Impedance

The total opposition to current flow in an AC circuit is known as impedance and is represented by the letter Z. The combined effects of resistance, inductive reactance, and capacitive reactance make up impedance (the total opposition to current flow in an AC circuit). In order to accurately calculate voltage and current in AC circuits, the effect of inductance and capacitance along with resistance must be considered. Impedance is measured in ohms.


Figure 9-22. Applying DC and AC to a circuit.

Figure 9-22. Applying DC and AC to a circuit.

The rules and equations for DC circuits apply to AC circuits only when that circuit contains resistance alone and no inductance or capacitance. In both series and parallel circuits, if an AC circuit consists of resistance only, the value of the impedance is the same as the resistance, and Ohm’s law for an AC circuit, I = E/Z, is exactly the same as for a DC circuit. Figure 9-22 illustrates a series circuit containing a heater element with 11 ohms resistance connected across a 110-volt source. To find how much current flows if 110 volts AC is applied, the following example is solved:

 

Figure 9-23. Two resistance values in parallel connected to an AC voltage. Impedance is equal to the total resistance of the circuit.

Figure 9-23. Two resistance values in parallel connected to an AC voltage. Impedance is equal to the total resistance of the circuit.

If there are two resistance values in parallel connected to an AC voltage, as seen in Figure 9-23, impedance is equal to the total resistance of the circuit. Once again, the calculations would be handled the same as if it were a DC circuit and the following would apply:

Since this is a pure resistive circuit RT = Z (resistance = Impedance)

To determine the current flow in the circuit use the equation:

Impedance is the total opposition to current flow in an AC circuit. If a circuit has inductance or capacitance, one must take into consideration resistance (R), inductive reactance (XL), and/or capacitive reactance (XC) to determine impedance (Z). In this case, Z does not equal RT. Resistance and reactance (inductive or capacitive) cannot be added directly, but they can be considered as two forces acting at right angles to each other. Thus, the relation between resistance, reactance, and impedance may be illustrated by a right triangle. [Figure 9-24] Since these quantities may be related to the sides of a right triangle, the formula for finding the impedance can be found using the Pythagorean Theorem. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Thus, the value of any side of a right triangle can be found if the other two sides are known.

Figure 9-24. Impedance triangle.

Figure 9-24. Impedance triangle.

Figure 9-25. A circuit containing resistance and inductance.

Figure 9-25. A circuit containing resistance and inductance.

In practical terms, if a series AC circuit contains resistance and inductance, as shown in Figure 9-25, the relation between the sides can be stated as:

The square root of both sides of the equation gives:

This formula can be used to determine the impedance when the values of inductive reactance and resistance are known. It can be modified to solve for impedance in circuits containing capacitive reactance and resistance by substituting XC in the formula in place of XL. In circuits containing resistance with both inductive and capacitive reactance, the reactances can be combined; but because their effects in the circuit are exactly opposite, they are combined by subtraction (the smaller number is always subtracted from the larger):

or

Figure 9-25 shows example 1. Here, a series circuit containing a resistor and an inductor are connected to a source of 110 volts at 60 cycles per second. The resistive element is a simple measuring 6 ohms, and the inductive element is a coil with an inductance of 0.021 henry. What is the value of the impedance and the current through the circuit?

Solution:

First, the inductive reactance of the coil is computed:

Next, the total impedance is computed:

Remember when making calculations for Z always use inductive reactance not inductance, and use capacitive reactance, not capacitance.

Once impedance is found, the total current can be calculated.

Since this circuit is resistive and inductive, there is a phase shift where voltage leads current.

Example 2 is a series circuit illustrated in which a capacitor of 200 μf is connected in series with a 10 ohm resistor. [Figure 9-26] What is the value of the impedance, the current flow, and the voltage drop across the resistor?

Figure 9-26. A circuit containing resistance and capacitance.

Figure 9-26. A circuit containing resistance and capacitance.

Solution:

First, the capacitance is changed from microfarads to farads. Since 1 million microfarads equal 1 farad, then 200 μf = 0.000200 farads

Next solve for capacitive reactance:

To find the impedance,

Since this circuit is resistive and capacitive, there is a phase shift where current leads voltage:

To find the current:

To find the voltage drop across the resistor (ER):

To find the voltage drop over the capacitor (EC):

The sum of these two voltages does not equal the applied voltage, since the current leads the voltage. Use the following formula to find the applied voltage:

When the circuit contains resistance, inductance, and capacitance, the following equation is used to find the impedance.

Example 3: What is the impedance of a series circuit consisting of a capacitor with a capacitive reactance of 7 ohms, an inductor with an inductive reactance of 10 ohms, and a resistor with a resistance of 4 ohms? [Figure 9-27]

Figure 9-27. A circuit containing resistance, inductance, and capacitance.

Figure 9-27. A circuit containing resistance, inductance, and capacitance.

Solution:

To find total current:

Remember that inductive and capacitive reactances can cause a phase shift between voltage and current. In this example, inductive reactance is larger than capacitive reactance, so the voltage leads current.

It should be noted that since inductive reactance, capacitive reactance, and resistance affect each other at right angles, the voltage drops of any series AC circuit should be added using vector addition. Figure 9-28 shows the voltage drops over the series AC circuit described in example 3 above.

Figure 9-28. Voltage drops.

Figure 9-28. Voltage drops.

To calculate the individual voltage drops, simply use the equations:

To determine the total applied voltage for the circuit, each individual voltage drop must be added using vector addition.

Parallel AC Circuits

When solving parallel AC circuits, one must also use a derivative of the Pythagorean Theorem. The equation for finding impedance in an AC circuit is as follows:

To determine the total impedance of the parallel circuit shown in Figure 9-29, one would first determine the capacitive and inductive reactances. (Remember to convert microfarads to farads.)

Figure 9-29. Total impedance of parallel circuit.

Figure 9-29. Total impedance of parallel circuit.

Next, the impedance can be found:

To determine the current flow in the circuit:

To determine the current flow through each parallel path of the circuit, calculate IR, IL, and IC.

It should be noted that the total current flow of parallel circuits is found by using vector addition of the individual current flows as follows:

Power in AC Circuits

Since voltage and current determine power, there are similarities in the power consumed by both AC and DC circuits. In AC however, current is a function of both the resistance and the reactance of the circuit. The power consumed by any AC circuit is a function of the applied voltage and both circuit’s resistance and reactance. AC circuits have two distinct types of power, one created by the resistance of the circuit and one created by the reactance of the circuit.

True Power

True power of any AC circuit is commonly referred to as the working power of the circuit. True power is the power consumed by the resistance portion of the circuit and is measured in watts (W). True power is symbolized by the letter P and is indicated by any wattmeter in the circuit. True power is calculated by the formula:

Apparent Power

Apparent power in an AC circuit is sometimes referred to as the reactive power of a circuit. Apparent power is the power consumed by the entire circuit, including both the resistance and the reactance. Apparent power is symbolized by the letter S and is measured in volt-amps (VA). Apparent power is a product of the effective voltage multiplied by the effective current. Apparent power is calculated by the formula:

Power Factor

As seen in Figure 9-30, the resistive power and the reactive power effect the circuit at right angles to each other. The power factor in an AC circuit is created by this right angle effect.

Figure 9-30. Power relations in AC circuit.

Figure 9-30. Power relations in AC circuit.

Power factor can be defined as the mathematical difference between true power and apparent power. Power factor (PF) is a ratio and always a measurement between 0 and 100. The power factor is directly related to the phase shift of a circuit. The greater the phase shift of a circuit the lower the power factor. For example, an AC circuit that is purely inductive (contains reactance only and no resistance) has a phase shift of 90° and a power factor of 0.0. An AC circuit that is purely resistive (has no reactance) has a phase shift of 0 and a power factor of 100. Power factor is calculated by using the following formula:

Example of calculating PF: Figure 9-31 shows an AC load connected to a 50 volt power supply. The current draw of the circuit is 5 amps and the total resistance of the circuit is 8 ohms. Determine the true power, the apparent power, and the power factor for this circuit.

Figure 9-31. AC load connected to a 50-volt power supply.

Figure 9-31. AC load connected to a 50-volt power supply.

Solution:

Power factor can also be represented as a percentage. Using a percentage to show power factor, the circuit in the previous example would have a power factor of 80 percent.

It should be noted that a low power factor is undesirable. Circuits with a lower power factor create excess load on the power supply and produce inefficiency in the system. Aircraft AC alternators must typically operate with a power factor between 90 percent and 100 percent. It is therefore very important to carefully consider power factor when designing the aircraft electrical system.