**Physics SS 1 Week 2**

**Topic**: **Resistors in Series and Parallel**

Introduction

Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Usually, we find circuits where more than two components are connected together.

There are two basic ways in which to connect more than two circuit components: *series* and *parallel*. First, an example of a series circuit:

Here, we have three resistors (labeled R_{1}, R_{2}, and R_{3}) connected in a long chain from one terminal of the battery to the other. (It should be noted that the subscript labeling — those little numbers to the lower-right of the letter “R” — are unrelated to the resistor values in ohms. They serve only to identify one resistor from another.) The defining characteristic of a series circuit is that there is only one path for electrons to flow. In this circuit the electrons flow in a counter-clockwise direction, from point 4 to point 3 to point 2 to point 1 and back around to 4.

Now, let’s look at the other type of circuit, a parallel configuration:

Again, we have three resistors but this time they form more than one continuous path for electrons to flow. There’s one path from 8 to 7 to 2 to 1 and back to 8 again. There’s another from 8 to 7 to 6 to 3 to 2 to 1 and back to 8 again. And then there’s a third path from 8 to 7 to 6 to 5 to 4 to 3 to 2 to 1 and back to 8 again. Each individual path (through R_{1}, R_{2}, and R_{3}) is called a *branch*.

The defining characteristic of a parallel circuit is that all components are connected between the same set of electrically common points. Looking at the schematic diagram, we see that points 1, 2, 3, and 4 are all electrically common. So are points 8, 7, 6, and 5. Note that all resistors as well as the battery are connected between these two sets of points.

And, of course, the complexity doesn’t stop at simple series and parallel either! We can have circuits that are a combination of series and parallel, too:

In this circuit, we have two loops for electrons to flow through: one from 6 to 5 to 2 to 1 and back to 6 again, and another from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how both current paths go through R_{1} (from point 2 to point 1). In this configuration, we’d say that R_{2} and R_{3} are in parallel with each other, while R_{1} is in series with the parallel combination of R_{2} and R_{3}.

This is just a preview of things to come. Don’t worry! We’ll explore all these circuit configurations in detail, one at a time!

The basic idea of a “series” connection is that components are connected end-to-end in a line to form a single path for electrons to flow:

The basic idea of a “parallel” connection, on the other hand, is that all components are connected across each other’s leads. In a purely parallel circuit, there are never more than two sets of electrically common points, no matter how many components are connected. There are many paths for electrons to flow, but only one voltage across all components:

Series and parallel resistor configurations have very different electrical properties. We’ll explore the properties of each configuration in the sections to come.

**Cells in Series and Parallel**

Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called **series** and **parallel** and occur very frequently. Components connected in series are connected along a single path, so the same current flows through all of the components. Components connected in parallel are connected so the same voltage is applied to each component.

A circuit composed solely of components connected in series is known as a **series circuit**; likewise, one connected completely in parallel is known as a **parallel circuit**.

In a series circuit, the current through each of the components is the same, and the voltage across the circuit is the sum of the voltages across each component. In a parallel circuit, the voltage across each of the components is the same, and the total current is the sum of the currents through each component.

Assume that a load requires a power supply of 6 volts and a current capacity of 1/8 ampere. Since a single cell normally supplies a voltage of only 1.5 volts, more than one cell is needed. To obtain the higher voltage, the cells are connected in series as shown

(A) Pictorial view of series-connected cells; (B) Schematic of series connection.

he load is shown by the resistance symbol and the battery is indicated by one long and one short line per cell.

In a series hookup, the negative electrode (cathode) of the first cell is connected to the positive electrode (anode) of the second cell, the negative electrode of the second to the positive of the third, etc. The positive electrode of the first cell and negative electrode of the last cell then serve as the terminals of the battery. In this way, the voltage is 1.5 volts for each cell in the series line. There are four cells, so the output terminal voltage is 1.5 x 4, or 6 volts. When connected to the load, 1/8 ampere flows through the load and each cell of the battery. This is within the capacity of each cell. Therefore, only four series-connected cells are needed to supply this particular load.

Note:

When connecting cells in series, connect alternate terminals together (- to +, – to +, etc.) Always have two remaining terminals that are used for connection to the load only. Do not connect the two remaining terminals together as this is a short across the battery and would not only quickly discharge the cells but could cause some types of cells to explode.

**Parallel-Connected Cells**

In this case, assume an electrical load requires only 1.5 volts, but will require 1/2 ampere of current. (Assume that a single cell will supply only 1/8 ampere.) To meet this requirement, the cells are connected in parallel, as shown in figure 2-7 view Aand schematically represented in 2-7 view B. In a parallel connection, all positive cell electrodes are connected to one line, and all negative electrodes are connected to the other. No more than one cell is connected between the lines at any one point; so the voltage between the lines is the same as that of one cell, or 1.5 volts. However, each cell may contribute its maximum allowable current of 1/8 ampere to the line. There are four cells, so the total line current is 1/8 x 4, or 1/2 ampere. In this case four cells in parallel have enough capacity to supply a load requiring 1/2 ampere at 1.5 volts.

(A) Pictorial view of parallel-connected cells; (B) Schematic of parallel connection.

**Series-Parallel-Connected Cells**

The figure depicts a battery network supplying power to a load requiring both a voltage and a current greater than one cell can provide. To provide the required 4.5 volts, groups of three 1.5-volt cells are connected in series. To provide the required 1/2 ampere of current, four series groups are connected in parallel, each supplying 1/8 ampere of current.

**Schematic of series-parallel connected cells**

The connections shown have been used to illustrate the various methods of combining cells to form a battery. Series, parallel, and series-parallel circuits will be covered in detail in the next chapter, “Direct Current.”

Some batteries are made from primary cells. When a primary-cell battery is completely discharged, the entire battery must be replaced. Because there is nothing else that can be done to primary cell batteries, the rest of the discussion on batteries will be concerned with batteries made of secondary cells.

**Example**

The first example is the easiest case – the resistors placed in parallel have the same resistance. The goal of the analysis is to determine the current in and the voltage drop across each resistor.

The first step is to simplify the circuit by replacing the two parallel resistors with a single resistor that has an equivalent resistance. Two 8 Ω resistors in series is equivalent to a single 4 Ω resistor. Thus, the two branch resistors (R_{2} and R_{3}) can be replaced by a single resistor with a resistance of 4 Ω. This 4 Ω resistor is in series with R_{1} and R_{4}. Thus, the total resistance is

R_{tot} = R_{1} + 4 Ω + R_{4} = 5 Ω + 4 Ω + 6 Ω

R_{tot} = 15 Ω

Now the Ohm’s law equation (ΔV = I • R) can be used to determine the total current in the circuit. In doing so, the total resistance and the total voltage (or battery voltage) will have to be used.

I_{tot} = ΔV_{tot} / R_{tot} = (60 V) / (15 Ω)

I_{tot} = 4 Amp

The 4 Amp current calculation represents the current at the battery location. Yet, resistors R_{1} and R_{4} are in series and the current in series-connected resistors is everywhere the same. Thus,

I_{tot} = I_{1} = I_{4} = 4 Amp

For parallel branches, the sum of the current in each individual branch is equal to the current outside the branches. Thus, I_{2} + I_{3} must equal 4 Amp. There are an infinite number of possible values of I_{2} and I_{3} that satisfy this equation. Since the resistance values are equal, the current values in these two resistors are also equal. Therefore, the current in resistors 2 and 3 are both equal to 2 Amp.

I_{2} = I_{3} = 2 Amp

Now that the current at each individual resistor location is known, the Ohm’s law equation (ΔV = I • R) can be used to determine the voltage drop across each resistor. These calculations are shown below.

ΔV_{1} = I_{1} • R_{1} = (4 Amp) • (5 Ω)

ΔV_{1} = 20 V

ΔV_{2} = I_{2} • R_{2} = (2 Amp) • (8 Ω)

ΔV_{2} = 16 V

ΔV_{3} = I_{3} • R_{3} = (2 Amp) • (8 Ω)

ΔV_{3} = 16 V

ΔV_{4} = I_{4} • R_{4} = (4 Amp) • (6 Ω)

ΔV_{4} = 24 V

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