## Introduction to Basic Circuit Theory

This information covers basic direct current theory by reviewing the three basic types of electrical circuits and the laws that apply to each type circuit:

- Series Circuits
- Parallel Circuits
- Series-Parallel Circuits

## Series Circuit

Illustration 1 | g01070312 |

A series circuit is the simplest kind of circuit. In a series circuit, each electrical device is connected to other electrical devices. There is only one path for current to flow. In Illustration 1, current flows from the battery (+) through a fuse (protection device) and a switch (control device) to the lamp (load) and then returns to frame ground. All circuit devices and components are connected in series. The following rules apply to all series circuits:

- At any given point in the circuit, the current value is the same.
- The total circuit resistance is equal to the sum of all the individual resistances. This is called an equivalent resistance.
- The voltage drop across all circuit loads is equal to the applied source voltage.

The following rules are for a series circuit:

- Voltage is the sum of all voltage drops.
- Current is the same at any given point in the circuit.
- Resistance is the sum of all individual resistances.

## Applying the Rules

Illustration 2 | g01070313 |

Illustration 2 shows that the circuit is made up of various devices and of various components. This includes a 24 volt power source. Since two of the circuit values are given (voltage and resistance), solving for the unknown value is simple.

The first step in solving the above circuit, is to determine the total circuit resistance.

The following equation is used for determining total resistance:

R_{t} = R1 + R2 + R3, or R_{t} = 3Ohms + 3Ohms + 6Ohms, or R_{t} = 12Ohms.

Since the value for the power source was given as 24 volts and the circuit resistance has been calculated as 12Ohms, the only value remaining to calculate is the current flow. Total circuit current is calculated by using the Ohm's Law Circle and by writing the following equation:

I = E/R, or I = 24V/12Ohms, or I = 2 amperes.

The remaining step is to plug the value for current flow into each of the resistive loads. One of the rules for series circuits stated that current was the same at any given point. The equation E = I × R for each resistor will determine the voltage drop across each load. The following equations are for voltage drops:

- E1 = 2A × 3Ohms = 6V
- E2 = 2A × 3Ohms = 6V
- E3 = 2A × 6Ohms = 12V

All of the circuit values have now been calculated. Verify each answer by using the Ohm's Law Circle.

## Parallel Circuit

Illustration 3 | g01071099 |

A parallel circuit is more complex than a series circuit because there is more than one path for current to flow. Each current path is called a branch. All branches connect to the same positive terminal and negative terminal. This causes the branches to have the same voltage. Each branch drops the same amount of voltage, regardless of resistance within the branch.

The current flow that is in each branch can be different. The difference depends on the resistance. Total current in the circuit equals the sum of the branch currents.

The total resistance is always less than the smallest resistance in any branch.

In the circuit shown in Illustration 3, current flows from the battery through a fuse and a switch. The current divides into two branches. Each branch contains a lamp. Each branch is connected to frame ground.

The following rules apply to parallel circuits:

- The voltage is the same in each parallel branch.
- The total current is the sum of each individual branch currents
- The equivalent resistance is equal to the applied voltage divided by the total current, and is always less than the smallest resistance in any one branch.

The following rules are for parallel circuits:

- Voltage is the same for all branches.
- Current is the sum of the individual branch currents.
- Equivalent resistance is smaller than the smallest resistance of any individual branch.

Illustration 4 | g01070318 |

The circuit is made up of various devices and various components. This includes a 24 volt power source. The resistance of each lamp is given along with the value of source voltage. Before you apply the basic laws of parallel circuits, it will be necessary to determine an equivalent resistance in order to replace the two 4 ohm parallel branches.

The first step in developing an equivalent circuit is to apply the basic rules for determining the total resistance of the two parallel branches. The total resistance of the combined branches will be smaller than the smallest resistance of an individual branch. The circuit above has two parallel branches, each with a 4Ohms lamp, therefore, the total resistance will be less than 4Ohms.

The following equation is used to solve for total resistance.

1/R_{t} = 1/R1 + 1/R2

1/R_{t} = 1/4 + 1/4 or

1/R_{t} = .25 + .25 = .50 or

R_{t} = 1/.50 or R_{t} = 2 ohms

One of the rules for parallel circuits states that the voltage is the same in all parallel branches. With 24 volts applied to each branch, the individual current flow can be calculated by using Ohm's Law. The equation I = E/R is used to calculate the current in each branch as 6 amps. In this particular case, the current flow in each branch is the same because the resistance values are the same.

## Solving Current Flow in a Parallel Circuit

Illustration 5 | g01070320 |

The circuit that is shown in Illustration 5 is a typical DC circuit with three parallel branches. The circuit also contains an ammeter connected in series with the parallel branches (all current flow in the circuit must pass through the ammeter).

Applying the basic rules for parallel circuits makes solving this problem very simple. The source voltage is given (24 volts) and each branch resistance is given (R1 = 4Ohms; R2= 4Ohms; R3 = 2Ohms). Applying the voltage rule for parallel circuits (voltage is the SAME in all branches) you can solve the unknown current value in each branch by using the Ohm's Law Circle, whereas, I = E/R.

I1 = E1/R1 or I1 = 24/4 or I1 = 6 amps

I2 = E2/R2 or I2 = 24/4 or I2 = 6 amps

I3 = E3/R3 or I3 = 24/2 or I3 = 12 amps

Since current flow in parallel branches is the sum of all branch currents, the equation for total current is I_{t} = I1 + I2 + I3 or 6+6+12 = 24 amp. With the source voltage given as 24 volts and the total current calculated at 24 amp, the total circuit resistance is calculated as 1 ohm. (R_{t} = E_{t}/I_{t}).

## Series-Parallel Circuits

Illustration 6 | g01070324 |

A series-parallel circuit is composed of a series section and a parallel section. All of the rules previously discussed regarding series circuits and parallel circuits are applicable in solving for unknown circuit values.

Although some series-parallel circuits appear to be very complex, the series parallel circuits are solved quite easily by using a logical approach. The following tips will make solving series-parallel circuits less complicated:

- Examine the circuit carefully. Then determine the path or paths that current may flow through the circuit before returning to the source.
- Redraw a complex circuit to simplify the appearance.
- When you simplify a series parallel circuit, begin at the farthest point from the voltage source. Replace the parallel resistor combinations one step at a time.
- A correctly redrawn series parallel (equivalent) circuit will contain only ONE series resistor in the end.
- Apply the simple series rules for determining the unknown values.
- Return to the original circuit and plug in the known values. Use Ohm's Law to solve the remaining values.

## Solving a Series-Parallel Problem

Illustration 7 | g01070325 |

The series parallel circuit, as shown in Illustration 7, shows a 2Ohms resistor in series with a parallel branch that contains a 6Ohms resistor and a 3Ohms resistor. To solve this problem it is necessary to determine the equivalent resistance for the parallel branch. Using the following equation, solve for the parallel equivalent (R_{e}) resistance:

1/R_{e} = 1/R_{2} + 1/R_{3}

1/R_{e} = 1/6 + 1/3 or

1/R_{e} = .1666 +.3333 = .50 or

1/R_{e} = 1/.50 or R_{e} = 2 ohms

Illustration 7 has been redrawn (See Illustration 8) with the equivalent resistance for the parallel branch. Solve circuit totals by using simple Ohm's Law rules for series circuits.

Illustration 8 | g01070328 |

Using the rules for series circuits, the total circuit resistance can now be calculated by using the equation R_{t } = R1 + Re or R_{t } = 2 + 2 or 4 ohms.

The remaining value that is unknown is current. Again, using Ohm's Law Circle, current can be calculated by the equation:

I = E/R or

I = 12/4 or

I = 3 amp

Illustration 9 shows all the known values.

Illustration 9 | g01070330 |

Circuit calculations indicate that the total current flow in the circuit is 3 amps. Since all current flow that leaves the source must return, you know that the 3 amps must flow through R1. It is now possible to calculate the voltage drop across R1 by using the equation E = I × R, or E = 3A × 2Ohms, or E1 = 6 volts.

If 6 volts is consumed by resistor R1, the remaining source voltage (6V) is applied to both parallel branches. Using Ohm's Law for the parallel branch reveals that 1 amp flows through R2 and 2 amps flow through R3 before combining into the total circuit current of 3 amps returning to the negative side of the power source.

## Other Methods and Tips for Solving Complex Series Parallel Circuits

As stated earlier, complex circuits can be easily solved by carefully examining the path for current flow and then draw the circuit again. No matter how complex a circuit appears, drawing an equivalent circuit and reducing the circuit to the lowest form (series circuit) will provide the necessary information to plug into the original circuit.

Illustration 10 | g01070333 |

- Trace current flow from the (+) side of the battery to the (-) side of the battery. All the current leaving the source is available at "TP1" (test point 1). At "TP1" the current is divided among the two parallel branches and then recombined at "TP2" before flowing through the series resistor "R3" and returning to ground. Now that you have identified the path of current flow, the next step is drawing an equivalent circuit for the parallel branches.

- Use Ohm's Law to calculate the equivalent resistance for the parallel branch. There are two equations that are available for solving parallel branch resistances. The following equations are used to solve for resistances.
- 1/R
_{e}= 1/R1 + 1/R2 - R
_{e}= R1 × R2/R1 + R2

The second equation is called product over sum method that is used for combining two parallel resistances. When the circuit contains only two branches the product over sum method is the easiest equation.

- 1/R

- Redraw the circuit substituting the R
_{e}value to represent the equivalent resistance. The circuit now has two resistors in series, shown as R_{e }and R3. Further reduce the circuit by adding R_{e }and R3 as a single resistance called R_{t}. The following circuits reflect those steps.

Illustration 11 | g01070335 |

Illustration 12 | g01070336 |

Illustration 13 | g01070337 |