Phase Relationships in a Series RL Circuit

A typical series RL circuit is shown in Figure 9.1. In it one resistor is connected in series with one inductor. In a series RL circuit, as in any series circuit, the current through all the components is the same. However, the sum of the voltage drop across the resistor and the voltage drop across the inductor, do not simply add algebraically to equal the applied voltage as they would in either a purely resistive or a purely reactive circuit. This is because of the combination of resistance and reactance and the different phase relationship between the voltage and current for each component.

Recall, as shown in Figure 9.2, that the voltage across a resistor is in phase with the current flowing through it; but for an inductor, however, the voltage leads the current by 90 degrees. Since the components are in series with one another, the current through each component is the same. Therefore, mathematically,

Using the current then as a common basis for comparison, the two individual diagrams shown in Figure 9.2 can be combined into one as illustrated in Figure 9.3. In this composite phasor diagram note that the voltage across the resistor is in phase with the current in the series circuit, while the voltage across the inductor leads this current by 90 degrees. If the phase relationships of the two voltage drops are compared, the voltage across the inductor leads the voltage across the resistor by 90 degrees.

Phasor Addition of Voltages

Because of their phase difference, these voltages cannot be added algebraically as one normally would add the voltage drops in a series circuit to obtain the total applied voltage. Instead, these voltages must be added vectorially, as show in Figure 9.4. This is like the voltages in a series RC circuit are added. Using the parallogram method, the vectorial sum of E_R and E_L is equal to the applied voltage of E_A. This phasor diagram is called the voltage phasor diagram for this circuit.

Derivation of Impedance Phasor Diagram

Now, by applying Ohm’s law,

(9–1)

Since the current, through the resistor in this series RL circuit is the total current,

(9–2)

Remember also that

(9–3)

and that the current through the inductor is the same as the total current. Therefore, equation 9–3 could also be written as

(9–4)

These IR and IX_L quantities can be substituted for the voltages they equal on the voltage phasor diagram as shown in Figure 9.5a. Since the total current is the common factor here, it may be factored out so that the phase relationship between R and X_L remains as shown in Figure 9.5b.

Recall that in a purely resistive series circuit the sum of all ohms of resistance equals the total ohms of resistance. And in a purely reactive series circuit the sum of all ohms of reactance equals the total ohms of reactance. In an RL circuit, however, there exists a combination of resistance and reactance like in a series RC circuit, and it is called impedance, Z, measured in ohms, as illustrated in Figure 9.6.

Calculating Impedance

Impedance of an RC circuit can be calculated by adding the resistance and reactance vectorially on the phasor diagram. This is accomplished by simply applying the Pythagorean theorem. As illustrated in Figure 9.6, the length of the reactance vector, X_L, is the same as the length between the tip of the resistance vector and the tip of the impedance vector Z. In fact, the phasor diagram can be drawn with the reactance vector placed as shown in Figure 9.7. Now, the right triangle of the Pythagorean theorem becomes clearly evident. Applying the Pythagorean theorem, the total impedance is equal to the square root of the resistance squared plus the inductive reactance squared.

(9–5)

For example, a typical series RL circuit is shown in Figure 9.8. If the resistance in that circuit is 60 ohms and the inductive reactance is 80 ohms, then the total impedance of the circuit can be determined as follows:

The total impedance of that circuit is 100 ohms.

When drawing voltage and impedance diagrams for circuits like this, it is often convenient to draw them in the form of a right triangle as shown in Figure 9.9.

Mathematical Relationship of Voltages

Using a voltage phasor diagram as shown in Figure 9.10, the same type of mathematical calculation can be used to show the relationships between the voltage drops in the circuit and the applied voltage.

Shifting the inductive voltage vector as shown in Figure 9.11, and using the Pythagorean theorem, the applied voltage is equal to the square root of the voltage across the resistor squared plus the voltage across the inductor squared:

(9–6)

Example of Voltage Calculation

Figure 9.12 is a typical series RL circuit and will be used to show you how to calculate E_Ain such a circuit. The voltage across the resistor in that circuit is 20 volts, and the voltage across the inductor is 15 volts. The total applied voltage can be determined using equation 9–6.

Figure 9.13 shows the right triangle relationship of the example circuit’s E_A, E_L, and E_R.

Solution of a Series RL Circuit

Now, if you are given an applied voltage and values for R and L for a series RL circuit, you should be able to determine the total impedance, the total current, and the individual voltage drops across the resistor and inductor. For example, in the series RL circuit of Figure 9.14, the value of the resistor is 75 ohms, and the value of the inductor is 4 millihenrys. The applied voltage is 250 volts with a frequency of 4 kilohertz. To determine the total impedance of this circuit, the value of the inductive reactance must first be calculated.

The impedance phasor diagram shown in Figure 9.15 can now be drawn to show the relationships between the values of resistance, reactance, and impedance in the circuit. The value of impedance is calculated using the Pythagorean theorem. The impedance is equal to the square root of the resistance squared plus the inductive reactance squared:

Remember that impedance is the total opposition of the resistive-reactive circuit to the flow of alternating current. Recall that in Chapter 7, this fact was used to determine a special form of Ohm’s law which is called Ohm’s law for ac circuits:

(9–7)

Rewriting this equation and solving for the total current, the total current equals the applied voltage divided by the total impedance.

(9–8)

If this equation is applied to the example series RL circuit:

The voltage drop across the resistor, ER, and inductor, E_L, can now be calculated as they would in any series circuit. The voltage drop across the resistor is equal to the current through it times the value of the resistance.

(9–9)

The current through the resistor is the same as the total current. Thus,

The voltage drop across the inductor is equal to the current through it times the value of the inductive reactance.

(9–10)

again, I_L is the same as the total current. Therefore,

If the voltage across the resistor, 150 volts, and the voltage across the inductor, 200 volts, are added, the result is 350 volts. That is more than the applied voltage. But remember that the voltage across the resistor and the voltage across the inductor are out of phase by 90 degrees. Therefore they must be added vectorially. Recall that vectorially, the applied voltage equals the square root of the voltage across the resistor squared, plus the voltage across the inductor squared. Therefore, for the example circuit

Thus, you can see that the vector sum of the circuit voltage drops does equal the applied voltage, 250 volts.

This technique is a valuable tool for circuit analysis because the calculation can be used to check the accuracy of calculations performed to determine the various individual voltage drops in the circuit. Simply be sure that the vector sum of the circuit voltage drops equals the applied voltage.

Phase Angle in a Series RL Circuit

Recall that when originally forming the voltage phasor diagram shown in Figure 9.16, the current in the circuit was used as a reference quantity since it is the same throughout the circuit. This total current is in phase with the voltage across the resistor. Notice, however, that the applied voltage and the total current are out of phase. More specifically, the applied voltage leads the total current by a number of degrees. Recall that this angle by which the total applied voltage and the total current are out of phase is called the phase angle of the circuit. The phase angle is the number of degrees by which the current being drawn from the ac voltage source and the voltage of the ac voltage source are out of phase. The phase angle in a series RL circuit can also be recognized as the angle between the voltage across the resistor and the applied voltage, as shown in Figure 9.16.

Since the impedance phasor diagram is proportional to the voltage phasor diagram by the common factor of total current which cancels, X_L can be substituted for E_L, Z can be substituted for E_A, and R can be substituted for E_R as shown in Figure 9.17. Therefore, the phase angle in a series RL circuit is also the angle between the resistance phasor and the impedance phasor. This is similar to a series RC circuit. Recall that when determining the value of this phase angle, a trigonometric function was used called the tangent function. Recall that the tangent of an angle in a right triangle is equal to the ratio of the length of the opposite side divided by the length of the adjacent side as shown in Figure 9.18.

The side opposite to the phase angle, θ, in the voltage phasor diagram of Figure 9.19 is the vector E_L. The length of the vector represents the value of the voltage drop across the inductor. The side adjacent to the phase angle, in the voltage phasor diagram, is the vector E_R whose length represents the value of the voltage drop across the resistor. The tangent of the phase angle, θ, is equal to the ratio of the opposite side divided by the adjacent side:

(9–11)

(9–12)

The value of the phase angle, θ, then is simply the arctangent of this ratio.

(9–13)

Remember that arctangent also can be abbreviated, tan to the minus one:

(9–14)

Calculating Phase Angle from Voltage Phasor Diagram

As shown in Figure 9.20, in the series RL circuit of Figure 9.14 previously solved, the voltage across the inductor EL was determined to be 200 volts, and the voltage across the resistor E_R was 150 volts. Therefore theta is:

Therefore, the phase angle of the example series RL circuit is 53 degrees, a positive 53 degrees.

Series RL Phase Angle Is Positive

In the chapter about RC circuits an arbitrary standard was established for the angle of rotation of vectors for series circuits. It was established that the angle of rotation of a phasor in a counter-clockwise direction from the zero degree reference forms a positive phase angle as shown in Figure 9.21. Therefore, the positive sign of the phase angle, as shown in Figure 9.22, indicates that the applied voltage is rotated 53 degrees counter-clockwise (or up) from the current vector direction. Phase angles in series RL circuits will always be positive phase angles.

Calculating Phase Angle from Impedance Phasor Diagram

Recall that earlier it was described how the impedance phasor diagram is proportional to the voltage phasor diagram by a factor of the total current for a series circuit. Since these two phasor diagrams are proportional, it is possible to determine the phase angle from the impedance phasor diagram as illustrated in Figure 9.23. The side opposite to the phase angle, θ, in the impedance phasor diagram, is the reactance vector, X_L. The length of this vector represents the value of the inductive reactance. The side adjacent to the phase angle in the impedance phasor diagram is the resistance vector, R. Its length represents the value of the resistor. The tangent of the phase angle, then, is equal to the ratio of the opposite side, X_L, divided by the adjacent side, R.

The arctangent of the ratio will yield the value of the phase angle. Recall that in the series RL circuit, the value of the inductive reactance, X_L, is 100 ohms, and the value of the resistance R is 75 ohms. Therefore theta is:

(9–14a)

Thus, it should be apparent that the value of the phase angle in a series RL circuit can be determined from either the voltage or impedance phasor diagram.

Alternative Series RL Circuit Solution Method

It might be instructive to note that the method of solution for values of impedance, current, voltages, and phase angle is not limited to the order of solutions illustrated in the previous section. Recall again from Chapter 2 that

(2–21)

and

(2–22)

In the impedance phasor diagram of Figure 9.17, the opposite side to the phase angle (θ) is the capacitive reactance vector (X_L). The adjacent side to the phase angle (θ) is the resistive vector (R). The hypotenuse is the impedance vector (Z). Substituting these values into the sine or cosine function equations shown above results in

(9–14b)

or

(9–14c)

Rewriting these equations produces

(9–14d)

or

(9–14e)

An alternative solution method follows using these new equations (9–14d and 9–14e). If you are given the values of R, C, applied voltage, and frequency in a series RC circuit such as the one shown in Figure 9.14, the values of X_C, phase angle, impedance, circuit current, and inductive and resistive voltages can be found using the following alternative method.

First, determine the inductive reactance (X_L) using the inductive reactance equation as done earlier.

Then, using equation 9–14a, the phase angle can be found using the tangent function.

Now, using equation 9–14d above:

or using equation 9–14e above:

Note the values of Z match the value calculated earlier. If current and voltage values are also desired, they can be found now using the following calculations.

and

What is illustrated here, then, is a different approach to determining the same values calculated previously.

Power Calculations in Series RL Circuit

Now, attention will be focused upon power calculations involved in the series RL circuit. These calculations are very similar to power calculations for series RC circuits.

The real power, P_R, dissipated by the resistor in Figure 9.24 can be calculated as it was in a purely resistive series circuit. Simply multiply the voltage drop across the resistor (150 volts in the example) times the current’s value flowing through it (2 amperes).

The reactive power, P_L, of the inductor is calculated as it was in a purely inductive circuit. Simply multiply the voltage across the inductor (200 volts in the example) times the current flowing through it (2 amperes).

Recall that the total power in a resistive-reactive circuit is called the apparent power, measured in volt-amperes. The total or apparent power can be found by multiplying the applied voltage (250 volts in the example) times the total current (2 amperes).

Remember, that the simple sum of the real power and the reactive power does not equal the apparent power. In the example, 500 volt-amperes does not equal 300 watts plus 400 VAR when they are added together.

or

This occurs because of the different phase relationships between the voltage and current for each component in the circuit. The phase relationships of the three power determinations are similar to the voltage phase relationships. Figure 9.25 shows the voltage phasor diagram with its values E_A, E_R, and E_L. If each voltage is multiplied by the total current, I_T, in the circuit, it will be found that the power phasor diagram is very similar to the voltage phasor diagram because as shown in Figure 9.26

(9–15)

(9–16)

and

(9–17)

In fact, the values are proportional to the voltage diagrams by the common factor of total current, I_T.

Using the Pythagorean theorem, the apparent power, P_A, is equal to the square root of the real power, P_R, squared plus the reactive power, P_L, squared:

(9–18)

Thus, the total apparent power in this circuit, 500 volt-amperes, should be equal to the square root of 300 watts squared plus 400 VAR squared. By completing the calculation, it is found that the apparent power is equal to 500 volt-amperes:

This is the same value obtained by multiplying the applied voltage by total current.

As shown in Figure 9.27, the angle 53 degrees between the real power, P_R, and apparent power, P_A, is the same as the angle between applied voltage, E_A, and resistive voltage, E_R. As you can see, the methods used to determine the voltages, current, impedance, and phase angle and power throughout a series RL circuit are very similar to the methods used to determine the same quantities for a series RC circuit.

Phase Relationships in a Parallel RL Circuit

In this circuit, as in any parallel circuit, the voltage across all components is the same:

However, the simple sum of the branch currents does not equal the total current in the circuit:

This occurs because of the different phase relationships between the voltage and current of each component.

Recall that the voltage across the resistor is in phase with the current through it; in the inductor, the voltage leads the current by 90 degrees as shown in Figure 9.29. Since the components in this circuit are in parallel with one another, the common factor in both phasor diagrams is the voltage across the components:

However, to illustrate the phase relationships in a parallel circuit, E_L and I_L must be rotated 90 degrees as shown in Figure 9.30a. Then the two individual phasor diagrams can be combined, as shown in Figure 9.30b. Not that the current through the resistor is shown in phase with the applied voltage across it while the voltage across the inductor leads the current through it by 90 degrees.

Phasor Addition of Currents

Comparing the phase relationships of the two branch currents, the current through the resistor leads the current through the inductor by 90 degrees. These individual branch currents can be calculated as they were in either a purely resistive or purely inductive circuit. Simply divide the voltage across the branch by the opposition to current in that branch.

In the resistive branch of the example circuit, the opposition to flow of current is measured in ohms of resistance; thus, in the example circuit, the resistive current is determined by dividing the applied voltage across the resistor by the value of the resistor:

(9–19)

In the inductive branch, the opposition to the flow of current is measured in ohms of reactance; thus, in the example circuit, the inductive branch current is determined by dividing the applied voltage across the inductor by the inductive reactance:

(9–20)

These currents cannot be added algebraically to obtain the total current as the branch currents in a parallel resistive circuit can be added. The currents must be added vectorially because of the different phase relationship in a parallel RL circuit. The same procedure is used that was used in other diagrams to form a right triangle between the tip of the inductive current vector and the tip of the resistive current vector, with the total current vector, I_T, completing the hypotenuse of the triangle. This is illustrated in Figure 9.31.

Now the Pythagorean theorem can be used to determine the total current in the circuit. By the Pythagorean theorem, the total current in a parallel RL circuit is equal to the square root of the resistive current squared plus the inductive current squared:

(9–21)

Parallel RL Circuit Example

As an example, the circuit in Figure 9.32 will be used. In it, the resistive current is 28 milliamperes and the inductive current is 21 milliamperes. Total current is calculated:

Thus, the total current in the circuit is 35 milliamperes.

Determining Impedance of a Parallel RL Circuit

Once the total current is known, the total impedance of the circuit can be determined using Ohm’s law for ac circuits:

(9–22)

However, this time the equation must be rewritten to solve for impedance. The total impedance of the circuit is equal to the applied voltage divided by the total current:

(9–23)

For example, if in the previous example circuit the total applied voltage was 70 volts as shown in Figure 9.33, then the circuit impedance is equal to the applied voltage, 70 volts, divided by the total current, 35 millamperes:

Phase Angle in a Parallel RL Circuit

Now the phase angle of the example RL circuit will be determined by utilizing the original current phasor diagram shown in Figure 9.34. Recall that the phase angle is the number of degrees of phase difference between the applied voltage and the total current. Also recall that the applied voltage, E_A, is also the voltage across the resistor, E_R, and that the current through the resistor is in phase with the voltage across it. Therefore, the applied voltage is in phase with the resistive current, I_R. The voltage across the inductor, E_L, is the same as the applied voltage and it leads the current in the inductor by 90 degrees. Therefore, I_L forms the right angle with I_R, and I_T is the hypotenuse of the right triangle.

Since the applied voltage, E_A, is in phase with the current through resisor I_R, the phase angle is the angle, θ, between I_R and I_T. And, as you can see, the phase angle is a parallel RL circuit is simply the angle between the resistive current and total current.

The value of the phase angle of a parallel RL circuit can be calculated by determining the arctangent of the ratio of the opposite and adjacent sides. The length of the opposite side represents the value of the inductive current. The length of the adjacent side represents the value of the resistive current. Mathematically, therefore:

(9–24)

or

(9–25)

or

(9–26)

Branch Currents

In the parallel RL circuit shown in Figure 9.35, the value of the resistor is 5 kilohms and the value of the inductor is 1.9 henrys. Therefore, at the applied frequency of 1,000 hertz, the inductor has an inductive reactance of 12 kilohms. This is determined by the inductive reactance equation:

(9–27)

The applied voltage is 120 volts. The resistive branch current is found by dividing the voltage across the resistor, 120 volts, by the value of the resistor, 5 kilohms, which equals 24 milliamperes:

The inductive branch current is found in a similar manner by dividing the voltage across the capacitor, 120 volts, by the inductive reactance, 12 kilohms, which equals 10 milliamperes:

The current phasor diagram shown in Figure 9.36 can now be drawn to show the relationships between the resistive and inductive branch currents, and the total circuit current.

Total Current

Because of the different phases, total current is a vector sum. Therefore, using the Pythagorean theorem, the total current is equal to the square root of the resistive branch current squared plus the inductive branch current squared. Since I_R is 24 milliamperes and I_L is 10 milliamperes, total current is calculated:

Thus, the value of the total current is 26 milliamperes. Since all of the current values were in milliamperes, the total current is measured in milliamperes also.

Impedance

The total impedance of the example parallel RL circuit can now be determined by dividing the applied voltage by the total current.

The total impedance is 4.6 kilohms.

Phase Angle

Using the current phasor diagram shown in Figure 9.37, it can be seen that the phase angle is the angle between the resistive current and the total current. The value of the phase angle is equal to the arctangent of the ratio of the inductive current divided by the resistive current.

The phase angle is a minus 23 degrees.

Parallel RL Circuit Phase Angle Is Negative

The negative sign is used with the phase angle of this particular example and the phase angle is said to be a negative 23 degrees. This is because the applied voltage in the parallel RL circuit is used as a reference at zero degrees. The negative sign is used to indicate that the total current phasor is rotated 23 degrees clockwise from the applied voltage phase, as shown in Figure 9.38. E_A leads I_T by 23 degrees. Or it could be said that I_T lags E_A by 23 degrees.

You should realize, therefore, that in a series RL circuit, the phase angle is positive as shown in Figure 9.39. However, in a parallel RL circuit, the phase angle is negative, as shown in Figure 9.40. The sign of the phase angle is used simply to indicate the direction of rotation from the reference at zero degrees. As a result, if you know that a circuit is an RL circuit and you know its phase angle. You can readily determine whether the resistor and inductor are connected in series or parallel by knowing the sign of the phase angle.

Alternative Parallel RL Circuit Solution Method

It might be instructive to note that the method of solution for values of currents, impedance, voltages, and phase angle is not limited to the order of solutions illustrated in the previous section. As illustrated previously in this chapter, and in Chapter 2, recall again that

(2–21)

and

(2–22)

In the current phasor diagram of Figure 9.34, the opposite side to the phase angle (θ) is the inductive current vector (I_L). The adjacent side to the phase angle (θ) is the resistive current vector (I_R). The hypotenuse is the total current vector (I_T). Substituting these values into the sine or cosine function equations shown above results in

(9–27a)

or

(9–27b)

Rewriting these equations produces

(9–27c)

or

(9–27d)

An alternative solution method follows using these new equations (9–27c and 9–27d).

If you are given the values of R, L, applied voltage, and frequency in a parallel RL circuit such as the one shown in Figure 9.41, the values of X_L, phase angle, circuit branch currents, total circuit current, and impedance can be determined.

First, determine the capacitive reactance (X_L) using the inductive reactance equation as done earlier.

Then, using equations 9–19 and 9–20, the individual branch currents can be found using Ohm’s law.

(9–19)

(9–20)

Now the phase angle of the circuit can be found using the arctangent function:

Now, using equation 9–27c above with 22.6° (the more exact value of the phase angle)

or using equation 9–27d above

Note that the value of total current matches the value calculated earlier.

If the value of total circuit impedance is desired, it can now be determined using the following relationship:

What is illustrated here, then, is a different approach to determining the same values calculated previously.

Parallel RL Circuit Power Calculations

In a parallel RL circuit, the power relationships are similar to those of a series RL circuit. To show you how to calculate power in a parallel RL circuit, the example parallel RL circuit used previously will be used. It is shown again in Figure 9.42 with some of its circuit values stated.

The real power, P_R, is equal to the voltage across the resistor, 120 volts, times the value of the current flowing through the resistor, 24 milliamperes.

The reactive power is equal to the voltage across the inductor times the value of the current through it:

The total apparent power is equal to the applied voltage times the total current.

Since each power determination is made by multiplying the current shown in the current phasor diagram of Figure 9.43 by the applied voltage, the power phasor diagram is proportional to the current phasor diagram by a factor of the applied voltage. The total apparent power, P_A, is the vector sum of the real and reactive power. That is, the apparent power is equal to the square root of the real power squared plus the reactive power squared:

(9–28)

Therefore, P_A for the example parallel RL circuit is calculated:

The total apparent power calculated this way is 3.12 volt-amperes, the same as it was calculated using the equation P_A = E_AI_T.

Concept of Q

The inductive reactance, X_L, of a coil is an indication of the ability of a coil to produce self-induced voltage. Recall that inductive reactance is expressed mathematically as

The expression takes into consideration the rate of change of current in the circuit in terms of frequency (f) and the size of the inductor in terms of its value (L). To manufacture a coil, however, many turns of wire are used. This results in a resistance in the coil due entirely to the length of the wire. This internal resistance was mentioned in Chapter 8 and is represented by r_i. Figure 9.44 shows that the internal resistance of a coil is represented schematically by showing a small resistor in series with the coil. The coil represents the inductor’s reactive qualities, while the resistor represents its resistive qualities.

When current is passed through a coil, energy is stored in the magnetic field as it expands around the coil, and then it is returned to the circuit when the magnetic field collapses (Figure 9.45). No heat energy is lost in a coil due to the coil’s inductive quality. But the resistance of the wire of the coil dissipates energy in the form of heat just like any other resistor would. This energy is lost to the circuit and is non-recoverable.

Basically, the quality of a coil is based on its ability to store energy in its magnetic field, then return that energy back to the circuit. Essentially, it is a measure of how efficient the coil is as far as its reactive, self-inductive qualities are concerned.

Derivation of Q Equation

Q is defined as the ratio of the reactive power in the inductance to the real power dissipated by its internal resistance. Mathematically,

(9–29)

Figure 9.46 is the equivalent circuit of an inductor to which an ac voltage is applied. The power for the reactive and resistive properties of the coil can be written as:

(9–30)

and

(9–31)

Substituting these power values into the equation for Q,

(9–32)

Factoring out I²,

(9–33)

And writing X_L as 2πfL yields,

(9–34)

In the equations, r_i is the resistance measured by an ohmmeter placed across the coil. L is the inductance of the coil being considered. And f is a standard frequency chosen to compare the Q of different coils. This frequency is usually one kilohertz for large values of L.

Sample Calculation of Q

Thus, to determine the Q of a coil of 8.5 henrys, r_i would be determined by measuring across it with an ohmmeter. In this example, r_i is about 400 ohms. Then using equation 9–34 and with f = one kilohertz, the coil’s Q is calculated:

Factors Affecting Q of a Coil

At low frequencies, r_i is simply the dc resistance of the wire used in the coil, as measured by an ohmmeter. However, as frequency increases, additional losses occur within the coil. In air-core coils, (the type used in high-frequency radio and radar circuits) this additional loss is only one, called skin effect. This is the tendency of high-frequency current to flow near the surface of a conductor. This results from current near the center of the conductor encountering slightly more inductive reactance due to the concentrated magnetic flux in the center compared to the surface, where part of the flux is in the air. This effect increases the effective resistance of the conductor since current flow is limited to a small cross-sectional area near the surface. Because of this effect, conductors for very high-frequency applications are often made of hollow tubing called waveguides.

In an iron core coil, greater losses occur because of eddy currents and hysteresis. These losses are due to the magnetic properties of the iron. These losses effectively increase r_i. Any increase in r_i tends to decrease the Q of the coil at high frequency even though X_L is increasing as frequency increases. At high frequency, therefore, coils with air cores, which do not have added magnetic losses, are normally used to maximize Q.

1. Calculate the voltage, current, power, and phase angle values shown for this series RL circuit.

Solution:

2. For the circuit of Example 1, sketch the voltage, impedance, and power phasor diagrams. Label all phasor lengths and locate the phase angle in each diagram.

    Solution:

3. Calculated the voltage, current, power, and phase angle values shown for this parallel RL circuit.

Solution:

4. For the circuit of Example 3, sketch the current and power phasor diagrams. Label all phasor lengths and locate the phase angle in each diagram.

    Solution:

5. Determine the Q of a 10 mH coil that has an internal resistance of 7 ohms at a frequency of 1 kHz.

    Solution:

1. For the circuit shown, determine the values specified for the circuit values given.

2. Sketch the voltage, impedance and power phasor diagrams for Problem 1c. Label all phasor lengths, locate and identify the phase angle for each diagram.

3. For the circuit shown, determine the values specified for the circuit values given.

4. Sketch the current and power phasor diagram for Problem 3c. Label all phasor lengths, locate and identify the phase angle for each diagram.

5. Determine the Q of a 65 mH coil at a frequency of 1 kHz if its resistance as measured with an ohmmeter is 25 Ω.

    Q = _________

1. For the circuit shown, determine the values specified, and sketch and label the impedance, voltage, and power phasor diagrams. Label all phasor lengths. Locate and identify the phase angle in each diagram.

2. For the circuit shown, determine the values specified and sketch the current and power phasor diagrams. Label all phasor lengths. Locate and identify the phase angle in each diagram.

3. Determine the Q of 2.5 mH coil at a

    frequency of 1 kHz if its internal resistance is 2 Ω.

    Q = ___________

4. Determine the internal resistance of a coil if its value is 8.5 H with a Q of 15 at a frequency of 1 kHz.

    r_i = _________

5. What is the phase relationship of the voltage across the coil versus the voltage across the resistor of a series RL circuit?