Define the term equivalent resistance Calculate the equivalent resistance of resistors connected in series Calculate the equivalent resistance of resistors connected in parallel

In Current and Resistance, we described the term ‘resistance’ and explained the basic design of a resistor. Basically, a resistor limits the flow of charge in a circuit and is an ohmic device where (V = IR). Most circuits have more than one resistor. If several resistors are connected together and connected to a battery, the current supplied by the battery depends on the **equivalent resistance** of the circuit.

You are watching: In the figure three resistors are connected to a voltage source

The equivalent resistance of a combination of resistors depends on both their individual values and how they are connected. The simplest combinations of resistors are series and parallel connections (Figure (PageIndex{1})). In a **series circuit**, the output current of the first resistor flows into the input of the second resistor; therefore, the current is the same in each resistor. In a **parallel circuit**, all of the resistor leads on one side of the resistors are connected together and all the leads on the other side are connected together. In the case of a parallel configuration, each resistor has the same potential drop across it, and the currents through each resistor may be different, depending on the resistor. The sum of the individual currents equals the current that flows into the parallel connections.

(c) The individual currents are easily calculated from Ohm’s law (left(I_i = frac{V_i}{R_i}

ight)), since each resistor gets the full voltage. The total current is the sum of the individual currents:

(d) The power dissipated by each resistor can be found using any of the equations relating power to current, voltage, and resistance, since all three are known. Let us use (P_i = V^2 /R_i), since each resistor gets full voltage.

(e) The total power can also be calculated in several ways, use (P = IV).

**Solution**

The total resistance for a parallel combination of resistors is found using Equation

ef{10.3}. Entering known values gives * Current I for each device is much larger than for the same devices connected in series (see the previous example). A circuit with parallel connections has a smaller total resistance than the resistors connected in series. The individual currents are easily calculated from Ohm’s law, since each resistor gets the full voltage. Thus, Similarly, *

**Significance**

Total power dissipated by the resistors is also 18.00 W:

Notice that the total power dissipated by the resistors equals the power supplied by the source.

Exercise (PageIndex{2A})

Consider the same potential difference ((V = 3.00 , V)) applied to the same three resistors connected in series. Would the equivalent resistance of the series circuit be higher, lower, or equal to the three resistor in parallel? Would the current through the series circuit be higher, lower, or equal to the current provided by the same voltage applied to the parallel circuit? How would the power dissipated by the resistor in series compare to the power dissipated by the resistors in parallel?

**Solution**

The equivalent of the series circuit would be (R_{eq} = 1.00 , Omega + 2.00 , Omega + 2.00 , Omega = 5.00 , Omega), which is higher than the equivalent resistance of the parallel circuit (R_{eq} = 0.50 , Omega). The equivalent resistor of any number of resistors is always higher than the equivalent resistance of the same resistors connected in parallel. The current through for the series circuit would be (I = frac{3.00 , V}{5.00 , Omega} = 0.60 , A), which is lower than the sum of the currents through each resistor in the parallel circuit, (I = 6.00 , A). This is not surprising since the equivalent resistance of the series circuit is higher. The current through a series connection of any number of resistors will always be lower than the current into a parallel connection of the same resistors, since the equivalent resistance of the series circuit will be higher than the parallel circuit. The power dissipated by the resistors in series would be (P = 1.800 , W), which is lower than the power dissipated in the parallel circuit (P = 18.00 , W).

Let us summarize the major features of resistors in parallel:

Equivalent resistance is found from Equation

ef{10.3}and is smaller than any individual resistance in the combination. The potential drop across each resistor in parallel is the same. Parallel resistors do not each get the total current; they divide it. The current entering a parallel combination of resistors is equal to the sum of the current through each resistor in parallel.

In this chapter, we introduced the equivalent resistance of resistors connect in series and resistors connected in parallel. You may recall from the Section onCapacitance, we introduced the equivalent capacitance of capacitors connected in series and parallel. Circuits often contain both capacitors and resistors. Table (PageIndex{1}) summarizes the equations used for the equivalent resistance and equivalent capacitance for series and parallel connections.

Table (PageIndex{1}): Summary for Equivalent Resistance and Capacitance in Series and Parallel Combinations Series combination Parallel combinationEquivalent capacitance | ||

Equivalent resistance |

## Combinations of Series and Parallel

More complex connections of resistors are often just combinations of series and parallel connections. Such combinations are common, especially when wire resistance is considered. In that case, wire resistance is in series with other resistances that are in parallel.

Combinations of series and parallel can be reduced to a single equivalent resistance using the technique illustrated in Figure (PageIndex{5}). Various parts can be identified as either series or parallel connections, reduced to their equivalent resistances, and then further reduced until a single equivalent resistance is left. The process is more time consuming than difficult. Here, we note the equivalent resistance as (R_{eq}).