Series AC 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. In Figure 126 a series circuit containing a lamp with 11 ohms resistance connected across a source is illustrated. To find how much current will flow if 110 volts DC is applied and how much current will flow if 110 volts AC are applied, the following examples are solved:

When AC circuits contain resistance and either inductance or capacitance, the impedance, Z, is not the same as the resistance, R. The impedance of a circuit is the circuit’s total opposition to the flow of current. In an AC circuit, this opposition consists of resistance and reactance, either inductive or capacitive or elements of both.

Resistance and reactance 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 127]

Since these quantities may be related to the sides of a right triangle, the formula for finding the impedance, or total opposition to current flow in an AC circuit, can be found by using the law of right triangles. This theorem, called the Pythagorean theorem, applies to any right triangle. 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. If an AC circuit contains resistance and inductance, as shown in Figure 128, 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 reactance’s can be combined, but because their effects in the circuit are exactly opposite, they are combined by subtraction:

In Figure 128, a series circuit consisting of resistance and inductance connected in series is connected to a source of 110 volts at 60 cycles per second. The resistive element is a lamp with 6 ohms resistance, 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 lamp and the coil?

Solution:

First, the inductive reactance of the coil is computed:

Next, the total impedance is computed:

Then the current flow,

The voltage drop across the resistance (E^R) is

The voltage drop across the inductance (EX_L) is

The sum of the two voltages is greater than the impressed voltage. This results from the fact that the two voltages are out of phase and, as such, represent the maximum voltage. If the voltage in the circuit is measured by a voltmeter, it will be approximately 110 volts, the impressed voltage. This can be proved by the equation,

In Figure 129, a series circuit is illustrated in which a capacitor of 200 μf is connected in series with a 10 ohm lamp. What is the value of the impedance, the current flow, and the voltage drop across the lamp?

Solution:

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

To find the impedance,

To find the current,

The voltage drop across the lamp (E^R) is

The voltage drop across the capacitor (EX_C) is

The sum of these two voltages does not equal the applied voltage, since the current leads the voltage. To find the applied voltage,

When the circuit contains resistance, inductance, and capacitance, the equation

is used to find the impedance.

Example: What is the impedance of a series circuit, consisting of a capacitor with a reactance of 7 ohms, an inductor with a reactance of 10 ohms, and a resistor with a resistance of 4 ohms? [Figure 130]

Solution:

Assuming that the reactance of the capacitor is 10 ohms and the reactance of the inductor is 7 ohms, then XC is greater than XL. Thus,