When two unequal charges are connected by a conductor, a complete pathway for current exists. An electric circuit is a complete conducting pathway. It consists not only of the conductor, but also includes the path through the voltage source. Inside the voltage source current flows from the positive terminal, through the source, emerging at the negative terminal.

## 1. Series Circuit Characteristics

A SERIES CIRCUIT is defined as a circuit that contains only ONE PATH for current flow. To compare the basic circuit that has been discussed and a more complex series circuit, Figure 1 shows two circuits. The basic circuit has only one lamp and the series circuit has three lamps connected in series.

Figure 1. Comparison of basic and series circuits.

### 1.1. Resistance in a Series Circuit

Referring to Figure 1, the current in a series circuit must flow through each lamp to complete the electrical path in the circuit. Each additional lamp offers added resistance. In a series circuit, THE TOTAL CIRCUIT RESISTANCE (RT) IS EQUAL TO THE SUM OF THE INDIVIDUAL RESISTANCES.

As an equation: RT = R1 + R2 + R3 + . . . Rn

 Note The subscript n denotes any number of additional resistances that might be in the equation.

Example: In Figure 2 a series circuit consisting of three resistors: one of 10 ohms, one of 15 ohms, and one of 30 ohms, is shown. A voltage source provides 110 volts. What is the total resistance?

Figure 2. Solving for total resistance in a series circuit.

In some circuit applications, the total resistance is known and the value of one of the circuit resistors has to be determined. The equation RT = R1 + R2
R3 can be transposed to solve for the value of the unknown resistance.

Example: In [fig-navy_mod1_00103] the total resistance of a circuit containing three resistors is 40 ohms. Two of the circuit resistors are 10 ohms each. Calculate the value of the third resistor (R3).

Figure 3. Calculating the value of one resistance in a series circuit.

### 1.2. Current in a Series Circuit

Since there is only one path for current in a series circuit, the same current must flow through each component of the circuit. To determine the current in a series circuit, only the current through one of the components need be known.

The fact that the same current flows through each component of a series circuit can be verified by inserting meters into the circuit at various points, as shown in Figure 4. If this were done, each meter would be found to indicate the same value of current.

Figure 4. Current in a series circuit.

### 1.3. Voltage in a Series Circuit

The voltage dropped across the resistor in a circuit consisting of a single resistor and a voltage source is the total voltage across the circuit and is equal to the applied voltage. The total voltage across a series circuit that consists of more than one resistor is also equal to the applied voltage, but consists of the sum of the individual resistor voltage drops. In any series circuit, the SUM of the resistor voltage drops must equal the source voltage. This statement can be proven by an examination of the circuit shown in Figure 5. In this circuit a source potential (ET) of 20 volts is dropped across a series circuit consisting of two 5-ohm resistors. The total resistance of the circuit (RT) is equal to the sum of the two individual resistances, or 10 ohms. Using Ohm’s law the circuit current may be calculated as follows:

Figure 5. Calculating individual voltage drops in a series circuit.

Since the value of the resistors is known to be 5 ohms each, and the current through the resistors is known to be 2 amperes, the voltage drops across the resistors can be calculated. The voltage (E1) across R1 is therefore:

By inspecting the circuit, you can see that R2 is the same ohmic value as R1 and carries the same current. The voltage drop across R2 is therefore also equal to 10 volts. Adding these two 10-volts drops together gives a total drop of 20 volts, exactly equal to the applied voltage. For a series circuit then:

ET = E1 = E2 + E3 = . . . En

Example: A series circuit consists of three resistors having values of 20 ohms, 30 ohms, and 50 ohms, respectively. Find the applied voltage if the current through the 30 ohm resistor is 2 amps. (The abbreviation amp is commonly used for ampere.)

To solve the problem, a circuit diagram is first drawn and labeled (Figure 6).

Figure 6. Solving for applied voltage in a series circuit.

Substituting:

 Note When you use Ohm’s law, the quantities for the equation MUST be taken from the SAME part of the circuit. In the above example the voltage across R2 was computed using the current through R2 and the resistance of R2.

The value of the voltage dropped by a resistor is determined by the applied voltage and is in proportion to the circuit resistances. The voltage drops that occur in a series circuit are in direct proportion to the resistances. This is the result of having the same current flow through each resistor—the larger the ohmic value of the resistor, the larger the voltage drop across it.

### 1.4. Power in a Series Circuit

Each of the resistors in a series circuit consumes power which is dissipated in the form of heat. Since this power must come from the source, the total power must be equal to the power consumed by the circuit resistances. In a series circuit the total power is equal to the SUM of the power dissipated by the individual resistors. Total power (PT) is equal to:

PT = P1 + P2 + P3 . . . Pn

Example: A series circuit consists of three resistors having values of 5 ohms, 10 ohms, and 15 ohms. Find the total power when 120 volts is applied to the circuit. (See Figure 7.)

Figure 7. Solving for total power in a series circuit.

Given: Solution: The total resistance is found first.

By using the total resistance and the applied voltage, the circuit current is calculated.

By means of the power formulas, the power can be calculated for each resistor:

To check the answer, the total power delivered by the source can be calculated:

The total power is equal to the sum of the power used by the individual resistors.

## 2. Summary of Characteristics

The important factors governing the operation of a series circuit are listed below. These factors have been set up as a group of rules so that they may be easily studied. These rules must be completely understood before the study of more advanced circuit theory is undertaken.

### 2.1. Rules for Series DC Circuits

1. The same current flows through each part of a series circuit.

2. The total resistance of a series circuit is equal to the sum of the individual resistances.

3. The total voltage across a series circuit is equal to the sum of the individual voltage drops.

4. The voltage drop across a resistor in a series circuit is proportional to the ohmic value of the resistor.

5. The total power in a series circuit is equal to the sum of the individual powers used by each circuit component.

## 3. Series Circuit Analysis

To establish a procedure for solving series circuits, the following sample problems will be solved.

Example: Three resistors of 5 ohms, 10 ohms, and 15 ohms are connected in series with a power source of 90 volts as shown in Figure 8. Find the total resistance, circuit current, voltage drop of each resistor, power of each resistor, and total power of the circuit.

Figure 8. Solving for various values in a series circuit.

In solving the circuit the total resistance will be found first. Next, the circuit current will be calculated. Once the current is known, the voltage drops and power dissipations can be calculated.

Example: Four resistors, R1 = 10 ohms, R2 = 10 ohms, R3 = 50 ohms, and R4 = 30 ohms, are connected in series with a power source as shown in Figure 9. The current through the circuit is 1/2 ampere.

1. What is the battery voltage?

2. What is the voltage across each resistor?

3. What is the power expended in each resistor?

4. What is the total power?

Figure 9. Computing series circuit values.

Given:

Solution (a):

Solution (b):

Solution (c):

Solution (d):

An important fact to keep in mind when applying Ohm’s law to a series circuit is to consider whether the values used are component values or total values. When the information available enables the use of Ohm’s law to find total resistance, total voltage, and total current, total values must be inserted into the formula. To find total resistance:

To find total voltage:

To find total current:

 Note IT is equal to I in a series circuit. However, the distinction between IT and I in the formula should be noted. The reason for this is that future circuits may have several currents, and it will be necessary to differentiate between IT and other currents.

To compute any quantity (E, I, R, or P) associated with a single given resistor, the values used in the formula must be obtained from that particular resistor. For example, to find the value of an unknown resistance, the voltage across and the current through that particular resistor must be used.

To find the value of a resistor:

To find the voltage drop across a resistor:

To find current through a resistor:

## 4. Kirchhoff’s Voltage Law

In 1847, G. R. Kirchhoff extended the use of Ohm’s law by developing a simple concept concerning the voltages contained in a series circuit loop. Kirchhoff’s voltage law states:

"The algebraic sum of the voltage drops in any closed path in a circuit and the electromotive forces in that path is equal to zero."

To state Kirchhoff’s law another way, the voltage drops and voltage sources in a circuit are equal at any given moment in time. If the voltage sources are assumed to have one sign (positive or negative) at that instant and the voltage drops are assumed to have the opposite sign, the result of adding the voltage sources and voltage drops will be zero.

 Note The terms electromotive force and emf are used when explaining Kirchhoff’s law because Kirchhoff’s law is used in alternating current circuits (covered in Module 2). In applying Kirchhoff’s law to direct current circuits, the terms electromotive force and emf apply to voltage sources such as batteries or power supplies.

Through the use of Kirchhoff’s law, circuit problems can be solved which would be difficult, and often impossible, with knowledge of Ohm’s law alone. When Kirchhoff’s law is properly applied, an equation can be set up for a closed loop and the unknown circuit values can be calculated.

### 4.1. Polarity of Voltage

To apply Kirchhoff’s voltage law, the meaning of voltage polarity must be understood.

In the circuit shown in Figure 10, the current is shown flowing in a counterclockwise direction. Notice that the end of resistor R1, into which the current flows, is marked NEGATIVE (í 7KHHQGRI51 at which the current leaves is marked POSITIVE (+). These polarity markings are used to show that the end of R1 into which the current flows is at a higher negative potential than the end of the resistor at which the current leaves. Point A is more negative than point B.

Figure 10. Voltage polarities.

Point C, which is at the same potential as point B, is labeled negative. This is to indicate that point C is more negative than point D. To say a point is positive (or negative) without stating what the polarity is based upon has no meaning. In working with Kirchhoff’s law, positive and negative polarities are assigned in the direction of current flow.

### 4.2. Application of Kirchhoff’s Voltage Law

Kirchhoff’s voltage law can be written as an equation, as shown below:

Ea + Eb + Ec + . . . En = 0

where Ea, Eb, etc., are the voltage drops or emf’s around any closed circuit loop. To set up the equation for an actual circuit, the following procedure is used.

1. Assume a direction of current through the circuit. (The correct direction is desirable but not necessary.)

2. Using the assumed direction of current, assign polarities to all resistors through which the current flows.

3. Place the correct polarities on any sources included in the circuit.

4. Starting at any point in the circuit, trace around the circuit, writing down the amount and polarity of the voltage across each component in succession. The polarity used is the sign AFTER the assumed current has passed through the component. Stop when the point at which the trace was started is reached.

5. Place these voltages, with their polarities, into the equation and solve for the desired quantity.

Example: Three resistors are connected across a 50-volt source. What is the voltage across the third resistor if the voltage drops across the first two resistors are 25 volts and 15 volts?

Solution: First, a diagram, such as the one shown in Figure 11, is drawn. Next, a direction of current is assumed (as shown). Using this current, the polarity markings are placed at each end of each resistor and also on the terminals of the source. Starting at point A, trace around the circuit in the direction of current flow, recording the voltage and polarity of each component. Starting at point A and using the components from the circuit:

Substituting values from the circuit:

Figure 11. Determining unknown voltage in a series circuit.

Using the same idea as above, you can solve a problem in which the current is the unknown quantity.

Example: A circuit having a source voltage of 60 volts contains three resistors of 5 ohms, 10 ohms, and 15 ohms. Find the circuit current.

Solution: Draw and label the circuit (Figure 12). Establish a direction of current flow and assign polarities. Next, starting at any point—point A will be used in this example—write out the loop equation.

Figure 12. Correct direction of assumed current.

Since the current obtained in the above calculations is a positive 2 amps, the assumed direction of current was correct. To show what happens if the incorrect direction of current is assumed, the problem will be solved as before, but with the opposite direction of current. The circuit is redrawn showing the new direction of current and new polarities in Figure 13. Starting at point A the loop equation is:

Figure 13. Incorrect direction of assumed current.

Notice that the AMOUNT of current is the same as before. The polarity, however, is NEGATIVE. The negative polarity simply indicates the wrong direction of current was assumed. Should it be necessary to use this current in further calculations on the circuit using Kirchhoff’s law, the negative polarity should be retained in the calculations.

#### 4.2.1. Series Aiding and Opposing Sources

In many practical applications a circuit may contain more than one source of emf. Sources of emf that cause current to flow in the same direction are considered to be SERIES AIDING and the voltages are added. Sources of emf that would tend to force current in opposite directions are said to be SERIES OPPOSING, and the effective source voltage is the difference between the opposing voltages. When two opposing sources are inserted into a circuit current flow would be in a direction determined by the larger source. Examples of series aiding and opposing sources are shown in Figure 14.

Figure 14. Aiding and opposing sources.

A simple solution may be obtained for a multiple-source circuit through the use of Kirchhoff’s voltage law. In applying this method, the same procedure is used for the multiple-source circuit as was used above for the single-source circuit. This is demonstrated by the following example.

Example: Using Kirchhoff’s voltage equation, find the amount of current in the circuit shown in Figure 15.

Figure 15. Solving for circuit current using Kirchhoff’s voltage equation.

Solution: As before, a direction of current flow is assumed and polarity signs are placed on the drawing. The loop equation will be started at point A.

E2 + ER1 + E1 + E3 + ER2 = 0

## 5. Circuit Terms and Characteristics

Before you learn about the types of circuits other than the series circuit, you should become familiar with some of the terms and characteristics used in electrical circuits. These terms and characteristics will be used throughout your study of electricity and electronics.

### 5.1. Reference Point

A reference point is an arbitrarily chosen point to which all other points in the circuit are compared. In series circuits, any point can be chosen as a reference and the electrical potential at all other points can be determined in reference to that point. In Figure 16 point A shall be considered the reference point. Each series resistor in the illustrated circuit is of equal value. The applied voltage is equally distributed across each resistor. The potential at point B is 25 volts more positive than at point A. Points C and D are 50 volts and 75 volts more positive than point A respectively.

Figure 16. Reference points in a series circuit.

When point B is used as the reference, as in Figure 17, point D would be positive 50 volts in respect to the new reference point. The former reference point, A, is 25 volts negative in respect to point B.

Figure 17. Determining potentials with respect to a reference point.

As in the previous circuit illustration, the reference point of a circuit is always considered to be at zero potential. Since the earth (ground) is said to be at a zero potential, the term GROUND is used to denote a common electrical point of zero potential. In Figure 18, point A is the zero reference, or ground, and the symbol for ground is shown connected to point A. Point C is 75 volts positive in respect to ground.

Figure 18. Use of ground symbols.

In most electrical equipment, the metal chassis is the common ground for the many electrical circuits. When each electrical circuit is completed, common points of a circuit at zero potential are connected directly to the metal chassis, thereby eliminating a large amount of connecting wire. The electrons pass through the metal chassis (a conductor) to reach other points of the circuit. An example of a chassis grounded circuit is illustrated in Figure 19.

Figure 19. Ground used as a conductor.

Most voltage measurements used to check proper circuit operation in electrical equipment are taken in respect to ground. One meter lead is attached to a grounded point and the other meter lead is moved to various test points. Circuit measurement is explained in more detail in NEETS Module 3.

### 5.2. Open Circuit

A circuit is said to be OPEN when a break exists in a complete conducting pathway. Although an open occurs when a switch is used to deenergize a circuit, an open may also develop accidentally. To restore a circuit to proper operation, the open must be located, its cause determined, and repairs made.

Sometimes an open can be located visually by a close inspection of the circuit components. Defective components, such as burned out resistors, can usually be discovered by this method. Others, such as a break in wire covered by insulation or the melted element of an enclosed fuse, are not visible to the eye. Under such conditions, the understanding of the effect an open has on circuit conditions enables a technician to make use of test equipment to locate the open component.

In Figure 20, the series circuit consists of two resistors and a fuse. Notice the effects on circuit conditions when the fuse opens.

Figure 20. Normal and open circuit conditions. (A) Normal current; (B) Excessive current.

Current ceases to flow; therefore, there is no longer a voltage drop across the resistors. Each end of the open conducting path becomes an extension of the battery terminals and the voltage felt across the open is equal to the applied voltage (EA).

An open circuit has INFINITE resistance. INFINITY represents a quantity so large it cannot be measured. The symbol for infinity is ∞. In an open circuit, RT = ∞.

### 5.3. Short Circuit

A short circuit is an accidental path of low resistance which passes an abnormally high amount of current. A short circuit exists whenever the resistance of a circuit or the resistance of a part of a circuit drops in value to almost zero ohms. A short often occurs as a result of improper wiring or broken insulation.

In Figure 21, a short is caused by improper wiring. Note the effect on current flow. Since the resistor has in effect been replaced with a piece of wire, practically all the current flows through the short and very little current flows through the resistor. Electrons flow through the short (a path of almost zero resistance) and the remainder of the circuit by passing through the 10-ohm resistor and the battery. The amount of current flow increases greatly because its resistive path has decreased from 10,010 ohms to 10 ohms. Due to the excessive current flow the 10-ohm resistor becomes heated. As it attempts to dissipate this heat, the resistor will probably be destroyed. Figure 22 shows a pictorial wiring diagram, rather than a schematic diagram, to indicate how broken insulation might cause a short circuit.

Figure 21. Normal and short circuit conditions.
Figure 22. Short due to broken insulation.

### 5.4. Source Resistance

A meter connected across the terminals of a good 1.5-volt battery reads about 1.5 volts. When the same battery is inserted into a complete circuit, the meter reading decreases to something less than 1.5 volts. This difference in terminal voltage is caused by the INTERNAL RESISTANCE of the battery (the opposition to current offered by the electrolyte in the battery). All sources of electromotive force have some form of internal resistance which causes a drop in terminal voltage as current flows through the source.

This principle is illustrated in Figure 23, where the internal resistance of a battery is shown as Ri. In the schematic, the internal resistance is indicated by an additional resistor in series with the battery. The battery, with its internal resistance, is enclosed within the dotted lines of the schematic diagram. With the switch open, the voltage across the battery terminals reads 15 volts. When the switch is closed, current flow causes voltage drops around the circuit. The circuit current of 2 amperes causes a voltage drop of 2 volts across Ri. The 1-ohm internal battery resistance thereby drops the battery terminal voltage to 13 volts. Internal resistance cannot be measured directly with a meter. An attempt to do this would damage the meter.

Figure 23. Effect of internal resistance.

The effect of the source resistance on the power output of a dc source may be shown by an analysis of the circuit in Figure 24. When the variable load resistor (RL) is set at the zero-ohm position (equivalent to a short circuit), current (I) is calculated using the following formula:

This is the maximum current that may be drawn from the source. The terminal voltage across the short circuit is zero volts and all the voltage is across the resistance within the source.

Figure 24. Effect of source resistance on power output.

If the load resistance (RL) were increased (the internal resistance remaining the same), the current drawn from the source would decrease. Consequently, the voltage drop across the internal resistance would decrease. At the same time, the terminal voltage applied across the load would increase and approach a maximum as the current approaches zero amps.

### 5.5. Power Transfer and Efficiency

Maximum power is transferred from the source to the load when the resistance of the load is equal to the internal resistance of the source. This theory is illustrated in the table and the graph of Figure 24. When the load resistance is 5 ohms, matching the source resistance, the maximum power of 500 watts is developed in the load.

The efficiency of power transfer (ratio of output power to input power) from the source to the load increases as the load resistance is increased. The efficiency approaches 100 percent as the load resistance approaches a relatively large value compared with that of the source, since less power is lost in the source. The efficiency of power transfer is only 50 percent at the maximum power transfer point (when the load resistance equals the internal resistance of the source). The efficiency of power transfer approaches zero efficiency when the load resistance is relatively small compared with the internal resistance of the source. This is also shown on the chart of Figure 24.

The problem of a desire for both high efficiency and maximum power transfer is resolved by a compromise between maximum power transfer and high efficiency. Where the amounts of power involved are large and the efficiency is important, the load resistance is made large relative to the source resistance so that the losses are kept small. In this case, the efficiency is high. Where the problem of matching a source to a load is important, as in communications circuits, a strong signal may be more important than a high percentage of efficiency. In such cases, the efficiency of power transfer should be only about 50 percent; however, the power transfer would be the maximum which the source is capable of supplying.

You should now understand the basic concepts of series circuits. The principles which have been presented are of lasting importance. Once equipped with a firm understanding of series circuits, you hold the key to an understanding of the parallel circuits to be presented next.