Chapter 7: Parallel DC Circuits

You have seen that the source voltage in a series circuit divides proportionately across each resistor in the circuit. IN A PARALLEL CIRCUIT, THE SAME VOLTAGE IS PRESENT IN EACH BRANCH. (A branch is a section of a circuit that has a complete path for current.) In Figure 1 this voltage is equal to the applied voltage (E_s). This can be expressed in equation form as:

E_S = E_R1 = E_R2

Voltage measurements taken across the resistors of a parallel circuit, as illustrated by Figure 2 verify this equation. Each meter indicates the same amount of voltage. Notice that the voltage across each resistor is the same as the applied voltage.

Example: Assume that the current through a resistor of a parallel circuit is known to be 4.5 milliamperes (4.5 mA) and the value of the resistor is 30,000 ohms (30 kΩ). Determine the source voltage. The circuit is shown in Figure 3.

Given:

Solution:

Since the source voltage is equal to the voltage of a branch:

To simplify the math operation, the values can be expressed in powers of ten as follows:

If you are not familiar with the use of the powers of 10 or would like to brush up on it, Mathematics, Vol. 1, NAVEDTRA 10069-C, will be of great help to you.

Ohm’s law states that the current in a circuit is inversely proportional to the circuit resistance. This fact is true in both series and parallel circuits.

There is a single path for current in a series circuit. The amount of current is determined by the total resistance of the circuit and the applied voltage. In a parallel circuit the source current divides among the available paths.

The behavior of current in parallel circuits will be shown by a series of illustrations using example circuits with different values of resistance for a given value of applied voltage.

Part (A) of Figure 4 shows a basic series circuit. Here, the total current must pass through the single resistor. The amount of current can be determined.

Given:

Solution:

Part (B) of Figure 4 shows the same resistor (R₁) with a second resistor (R₂) of equal value connected in parallel across the voltage source. When Ohm’s law is applied, the current flow through each resistor is found to be the same as the current through the single resistor in part (A).

Given:

Solution:

It is apparent that if there is 5 amperes of current through each of the two resistors, there must be a TOTAL CURRENT of 10 amperes drawn from the source.

The total current of 10 amperes, as illustrated in Figure 4(B), leaves the negative terminal of the battery and flows to point a. Since point a is a connecting point for the two resistors, it is called a JUNCTION. At junction a, the total current divides into two currents of 5 amperes each. These two currents flow through their respective resistors and rejoin at junction b. The total current then flows from junction b back to the positive terminal of the source. The source supplies a total current of 10 amperes and each of the two equal resistors carries one-half the total current.

Each individual current path in the circuit of Figure 4(B) is referred to as a BRANCH. Each branch carries a current that is a portion of the total current. Two or more branches form a NETWORK.

From the previous explanation, the characteristics of current in a parallel circuit can be expressed in terms of the following general equation:

I_T = I₁ + I₂ + . . . I_n

Compare part (A) of Figure 5 with part (B) of the circuit in Figure 4. Notice that doubling the value of the second branch resistor (R₂) has no effect on the current in the first branch (IR₁), but does reduce the second branch current (IR₂) to one-half its original value. The total circuit current drops to a value equal to the sum of the branch currents. These facts are verified by the following equations.

Given:

Solution:

The amount of current flow in the branch circuits and the total current in the circuit shown in Figure 5(B) are determined by the following computations.

Given:

Solution:

Notice that the sum of the ohmic values in each circuit shown in Figure 5 is equal (30 ohms), and that the applied voltage is the same (50 volts). However, the total current in 3-41(B) (15 amps) is twice the amount in 3-41(A) (7.5 amps). It is apparent, therefore, that the manner in which resistors are connected in a circuit, as well as their actual ohmic values, affect the total current.

The division of current in a parallel network follows a definite pattern. This pattern is described by KIRCHHOFF’S CURRENT LAW which states:

"The algebraic sum of the currents entering and leaving any junction of conductors is equal to zero."

This law can be stated mathematically as:

I_a + l_b + . . . I_n + 0

where: I_a, I_b, etc., are the currents entering and leaving the junction. Currents ENTERING the junction are considered to be POSITIVE and currents LEAVING the junction are considered to be NEGATIVE. When solving a problem using Kirchhoff’s current law, the currents must be placed into the equation WITH THE PROPER POLARITY SIGNS ATTACHED.

Example: Solve for the value of I3 in Figure 6.

Given:

Solution:

I_a + l_b + . . . I_a + 0

The currents are placed into the equation with the proper signs.

I₃ has a value of 2 amperes, and the negative sign shows it to be a current LEAVING the junction.

Example. Using Figure 7, solve for the magnitude and direction of I₃.

Given:

Solution:

I₃ is 2 amperes and its positive sign shows it to be a current entering the junction.

In the example diagram, Figure 8, there are two resistors connected in parallel across a 5-volt battery. Each has a resistance value of 10 ohms. A complete circuit consisting of two parallel paths is formed and current flows as shown.

Computing the individual currents shows that there is one-half of an ampere of current through each resistance. The total current flowing from the battery to the junction of the resistors, and returning from the resistors to the battery, is equal to 1 ampere.

The total resistance of the circuit can be calculated by using the values of total voltage (ET) and total current (I_T).

Given:

Solution:

This computation shows the total resistance to be 5 ohms; one-half the value of either of the two resistors.

Since the total resistance of a parallel circuit is smaller than any of the individual resistors, total resistance of a parallel circuit is not the sum of the individual resistor values as was the case in a series circuit. The total resistance of resistors in parallel is also referred to as EQUIVALENT RESISTANCE (R_eq). The terms total resistance and equivalent resistance are used interchangeably.

There are several methods used to determine the equivalent resistance of parallel circuits. The best method for a given circuit depends on the number and value of the resistors. For the circuit described above, where all resistors have the same value, the following simple equation is used:

This equation is valid for any number of parallel resistors of EQUAL VALUE.

Example: Four 40-ohm resistors are connected in parallel. What is their equivalent resistance?

Given:

Solution:

Figure 9 shows two resistors of unequal value in parallel. Since the total current is shown, the equivalent resistance can be calculated.

Given:

Solution:

The equivalent resistance of the circuit shown in Figure 9 is smaller than either of the two resistors (R₁, R₂). An important point to remember is that the equivalent resistance of a parallel circuit is always less than the resistance of any branch.

Equivalent resistance can be found if you know the individual resistance values and the source voltage. By calculating each branch current, adding the branch currents to calculate total current, and dividing the source voltage by the total current, the total can be found. This method, while effective, is somewhat lengthy. A quicker method of finding equivalent resistance is to use the general formula for resistors in parallel:

If you apply the general formula to the circuit shown in Figure 9 you will get the same value for equivalent resistance (2Ω) as was obtained in the previous calculation that used source voltage and total current.

Given:

Solution:

Convert the fractions to a common denominator.

Since both sides are reciprocals (divided into one), disregard the reciprocal function.

The formula you were given for equal resistors in parallel

is a simplification of the general formula for resistors in parallel

There are other simplifications of the general formula for resistors in parallel which can be used to calculate the total or equivalent resistance in a parallel circuit.

RECIPROCAL METHOD.—This method is based upon taking the reciprocal of each side of the equation. This presents the general formula for resistors in parallel as:

This formula is used to solve for the equivalent resistance of a number of unequal parallel resistors. You must find the lowest common denominator in solving these problems. If you are a little hazy on finding the lowest common denominator, brush up on it in Mathematics Volume 1, NAVEDTRA 10069 (Series).

Example: Three resistors are connected in parallel as shown in Figure 10. The resistor values are: R₁ = 20 ohms, R₂ = 30 ohms, R₃ = 40 ohms. What is the equivalent resistance? (Use the reciprocal method.)

Given:

Solution:

PRODUCT OVER THE SUM METHOD.—A convenient method for finding the equivalent, or total, resistance of two parallel resistors is by using the following formula.

This equation, called the product over the sum formula, is used so frequently it should be committed to memory.

Example: What is the equivalent resistance of a 20-ohm and a 30-ohm resistor connected in parallel, as in Figure 11?

Given:

Solution:

Power computations in a parallel circuit are essentially the same as those used for the series circuit. Since power dissipation in resistors consists of a heat loss, power dissipations are additive regardless of how the resistors are connected in the circuit. The total power is equal to the sum of the power dissipated by the individual resistors. Like the series circuit, the total power consumed by the parallel circuit is:

Example: Find the total power consumed by the circuit in Figure 12.

Given:

Solution:

Since the total current and source voltage are known, the total power can also be computed by:

Given:

Solution:

In the study of electricity, it is often necessary to reduce a complex circuit into a simpler form. Any complex circuit consisting of resistances can be redrawn (reduced) to a basic equivalent circuit containing the voltage source and a single resistor representing total resistance. This process is called reduction to an EQUIVALENT CIRCUIT.

Figure 13 shows a parallel circuit with three resistors of equal value and the redrawn equivalent circuit. The parallel circuit shown in part A shows the original circuit. To create the equivalent circuit, you must first calculate the equivalent resistance.

Given:

Solution:

Once the equivalent resistance is known, a new circuit is drawn consisting of a single resistor (to represent the equivalent resistance) and the voltage source, as shown in part B.