193. General

In circuits using ac electricity1, the current is affected by inductance and capacitance as well as resistance. In addition, certain combinations of these loads will produce unusual effects, such as resonance (par. 202), not experienced in dc circuits. These phenomena are used extensively in electrical and electronic circuits. Consequently, problems in ac electricity are more complex than corresponding problems in dc electricity.

194. Application of Vectors and Trigonometry in Solving Ac Circuit Problems

a. As discussed in chapter 12, a vector is a line whose length and direction represent accurately a given quantity; the quantity thus represented is a vector quantity. Because the magnitude of ac currents and voltages varies from instant to instant, the magnitude is a function of time, and the current and voltage can be expressed as vectors: The length of the vector represents the magnitude of the current or voltage, and the direction represents its relationship in time to another vector (b below).

b. When a circuit contains inductance or capacitance, the current in the circuit is not in phase with the voltage that produces it. In other words, the instant the voltage is zero, the current that it produces has a value other than zero, or when the voltage is at its maximum, the current has a value different from its maximum value. The current is said to lead the voltage if the current reaches its maximum before the voltage maximum occurs; the current is said to lag the voltage if the current

1 This chapter is limited to the application of mathematics to single-phase, sinusoidal ac. The electrical phenomena of this type of ac are treated briefly. See TM 11-681 for a complete treatment of single-phase, sinusoidal ac.

reaches its maximum after the voltage maximum occurs. The relationship between current and voltage can be represented by vectors, with one vector representing current, another voltage, and with the angle between them indicating the amount of lag or lead. Figure 67 shows a vector representation of leading and lagging current. The angle is called the phase angle.

c. The voltage drop across a resistor also may be represented by a vector having the same direction as the vector representing the current flowing through the resistor. In other words, the voltage across the resistor and the current flowing through it are in phase.

d. The voltage drop across a capacitor may be represented by a vector making an angle of 90° with the vector representing the current flowing through the capacitor. In a purely capacitive circuit, the current will lead the applied voltage by an angle of 90°.

e. The voltage drop across an inductor may be represented by a vector making an angle of 90° with the vector representing the current flowing through the inductance. In a purely inductive circuit, the current will lag the applied voltage by an angle of 90°.

f. In a circuit that contains inductance, capacitance, and resistance, the current will lead or lag the applied voltage by a phase angle of less than 90°.

g. The example below illustrates the use of vectors in the solution of a typical ac circuit problem. Paragraphs 199 through 201 give a more detailed coverage of problems of this type. Example:

In a series circuit (fig. 68), the voltage drop across the capacitor (Ec) is 10 volts, the voltage drop across the inductance (EL) is 50 volts, and the voltage drop across the resistance (ER) is 30 volts. Determine the magnitude of the applied voltage. By what phase angle (A) does the current lead or lag the applied voltage in the circuit?

не

R

TM684-65

Figure 68. An ac series circuit containing inductance,

Step 1.

Step 2.

Step 3.

capacitance, and resistance.

The vector diagram for this circuit is shown in figure 69. In a series circuit, the same current flows through each element. Draw the vector representing the current (I) in a horizontal position. The angles of all vectors representing voltage drops are given with respect to the current.

Draw the vector EL, representing the voltage drop across the inductance, at an angle of 90° with the vector I.

Draw the vector Ec, representing the voltage drop across the capacitor, at a angle of -90° with the vector 1.

Step 6.

Step 7.

The vector sum of these voltage drops is equal to the applied voltage.

Along the horizontal:

EL + Ec + ER = 50 +

(−10) + 0 = 40

Because the vectors form a right triangle, with the applied voltage E as the hypotenuse and ER and Ex as the sides (fig. 69), the law of right triangles (par. 133) can be used to solve for one of the quantities when the other two are known. From this law, the relationship between E, ER, and Ex is expressed by the formula

E = VER2 + Ex2.

E = VER2 + Ex2

= √(30)2 + (40)2
= √900 + 1600
= √2500

= 50 volts

Step 8.

Step 9.

=

40

or

30 3

= 1.33333

A 53° 7' 48"

The circuit is predominately inductive; therefore, the current lags the applied voltage by a phase angle of 53° 7′ 48′′.

195. Ohm's Law Applied to Ac Circuits

Because of the effects of inductance and capacitance in ac circuits, Ohm's law (par. 186) must be modified to take these added effects into consideration.

a. If the circuit contains a combination of resistance and inductive reactance (par. 196) or capacitive reactance (par. 197), or both, the overall effect is called impedance (par. 198), and Ohm's law is modified to read:

E 1= Z

196. Inductive Reactance

Inductance enables an electric circuit to build up a voltage by electromagnetic induction whenever the current strength changes. The induced voltage always opposes the applied voltage and thus retards the change in the current. Inductive reactance is the effect of inductance expressed in ohms. The formula for finding inductive reactance is:

XL = 2πfL

where X is the inductive reactance in ohms, L is the inductance in henrys, and ƒ is the frequency in cps.

Example 1: Determine the inductive reac

tance of a coil if the ac in the circuit has a frequency of 100

cps, and the inductance of the

coil is 0.036 henry.

XL = 2πfL

=

XL

220

75.36

= 2.92 amperes

197. Capacitive Reactance

Capacitance enables a capacitor to retain an electric charge which opposes any changes in the voltage of the circuit in which the capacitor is connected. Capacitive reactance is the effect of the capacitance expressed in ohms.