on its resistance, its reactance, and the phase relationships between them. Impedance is defined as the total opposition to current flow in a circuit. Equation (8-6) is the mathematical representation for the magnitude of impedance in an AC circuit.
The relationship between resistance, reactance, and impedance is shown in Figure 5.
Figure 5 Relationship Between Resistance, Reactance, and Impedance
The current through a certain resistance is always in phase with the applied voltage. Resistance is shown on the zero axis. The current through an inductor lags applied voltage by 90°; inductive reactance is shown along the 90° axis. Current through a capacitor leads applied voltage by 90°; capacitive reactance is shown along the -90° axis. Net reactance in an AC circuit is the difference between inductive and capacitive reactance. Equation (8-7) is the mathematical representation for the calculation of net reactance when XL is greater than Xc.
Equation (8-8) is the mathematical representation for the calculation of net reactance when Xc is greater than XL.
Impedance is the vector sum of the resistance and net reactance (X) in a circuit, as shown in Figure 5. The angle 0 is the phase angle and gives the phase relationship between the applied voltage and the current. Impedance in an AC circuit corresponds to the resistance of a DC circuit. The voltage drop across an AC circuit element equals the current times the impedance. Equation (8-9) is the mathematical representation of the voltage drop across an AC circuit.
where
The phase angle 0 gives the phase relationship between current and the voltage.
Impedance in an R-C-L series circuit is equal to the phasor sum of resistance, inductive reactance, and capacitive reactance (Figure 8).
Figure 8 Series R-C-L Impedance-Phasor
Equations (8-12) and (8-13) are the mathematical representations of impedance in an R-C-L circuit. Because the difference between XL and Xc is squared, the order in which the quantities are subtracted does not affect the answer.
Example: Find the impedance of a series R-C-L circuit, when R = 6 , XL = 20 , and Xc = 10 (Figure 9).
Figure 9 Simple R-C-L Circuit
Solution:
Impedance in a parallel R-C-L circuit equals the voltage divided by the total current. Equation (8-14) is the mathematical representation of the impedance in a parallel R-C-L circuit.
where
Total current in a parallel R-C-L circuit is equal to the square root of the sum of the squares of the current flows through the resistance, inductive reactance, and capacitive reactance branches of the circuit. Equations (8-15) and (8-16) are the mathematical representations of total current in a parallel R-C-L circuit. Because the difference between IL and Ic is squared, the order in which the quantities are subtracted does not affect the answer.
where
Example:
A 200 resistor, a 100 XL,and an 80 Xc are placed in parallel across a 120V AC source (Figure 10). Find: (1) the branch currents, (2) the total current, and (3) the impedance.
Figure 10 Simple Parallel R-C-L Circuit Solution:
Summary
Impedance (Z) is the total opposition to current flow in an AC circuit. The formula for impedance in a series AC circuit is:
The formula for impedance in a parallel R-C-L circuit is:
The formulas for finding total current (IT) in a parallel R-C-L circuit are:
Impedance is the resultant of phasor addition of R and XL. The symbol for impedance is Z. Impedance is the total opposition to the flow of current and is expressed in ohms. Equation (8-10) is the mathematical representation of the impedance in an RL circuit.
Example: If a 100 resistor and a 60 XL are in series with a 115V applied voltage (Figure 6), what is the circuit impedance?
Figure 6 Simple R-L Circuit
Solution:
Impedance in R-C Circuits
In a capacitive circuit, as in an inductive circuit, impedance is the resultant of phasor addition of R and XC. Equation (8-11) is the mathematical representation for impedance in an R-C circuit.
Example: A 50 52 Xc and a 60 52 resistance are in series across a 110V source (Figure 7). Calculate the impedance.
Figure 7 Simple R-C Circuit
Solution:
Analysis the Power Generation Process.
Any device relying on magnetism or magnetic fields to operate is a form of inductor. Motors, generators, transformers, and coils are inductors. The use of an inductor in a circuit can cause current and voltage to become out-of-phase and inefficient unless corrected.
EO1.1 DESCRIBE inductive reactance (XL).
EO1.2 Given the operation frequency (f) and the value of inductance (L), CALCULATE the inductive reactance (XL) of a simple circuit.
EO1.3 DESCRIBE the effect of the phase relationship between current and voltage in an inductive circuit.
EO1.4 DRAW a simple phasor diagram representing AC current (I) and voltage (E) in an inductive circuit.
Inductive Reactance
In an inductive AC circuit, the current is continually changing and is continuously inducing an EMF. Because this EMF opposes the continuous change in the flowing current, its effect is measured in ohms. This opposition of the inductance to the flow of an alternating current is called inductive reactance (XL). Equation (8-1) is the mathematical representation of the current flowing in a circuit that contains only inductive reactance.
The value of XL in any circuit is dependent on the inductance of the circuit and on the rate at which the current is changing through the circuit. This rate of change depends on the frequency of the applied voltage. Equation (8-2) is the mathematical representation for XL.
The magnitude of an induced EMF in a circuit depends on how fast the flux that links the circuit is changing. In the case of self-induced EMF (such as in a coil), a counter EMF is induced in the coil due to a change in current and flux in the coil. This CEMF opposes any change in current, and its value at any time will depend on the rate at which the current and flux are changing at that time. In a purely inductive circuit, the resistance is negligible in comparison to the inductive reactance. The voltage applied to the circuit must always be equal and opposite to the EMF of self-induction.
Voltage and Current Phase Relationships in an Inductive Circuit
As previously stated, any change in current in a coil (either a rise or a fall) causes a corresponding change of the magnetic flux around the coil. Because the current changes at its maximum rate when it is going through its zero value at 90° (point b on Figure 1) and 270° (point d), the flux change is also the greatest at those times. Consequently, the self-induced EMF in the coil is at its maximum (or minimum) value at these points, as shown in Figure 1. Because the current is not changing at the point when it is going through its peak value at 0° (point a), 180° (point c), and 360° (point e), the flux change is zero at those times. Therefore, the selfinduced EMF in the coil is at its zero value at these points.
Figure 1 Current, Self-Induced EMF, and Applied Voltage in an Inductive Circuit
According to Lenz’s Law (refer to Module 1, Basic Electrical Theory), the induced voltage always opposes the change in current. Referring to Figure 1, with the current at its maximum negative value (point a), the induced EMF is at a zero value and falling. Thus, when the current rises in a positive direction (point a to point c), the induced EMF is of opposite polarity to the applied voltage and opposes the rise in current. Notice that as the current passes through its zero value (point b) the induced voltage reaches its maximum negative value. With the current now at its maximum positive value (point c), the induced EMF is at a zero value and rising. As the current is falling toward its zero value at 180° (point c to point d), the induced EMF is of the same polarity as the current and tends to keep the current from falling. When the current reaches a zero value, the induced EMF is at its maximum positive value. Later, when the current is increasing from zero to its maximum negative value at 360° (point d to point e), the induced voltage is of the opposite polarity as the current and tends to keep the current from increasing in the negative direction. Thus, the induced EMF can be seen to lag the current by 90°.
The value of the self-induced EMF varies as a sine wave and lags the current by 90°, as shown in Figure 1. The applied voltage must be equal and opposite to the self-induced EMF at all times; therefore, the current lags the applied voltage by 90° in a purely inductive circuit.
If the applied voltage (E) is represented by a vector rotating in a counterclockwise direction (Figure lb), then the current can be expressed as a vector that is lagging the applied voltage by 90°. Diagrams of this type are referred to as phasor diagrams.
Example: A 0.4 H coil with negligible resistance is connected to a 115V, 60 Hz power source (see Figure 2). Find the inductive reactance of the coil and the current through the circuit. Draw a phasor diagram showing the phase relationship between current and applied voltage.
Figure 2 Coil Circuit and Phasor Diagram
Solution:
1. Inductive reactance of the coil
2. Current through the circuit
3. Draw a phasor diagram showing the phase relationship between current and applied voltage.
Phasor diagram showing the current lagging voltage by 90° is drawn in Figure 2b.
Summary
Opposition to the flow of alternating current caused by inductance is called Inductive Reactance (XL).
The formula for calculating XL is:
Current (I) lags applied voltage (E) in a purely inductive circuit by 90° phase angle. The phasor diagram shows the applied voltage (E) vector leading (above) the current (I) vector by the amount of the phase angle differential due to the relationship between voltage and current in an inductive circuit.
Resonance in a parallel R-C-L circuit will occur when the reactive current in the inductive branches is equal to the reactive current in the capacitive branches (or when XL = Xc). Because inductive and capacitive reactance currents are equal and opposite in phase, they cancel one another at parallel resonance.
If a capacitor and an inductor, each with negligible resistance, are connected in parallel and the frequency is adjusted such that reactances are exactly equal, current will flow in the inductor and the capacitor, but the total current will be negligible. The parallel C-L circuit will present an almost infinite impedance. The capacitor will alternately charge and discharge through the inductor. Thus, in a parallel R-C-L, as in Figure 10, the net current flow through the circuit is at minimum because of the high impendence presented by XL and Xc in parallel.
Summary
Resonance is a state in which the inductive reactance equals the capacitive reactance (XL = Xc) at a specified frequency (fRes).
Resonant frequency is:
R-C-L series circuit at resonance is when net reactance is zero and circuit current output is determined by the series resistance of the circuit.
R-C-L parallel circuit at resonance is when net reactance is maximum and circuit current output is at minimum.
Under Construction
Under Construction
Under Construction
Under Construction
Under Construction
Under Construction
Under Construction
Under Construction
Under Construction