Phase Differences

As in any series resistive-reactive circuit, the simple sum of the voltage drops in the circuit does not equal the applied voltage.

(11–1)

Recall that this occurs because of the different phase relationships between the voltage and current for each component.

Current as a Reference

Recall that in a series circuit as shown in Figure 11.1, the current is the same throughout the circuit. Therefore, as shown in Figure 11.2, it will be used as the reference quantity when discussing the phase relationships of the voltages in the circuit.

Development of a Circuit Phasor Diagram

Also recall that as shown in Figure 11.3 the voltage across the resistor, E_R, is in phase with the current passing through it. This is the circuit current so the voltage across the resistor is in phase with the circuit current.

The voltage across a capacitor, E_C, lags the current through it by 90 degrees, as in any capacitive circuit, and as shown in Figure 11.4. The current again is the circuit current so that E_Clags the circuit current by 90 degrees. Finally, the voltage across the inductor, E_L, leads the current through it, the circuit current, by 90 degrees as shown in Figure 11.5. Plotting all of these on the same diagram, the phase relationships of the voltages and current in this series RLC circuit can be compared as shown in Figure 11.6.

Voltage Phasor Comparisons

As you can see in Figure 11.6, the voltage across the inductor, E_L, leads the voltage across the resistor, E_R, by 90 degrees and leads the voltage across the capacitor, E_Cby 180 degrees.

Because of the 180-degree phase difference between the voltage across the inductor, E_L, and the voltage across the capacitor, E_C, these two reactive voltages are opposite in phase, as shown in Figure 11.7, and one value partially cancels the effect of the other.

Calculations of E_X

Because of the partial cancellation, a net reactive voltage, E_X, exists which is equal to the difference between the two reactive voltages, E_L, and E_C, as shown in Figure 11.8. If E_L, is larger than E_C, the net reactive voltage is in phase with E_L. A positive E_X, indicates that it is in phase with E_L. If E_C, is larger than E_L, the net reactive voltage is in phase with E_C. A negative E_X, indicates that it is in phase with E_C. The sign of E_X, indicates its direction on the Y(reactive) axis.

Calculation of E_A

If E_L, and E_C are known, it is possible to calculate the applied voltage, E_A. For example, assume that E_L, is larger than E_C. The net reactive voltage, E_X, will, therefore, be in phase with E_L as shown in Figure 11.8. E_A is then calculated by extending the phasor E_A from the origin as shown in Figure 11.9.

Note that if the phasor E_X, is shifted to the right so that it extends between the tip of E_R and the tip of E_A the right triangle for vector addition becomes apparent. By the Pythagorean theorem, E_A, equals the square root of E_R squared plus E_Xsquared:

(11–2)

Since E_X is the net difference between E_Land E_C,

(11–3)

Substituting this expression for E_X, into equation 11–2, the applied voltage is equal to the square root of E_R, squared plus E_L, minus E_C, quantity squared as shown in equation 11–4.

(11–4)

This equation described the relationship between the voltages present in the series RLC circuit.

Calculation of Circuit Voltage Values

Since, by Ohm’s law,

(11–5)

the voltage across the resistor is the current through it times the resistance.

(11–6)

Because

in a series RLC circuit, E_R, can be expressed:

(11–7)

Also, recall that the voltage across an inductor, E_L, is equal to the current through the inductance times the inductive reactance. Therefore

(11–8)

and substituting

(11–9)

The voltage across the capacitor equals the capacitive current times its capacitive reactance:

(11–10)

and substituting

(11–11)

These equivalent IR and IX quantities can be substituted for the voltages of Figure 11.6 they represent on the voltage phasor diagram as shown in Figure 11.10. Then by factoring out the common term of total current, the impedance phasor diagram is formed as shown in Figure 11.11.

Calculations of Total Reactance

In a series RLC circuit, the inductive reactance and capacitive reactance are 180 degrees out of phase as shown in Figure 11.11. Because of this, the two reactive quantities are opposite in phase and one partially cancels the effect of the other. The result is a total reactance, called X_T, which is equal to the difference between the inductive reactance and the capacitive reactance values:

(11–12)

If X_Lis larger than X_C, the net total reactance is in phase with X_L. This would be indicated by a positive quantity for X_T. On the other hand, if X_Cis larger than X_L, the net total reactance is in phase with X_C. This would be indicated by a negative X_T. The sign of X_T indicates its direction on the Y(reactive) axis. If in the voltage phasor diagram, E_L, is larger than E_C, to make E_X, positive, then to correspond, X_L would be larger than X_C to make X_T positive on the impedance phasor diagram.

Calculation of Z_T

Recall that the total opposition to the flow of ac current in a resistive-reactive circuit is called impedance, Z, as shown in Figure 11.12. The total impedance of a series RLC circuit is equal to the vector sum of the total resistance and net total reactance.

(11–13)

Note in Figure 11.12 that if phasor X_T is shifted to the right so that it extends between the tip of Z and the tip of R, the right triangle for vectorial addition appears. By the Pythagorean theorem, the total impedance of the circuit is:

(11–14)

Since X_T is the net difference between X_Land X_C, per equation 11–12, substituting this expression for X_Tinto equation 11–14 gives equation 11–15 for the total impedance.

(11–15)

Calculation of I_T

Recall that the total current in a series RLC circuit is simply equal to the applied voltage divided by the total impedance of the circuit (Ohm’s law for ac circuits).

(11–16)

Once the total current is known, the voltage drops across each component may be determined by Ohm’s law as stated in equations 11–7, 11–9, and 11–11.

Calculation of the Phase Angle

Recall that the phase angle is defined as the phase difference between the total applied voltage and the total current being drawn from that voltage supply. Returning to the voltage phasor diagram for the series RLC circuit, shown in Figure 11.13, remember that the total current in the circuit is used as a reference for determining phase relationships in the circuit. This total current is in phase with the voltage across the resistor. The circuit phase angle between the total applied voltage and the total current drawn from the voltage supply is represented by the angle theta. The tangent of any angle, theta, of a right triangle is:

(11–17)

For the specific angle of Figure 11.13,

(11–18)

The arctangent of this ratio equals the value of the phase angle.

(11–19)

Alternative Methods of Calculating the Phase Angle

Since the impedance phasor diagram is proportional to the voltage phasor diagram by a factor of the total current, the phase angle is also equal to the phase difference between the total impedance of the circuit and the resistance as shown in Figure 11.14. For the impedance phasor diagram, the tangent of the phase angle is, using equation 11–17 again and substituting for opposite and adjacent,

(11–20)

The arctangent of this ratio also equals the value of the phase angle:

(11–21)

Calculation of Series RLC Circuit Power Values

Power calculations in series and parallel RLC circuits are performed in a similar manner to power calculations in series and parallel RL and RC circuits. The primary difference is that both inductive and capacitive reactive power are involved.

As you know, power is voltage times current or P= EI. The real power, P_R, in watts dissipated by the resistor, is equal to E_R, times I_R. I_R is equal to I_T, the total circuit current of the series circuit. Similarly, the reactive power of the inductor, P_L in VAR, equals E_L, times I_T and the reactive power of the capacitor, P_C in VAR, equals E_C, times I_T.

In series RLC circuits, the power phasor diagram is proportional to the voltage phasor diagram by a factor of the total current as shown in Figure 11.15. The inductive power phasor and the capacitive power phasor are out of phase by 180 degrees. Thus, the resultant reactive power, which is designated P_X, is equal to the difference between the inductive reactive power and the capacitive reactive power as shown in Figure 11.16.

If P_L is larger than P_C, the net reactive power is in phase with P_L. If P_C is larger than P_L, the net reactive power is in phase with P_C. The sign of P_X will determine its ultimate vectorial direction.

Recall that the total power in a resistive-reactive circuit is the apparent power measured in volts-amperes, and is the vector sum of the resistive or real power and the net reactive power.

Note in Figure 11.17 that if the phasor P_X is shifted to the right so that it extends between the tip of the total apparent power and the tip of the real power, the right triangle for vector addition appears. By the Pythagorean theorem:

(11–22)

Since P_X is equal to the difference between P_L and P_C,

(11–23)

Substituting this expression for P_X into equation 11–22, the total apparent power is:

(11–24)

which is the relationship between the different types of power in a series RLC circuit.

The total apparent power can also be expressed in P= EI form as

(11–25)

where E_A, is the applied voltage and IT the circuit current. The apparent power calculated by using one of these two methods should be identical to the apparent power calculated by using the other method.

Resistance and Reactance Phasor Diagram

At this point, it is useful to sketch the resistance and reactance phasor diagram to help visualize the relationships between the resistive and reactive quantities of the circuit.Figure 11.19 shows such a diagram. Note that X_L is larger than X_C; thus, the net total reactance is in phase with X_Land is, using equation 11–12,

Calculation of Z_T

The total impedance, Z_T, is equal to

Calculation of I_T

By using Ohm’s law for ac circuits the total current, I_T, can be calculated.

Calculation of Circuit Voltage Values

Using Ohm’s law again, the voltage drops across each of the components in the circuit can be calculated.

The voltage across the resistor, E_R, is calculated:

The voltage across the inductor, E_L, is calculated:

The voltage across the capacitor, E_C, is calculated:

Calculation of E_T

At first, as shown in Figure 11.20, this result appears impossible since there seems to be more voltage in the circuit than the applied voltage, E_A. However, you must keep in mind that the voltage across the inductor and the voltage across the capacitor are 180 degrees out of phase and one partially cancels the other.

Therefore, the applied voltage, E_A, is calculated using equation 11–4.

This calculation confirms the original applied voltage value given in the example. And this method gives you a valuable check of the accuracy of the calculation of the individual voltage drops that were calculated for each component in the circuit.

Calculation of Phase Angle

Using the voltage phasor diagram of Figure 11.13, the phase angle of the example series RLC circuit can be calculated:

The angle whose tangent is 0.75 is about 37 degrees.

Alternate Method of Calculating Phase Angle

The same result may be obtained using the impedance phasor diagram. In this case the arctangent is X_T divided by R.

Positive and Negative Phase Angles

The phase angle calculated for the example series RLC circuit is considered to be a positive phase angle since the phase angle for the circuit is measured counter-clockwise. Thus, the phase angle for the circuit is stated as being a positive 37 degrees.

The phase angle in a series RLC circuit can either be positive or negative. If X_L is greater than X_C, the angle is positive. On the other hand, if X_C is greater than X_L, the angle is negative.

Calculation of Component Power Values

In the example series RLC circuit, the voltage drops in the circuit are: 40 volts across the resistor, 90 volts across the inductor and 60 volts across the capacitor. The applied voltage is 50 volts and a total current of 2 amperes is flowing.

Multiplying the voltage times the current for each component yields these individual power values:

Calculation of Total Reactive Power

The net reactive power, P_X, is equal to the difference between P_Land P_C, or 180 VAR minus 120 VAR which equals 60 VAR. Since P_Lis larger than P_C, P_X is positive and is in phase with P_L.

Calculation of Total Apparent Power

By equation 11–22, the apparent power, P_A, is equal to the square root of P_R squared plus P_X squared:

The total apparent power of the example series RLC circuit is 100 volt-amperes.

This should be the same as the value obtained by multiplying the applied voltage, E_A, by the total current, I_T.

It is, and proves the accuracy of the first calculation.

Voltage as a Reference

Since the voltage across all components of a parallel circuit is the same as the applied voltage, E_A, it will be used as the reference quantity in discussing the phase relationships of the currents in the circuit. Therefore,

Development of a Circuit Phasor Diagram

Recall that the current through a resistor, I_R, is in phase with the voltage across it, E_R. The capacitive current, I_C, leads the voltage across the capacitor, E_A, by 90 degrees; and the current through an inductor, I_L, lags the voltage across it, E_A, by 90 degrees. Thus, the phase relationship of the applied voltage and currents in a parallel RLC circuit are as shown in Figure 11.22.

Calculations of Branch Currents

The individual branch currents in the example RLC circuit can be calculated as they are in either a purely resistive, purely capacitive, or purely inductive circuit. Simply divide the voltage across the branch by the opposition to current in the branch.

In the resistive branch the opposition to the flow of current is measured in ohms of resistance. The resistive current is determined by dividing the applied voltage by the value of the resistor:

(11–27)

In the inductive branch, the opposition to the flow of current is measured in ohms of inductive reactance. The inductive current is determined by dividing the applied voltage by the reactance of the inductor:

(11–28)

In the capacitive branch, the opposition to the flow of current is measured in ohms of capacitive reactance. The capacitive current is determined by dividing the applied voltage by the reactance of the capacitor:

(11–29)

Current Phasor Comparisons

In the parallel RLC circuit, since E_A= E_R= E_L= E_C, and since I_Ris in phase with E_Rand thus all the voltages E_L, E_C, and E_A, I_Rbecomes the reference vector for the current phasor diagram of Figure 11.23. Comparing the current phase relationships using the current phasor diagram of Figure 11.23, you can see that the capacitive branch current, I_C, leads the resistive branch current, I_R, by 90 degrees; the inductive branch current, I_L, lags I_Rby 90 degrees. Because of the 180-degree difference between the capacitive branch current and the inductive branch current, these two reactive current values are opposite in phase, and one partially cancels the effect of the other.

Calculation of I_X

The difference between I_C and I_L is I_X, which is the net reactive current. If I_C is larger than I_L, the net reactive current is in phase with I_C and the value of I_X is positive. If, on the other hand, I_L is larger than I_C, the net reactive current is in phase with I_L and the value of I_X is negative. The sign of I_X indicates its direction on the Y(reactive) axis.

By adding the total current vector, I_T, to the current phasor diagram, and shifting vector I_X to the right so that it extends between the tip of I_R and the tip of I_T, a right triangle is produced. This is shown in Figure 11.24. The right triangle can then be used to add the phasors I_Rand I_Xvectorially.

Applying the Pythagorean theorem,

(11–30)

Sine I_X is the net difference between I_C and I_L, as shown in equation 11–31,

(11–31)

I_X is equal to I_C minus I_L. Substituting this expression for I_Xinto equation 11–30, the total current is equal to

(11–32)

which is the relationship between the branch currents in a parallel RLC circuit.

Calculation of Z_T

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

(11–33)

Calculation of Phase Angle

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 is in phase with the resistive current. Therefore, the phase angle is measured from the horizontal vector I_R(which is also E_A) to the vector I_T; it is identified by the angle, theta, on the current phasor diagram of Figure 11.25.

The tangent of the phase angle, theta, is equal to the ratio of the net reactive current divided by the resistive current as shown in equation 11–34:

(11–34)

Therefore,

(11–35)

the arctangent of the net reactive current divided by the resistive current equals the value of the phase angle.

Calculation of Power

Power calculations in parallel RLC circuits are the same as power calculations in series RLC circuits. The individual power values are calculated by multiplying current through a component times voltage across a component.

Therefore,

(11–36)

(11–37)

(11–38)

The total apparent power is found by using the Pythagorean theorem solution of resistive and net reactive power shown in the power phasor diagram of Figure 11.26 defined by these equations:

(11–39)

(11–40)

(11–41)

Calculation of Branch Currents

The resistive branch current, I_R, is calculated by dividing the voltage across the resistor, 180 volts, by the value of the resistor, 60 kilohms.

The inductive branch current is determined in a similar manner by dividing the voltage across the inductor by the inductive reactance.

The capacitive branch current is calculated by dividing the voltage across the capacitor by the capacitive reactance.

Current Phasor Diagrams

The current phasor diagram can now be drawn to show the relationships between the resistive, inductive, and capacitive branch currents as shown in Figure 11.28. Note that I_Cis larger than I_L; therefore, the net reactive current is in phase with I_C. The net reactive current, I_X, equals

Calculation of I_T

As you can see in Figure 11.28, by shifting the phasor I_X to the right so that it extends from the tip of the vector representing I_Rto the tip of the vector for I_T, a right triangle is formed. The right triangle can then be used to determine the value of I_T. Using the Pythagorean theorem, the total current, I_T, is now equal to the vector sum of the values of I_Xand I_R.

The total current in the circuit is 5 milliamperes.

Calculation of Z

The total impedance of this circuit can now be calculated by dividing the applied voltage by the total current.

Calculations of the Phase Angle

The phase angle, theta is the angle between the resistive current and the total current as shown on the current phasor diagram in Figure 11.29. The value of the phase angle is equal to the arctangent of the ratio of net reactive current to the resistive current. Therefore,

Thus, the phase angle of this parallel RLC circuit is approximately 53 degrees.

The phase angle is described as a positive phase angle because the total current phasor is rotated counter-clockwise from the reference. However, the phase angle of parallel RLC circuit may be either positive or negative depending upon the relationships of I_Land I_C. If I_Cis greater than I_L, the phase angle theta will be positive as shown in Figure 11.30a. If I_Lis greater than I_C, theta will be negative as shown in Figure 11.30b.

Calculation of Component Power Values

As you know power values depend on the various voltages and currents in the circuit. In the parallel RLC circuit example shown in Figure 11.31, currents are: I_Requals 3 milliamperes, I_L equals 6 milliamperes, and I_C equals 10 milliamperes. The applied voltage is 180 volts; the total current is 5 milliamperes.

The real power dissipated by the resistor, P_R, is calculated:

The inductive reactive power, P_Lis calculated:

The capacitive reactive power, P_C, is calculated:

Calculation of Total Reactive Power

The net reactive power, P_Xis calculated:

The power phasor diagram will look like the one in Figure 11.26.

Since P_Cis larger than P_L, P_Xis positive and is in phase with P_C.

Calculation of Total Power

The apparent total power, P_A, is calculated using equation 11–39:

This value should be the same as the apparent power value obtained by multiplying the applied voltage by the total current:

1. Draw a phasor diagram for this circuit showing current and impedance phasors.

    Solution:

    X_L> X_C; therefore, X_Tis plotted in phase with X_L

2. Draw voltage phasor diagrams for the circuit in Example 1. Calculate the value and sign of the phase angle.

    Solution:

    The angle is rotated counter-clockwise; therefore, the sign of the angle is positive.

3. Draw phasor diagrams showing Pythagorean theorem relationships in parallel RLC circuits. Write equations for each solution.

    Solution:

4. Given the circuit and typical circuit values shown, calculate the circuit values specified.

Solution:

5. Given this circuit and typical circuit values shown, calculate the circuit values specified.

Solution:

1. Draw a phasor diagram for this circuit showing voltage and current phasors.

2. Draw power phasor diagrams for the circuit in problem 1. Calculate the value and sign of the phase angle.

3. Draw phasor diagrams showing Pythagoren theorem relationships in series RLC circuits. Write equations for each solution.

4. Given the circuit and typical circuit values below, calculate the circuit values specified.

5. Given this circuit and typical circuit values shown, calculate the circuit values specified.

1. Draw a phasor diagram for this circuit showing power and current phasors.

2. Draw a phasor diagram for this circuit showing power and voltage phasors.

3. Calculate the value and sign of the phase angle in the circuit of Problem 1.

4. Calculate the value and sign of the phase angle in the circuit of Problem 2.

5. Calculate the value of Z_Tin Problem 1.

6. Calculate the value of I_Tin Problem 2.

7. Given this circuit and typical circuit values shown, calculate the circuit values specified.

8. Given this circuit and typical values shown, calculate the circuit values specified.

9. In a series RCL circuit, which component will have the most voltage across it?

10. In a parallel RCL circuit, which component will have the most current through it?