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Parallel DC Circuits and Series-Parallel DC Circuits and Alternating Current (AC) and Voltage

Filed Under: Fundamentals of Electricity and Electronics

Parallel DC Circuits

A circuit in which two of more electrical resistances or loads are connected across the same voltage source is called a parallel circuit. The primary difference between the series circuit and the parallel circuit is that more than one path is provided for the current in the parallel circuit. Each of these parallel paths is called a branch. The minimum requirements for a parallel circuit are the following:

  • A power source
  • Conductors
  • A resistance or load for each current path
  • Two or more paths for current flow

Figure 12-96 depicts the most basic parallel circuit. Current flowing out of the source divides at point A in the diagram and goes through R1 and R2. As more branches are added to the circuit, more paths for the source current are provided.

Figure 12-96. Basic parallel circuit.
Figure 12-96. Basic parallel circuit.

Voltage Drops

The first point to understand is that the voltage across any branch is equal to the voltage across all of the other branches.

Total Parallel Resistance

The parallel circuit consists of two or more resistors connected in such a way as to allow current flow to pass through all of the resistors at once. This eliminates the need for current to pass one resistor before passing through the next. When resistors are connected in parallel, the total resistance of the circuit decreases. The total resistance of a parallel combination is always less than the value of the smallest resistor in the circuit. In the series circuit, the current has to pass through the resistors one at a time. This gave a resistance to the current equal the sum of all the resistors. In the parallel circuit, the current has several resistors that it can pass through, actually reducing the total resistance of the circuit in relation to any one resistor value.

The amount of current passing through each resistor varies according to its individual resistance. The total current of the circuit is the sum of the current in all branches. It can be determined by inspection that the total current is greater than that of any given branch. Using Ohm’s Law to calculate the total resistance based on the applied voltage and the total current, it can be determined that the total resistance is less than any branch.

An example of this is if there was a circuit with a 100 Ω resistor and a 5 Ω resistor; while the exact value must be calculated, it still can be said that the combined resistance between the two is less than the 5 Ω.

Resistors in Parallel

The formula for the total parallel resistance is as follows:

If the reciprocal of both sides is taken, then the general formula for the total parallel resistance is:

Two Resistors in Parallel

Typically, it is more convenient to consider only two resistors at a time because this setup occurs in common practice. Any number of resistors in a circuit can be broken down into pairs. Therefore, the most common method is to use the formula for two resistors in parallel.Combining the terms in the denominator and rewriting:Put in words, this states that the total resistance for two resistors in parallel is equal to the product of both resistors divided by the sum of the two resistors. In the formula below, calculate the total resistance.

Current Source

A current source is an energy source that provides a constant value of current to a load even when the load changes in resistive value. The general rule to remember is that the total current produced by current sources in parallel is equal to the algebraic sum of the individual sources.

Kirchhoff’s Current Law

Kirchhoff’s Current Law can be stated as: the sum of the currents into a junction or node is equal to the sum of the currents flowing out of that same junction or node. A junction can be defined as a point in the circuit where two or more circuit paths come together. In the case of the parallel circuit, it is the point in the circuit where the individual branches join.Refer to Figure 12-97 for an example. Point A and point B represent two junctions or nodes in the circuit with three resistive branches in between.

Figure 12-97. Kirchhoff’s Current Law.
Figure 12-97. Kirchhoff’s Current Law.

The voltage source provides a total current IT into node A. At this point, the current must divide, flowing out of node A into each of the branches according to the resistive value of each branch. Kirchhoff’s Current Law states that the current going in must equal that going out. Following the current through the three branches and back into node B, the total current IT entering node B and leaving node B is the same as that which entered node A. The current then continues back to the voltage source. Figure 12-98 shows that the individual branch currents are:

Figure 12-98. Individual branch currents.
Figure 12-98. Individual branch currents.

The total current flow into the node A equals the sum of the branch currents, which is: IT = I1 + I2The total current entering node B is also the same.

Figure 12-99 illustrates how to determine an unknown current in one branch. Note that the total current into a junction of the three branches is known. Two of the branch currents are known. By rearranging the general formula, the current in branch two can be determined.

 

Figure 12-99. Determining an unknown circuit in branch 2.
Figure 12-99. Determining an unknown circuit in branch 2.

Current Dividers

It can now be easily seen that the parallel circuit is a current divider. As shown in Figure 12-96, there is a current through each of the two resistors.

Figure 12-96. Basic parallel circuit.
Figure 12-96. Basic parallel circuit.

Because the same voltage is applied across both resistors in parallel, the branch currents are inversely proportional to the ohmic values of the resistors. Branches with higher resistance have less current than those with lower resistance. For example, if the resistive value of R2 is twice as high as that of R1, the current in R2 is half of that of R1. All of this can be determined with Ohm’s Law. By Ohm’s Law, the current through any one of the branches can be written as:The voltage source appears across each of the parallel resistors and RX represents any one the resistors. The source voltage is equal to the total current times the total parallel resistance.

This formula is the general current divider formula. The current through any branch equals the total parallel resistance divided by the individual branch resistance, multiplied by the total current.

Series-Parallel DC Circuits

Most of the circuits that the technician encounters will not be a simple series or parallel circuit. Circuits are usually a combination of both, known as series-parallel circuits, which are groups consisting of resistors in parallel and in series. An example of this type of circuit can be seen in Figure 12-100. While the series-parallel circuit can initially appear to be complex, the same rules that have been used for the series and parallel circuits can be applied to these circuits.

Figure 12-100. Series-parallel circuits.
Figure 12-100. Series-parallel circuits.

The voltage source provides a current out to resistor R1, then to the group of resistors R2 and R3 and then to the next resistor R4 before returning to the voltage source. The first step in the simplification process is to isolate the group R2 and R3 and recognize that they are a parallel network that can be reduced to an equivalent resistor. Using the formula for parallel resistance,

R2 and R3 can be reduced to R23. Figure 12-101 now shows an equivalent circuit with three series connected resistors. The total resistance of the circuit can now be simply determined by adding up the values of resistors R1, R23, and R4.

Figure 12-101. Equivalent circuit with three series connected resistors.
Figure 12-101. Equivalent circuit with three series connected resistors.

Determining the Total Resistance

A more quantitative example for determining total resistance and the current in each branch in a combination circuit is shown in the following example. [Figure 12-102]

Figure 12-102. Determining total resistance.
Figure 12-102. Determining total resistance.

The first step is to determine the current at junction A, leading into the parallel branch. To determine the IT, the total resistance RT of the entire circuit must be known. The total resistance of the circuit is given as:

With the total resistance RT now determined, the total IT can be determined. Using Ohm’s Law:

The current through the parallel branches of R2 and R3 can be determined using the current divider rule discussed earlier in the text. Recall that:

Now using Kirchhoff’s Current Law, the current in branch with R3 can be determined.

Alternating Current (AC) and Voltage

Alternating current (AC) has largely replaced direct current (DC) in commercial power systems for a number of reasons. It can be transmitted over long distances more readily and more economically than DC, since AC voltages can be increased or decreased by means of transformers.

Because more and more units are being operated electrically in airplanes, the power requirements are such that a number of advantages can be realized by using AC. Space and weight can be saved since AC devices, especially motors, are smaller and simpler than DC devices. In most AC motors, no brushes are required, and commutation trouble at high altitude is eliminated. Circuit breakers operate satisfactorily under load at high altitudes in an AC system, whereas arcing is so excessive on DC systems that circuit breakers must be replaced frequently. Finally, most airplanes using a 24-volt DC system have special equipment that requires a certain amount of 400-cycle AC current.

AC and DC Compared

Many of the principles, characteristics, and effects of AC are similar to those of DC. Similarly, there are a number of differences. DC flows constantly in only one direction with a constant polarity. It changes magnitude only when the circuit is opened or closed, as shown in the DC waveform in Figure 12-103. AC changes direction at regular intervals, increases in value at a definite rate from zero to a maximum positive strength, and decreases back to zero; then it flows in the opposite direction, similarly increasing to a maximum negative value, and again decreasing to zero. DC and AC waveforms are compared in Figure 12-103.

Figure 12-103. DC and AC voltage curves.
Figure 12-103. DC and AC voltage curves.

Since AC constantly changes direction and intensity, the following two effects (to be discussed later) take place in AC circuits that do not occur in DC circuits:

  1. Inductive reactance
  2. Capacitive reactance

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