Easy and Simple Formula for Parallel Resistor

Formula for parallel resistor will be our main focus here. After discussing series resistor and voltage divider, let us learn about parallel resistors and current division. Parallel resistor belongs to one of the passive elements.

You will know whether resistors are in parallel or series simply from its terminal connection. We call resistors in parallel connection if their terminal is connected together respectively from resistor to resistor.

Parallel Circuit Definition

This parallel resistor is different from the series resistor where there is only one straight path passing the resistors. In the parallel resistor there will be many paths from head to head and tail to tail. From this term, the parallel resistor is used as a current divider, while the series resistor is voltage divider.

Since the current is divided to multiple paths or branches, the current for each branch may be different from each other. But, the voltage drop for each resistor will be the same for each other. This concludes that for resistors in parallel they may have different current through each of them but always have the same voltage drop across each of them.

Why do we need to learn formula for parallel resistor? You should know after reading this post until finished.

Parallel resistors definition is a resistive circuit where the resistors are connected together to the same nodes and produce more than one current paths connected to the same voltage source. If you are still confused how we make a parallel resistive circuit, we can find it below.

Parallel Circuit Diagram

The example below is the simple parallel resistor in a circuit. We use three resistors R1, R2, and R3. From the explanation above, the voltage drop across the resistors in the parallel connection will be the same for each other and equal to the voltage source.

Hence,

    \begin{align*}V_{R1}=V_{R2}=V_{R3}=V_{AB}=12V\end{align*}

All of the three resistors are connected together between A and B.

parallel circuit diagram

Different from series resistive circuits where the equivalent resistance is the sum of all the resistors, the parallel resistor is calculated differently. We are using the reciprocal of resistance (1/R) for each resistor added together.

Formula for Parallel Resistor

Just like we mentioned above, the equivalent resistance of parallel resistors is the sum of reciprocal of each resistor. If the resistors connected together in parallel have the same value then it will be very easy.

Just as we mentioned above, we can look at the example below:

formula for parallel resistor

 

If two resistors connected in parallel have the same resistance then the equivalent resistance, RT is the half the resistance of one resistor. Hence the equivalent resistance of two resistors connected in parallel if they have the same resistance is R/2. If there are three resistors in parallel then the equivalent resistance is R/3, and so on.

Aside from series resistors, this type of circuit is the most common type of electric circuit. For analyzing a parallel resistor circuit we can use the same method for the series resistor, using Kirchhoff’s laws and Ohm’s law.

Consider the circuit in Figure.(1),

parallel resistor connection
Figure 1. Parallel resistor connection

where two resistors are connected in parallel and therefore have the same voltage across them.

From Ohm’s law

(1)   \begin{equation*}v=i_{1}R_{1}=i_{2}R_{2}\\i_{1}=\frac{v}{R_{1}}, \quad i_{2}=\frac{v}{R_{2}}\end{equation*}

Applying KCL at node a gives the total current i as

(2)   \begin{equation*}i=i_{1}+i_{2}\end{equation*}

Substituting Equation.(1) into (2), we have

(3)   \begin{align*}i&=\frac{v}{R_{1}}+\frac{v}{R_{2}}\\ &=v(\frac{1}{R_{1}}+\frac{1}{R_{2}})\\ &=\frac{v}{R_{eq}}\end{align*}

where Req is the equivalent resistance of the resistors in parallel :

(4)   \begin{align*}\frac{1}{R_{eq}}&=\frac{1}{R_{1}}+\frac{1}{R_{2}}\\\frac{1}{R_{eq}}&=\frac{R_{1}+R_{2}}{R_{1}R_{2}}\end{align*}

or

(5)   \begin{align*}{R_{eq}}=\frac{R_{1}R_{2}}{R_{1}+R_{2}}\end{align*}

Hence,

The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum.

Above is the simplest parallel resistor equation we can use every time we need it.

Must be noted that the Equation.(5) only works for two resistors in parallel.

We can expand the parallel resistor equation in Equation.(4) to the general case of a circuit with N resistors in parallel. The equivalent resistance is

(6)   \begin{align*}\frac{1}{R_{eq}}&=\frac{1}{R_{1}}+\frac{1}{R_{2}}+...+\frac{1}{R_{N}}\end{align*}

Take a note that Req is always smaller than the resistance of the smallest resistor in the parallel connection. If R1 =R2 = … = RN = R, then

(7)   \begin{align*}R_{eq}=\frac{R}{N}\end{align*}

For example, if 4 resistors with 100 Ω are connected in parallel, their equivalent resistance is 25 Ω.

Keep in mind that,

The equivalent resistance of parallel resistors is always less than the smallest resistor connected to that network. Hence, the equivalent resistance, RT will decrease every time we have additional parallel resistors.

Conductance Formula for Parallel Circuit

Because of this simple yet complicated parallel resistor equation for the equivalent resistance we will learn about a new value known as Conductance (G), measured in Siemens (S). The conductance is the inverse of resistance, where G = 1/R. After we get the conductance then we convert it back inversely to get the equivalent resistance RT of the parallel resistors.

It is easier to use conductance than resistance when dealing with resistors connected in parallel.

From Equation.(6) the equivalent conductance for N resistors is

(8)   \begin{align*}G_{eq}=G_{1}+G_{2}+...+G_{N}\end{align*}

where:

    \begin{align*}G_{eq}&=\frac{1}{R_{eq}}\\G_{1}&=\frac{1}{R_{1}}\\G_{2}&=\frac{1}{R_{2}}\\G_{N}&=\frac{1}{R_{N}}\\\end{align*}

Equation.(8) states :

The equivalent conductance of resistors connected in parallel is the sum of their individual conductances.

It means we can redraw Figure.(1) with (2) where we replace the resistances to conductances.

The equivalent conductances of parallel resistors are obtained the same way as equivalent resistances of series resistors.

In opposite, the equivalent conductances of series resistors are obtained the same way as equivalent resistances of parallel resistors.

equivalent resistance or conductance
Figure 2. Equivalent resistance or conductance

Hence the equivalent conductance Geq of N resistors in series is

(9)   \begin{align*}\frac{1}{G_{eq}}=\frac{1}{G_{1}}+\frac{1}{G_{2}}+...+\frac{1}{G_{N}}\end{align*}

Given the total current i entering node a in Figure.(1) with the same values of voltage, we get

(10)   \begin{align*}v=iR_{eq}=i\frac{R_{1}R_{2}}{R_{1}+R_{2}}\end{align*}

We define parallel resistors as the resistors connected together between the same two point. The parallel resistor itself has various forms of circuit.

How to Find Current in a Parallel Circuit

Because the current in parallel circuit is based on its resistance, now we will learn about how to find current in a parallel. The currents I1, I2, …, In entering the parallel path of resistors depend on the resistance of that branch. The total current, IT is the sum of the currents in parallel branches. If the resistance between branches is equal then the currents will be also equally divided.

If R1 = R2 then the I1 = I2 = 0.5IT. It means the total current IT is divided equally for two branches. If the R1 has different resistance from R2 then we need to calculate the I1 and I2 differently. Even if the voltage across the branches are equal, the current may differ according to Ohm’s law.

For parallel circuit example let us see the circuit below and try to find all the currents,

how to find current in a parallel circuit

Since the R1 and R2 have different values, then the currents I1 and I2 are guaranteed to have different values. Remember one of the Kirchhoff laws?

Kirchhoff’s current laws states:

The total current leaving a node is equal to the current entering that same node.

Hence,
The total current in the circuit can be expressed as:

    \begin{align*}I_{T}=I_{R1}+I_{R2}\end{align*}

After that, we will use Ohm’s law to calculate the current entering each branch through the resistors. The I1 is the current entering R1 while the I2 is the current entering R2. The voltage source Vs has 12V and we get:

    \begin{align*}I_{1}&=\frac{V_{s}}{R_{1}}=\frac{12}{3}=4A\\I_{2}&=\frac{V_{s}}{R_{2}}=\frac{12}{6}=2A\\\end{align*}

And we get the total current,

    \begin{align*}I_{T}=I_{R1}+I_{R2}=4+2=6A\end{align*}

To clarify this, we use Ohm’s law to calculate IT from Vs and RT.
The total resistance RT is

    \begin{align*}\frac{R_{1}R_{2}}{R_{1}+R_{2}}=\frac{3\times6}{3+6}=2\Omega\end{align*}

Then the total current IT is

    \begin{align*}I_{T}=\frac{V_{s}}{R_{T}}=\frac{12}{2}=6A\end{align*}

Hence this clarifies our calculations.

We conclude that

    \begin{align*}I_{total}=I_{1}+I_{2}+...+I_{n}\end{align*}

Current Divider Equation

Combining Equations.(1) and (10) we get the current divider equation.

(11)   \begin{align*}i_{1}&=\frac{R_{2}i}{R_{1}+R_{2}}\\i_{2}&=\frac{R_{1}i}{R_{1}+R_{2}}\end{align*}

which shows the total current i is shared by the resistors in inverse proportion to their resistances.

This is known as the principal of current division, and the circuit in Figure.(1) is known as current divider.

Take a note that the larger current flows through the smaller resistance.

short and open circuit in parallel connection
Figure 3. Short and open circuit in parallel connection

Suppose one of the resistor in Figure.(1) is zero, say R2 = 0; so R2 is a short circuit, as can be seen in Figure.(3a).

From Equation.(11), R2 = 0 implies that i1 = 0, i2 = i. This means that the entire current i bypasses R1 and flows through the short circuit R2 = 0, the path with least resistance.
When a circuit is short-circuited as can be seen in Figure.(3a), take note that :

  1. The equivalent resistance Req = 0
  2. The entire current flows through the short circuit.

For another extreme example where R2 = ∞ , that is, R2 is an open circuit as can be seen in Figure.(3b).
The current still flows through a path with least resistance, R1.
Equation.(11) becomes

(12)   \begin{align*}i_{1}&=\frac{G_{1}}{G_{1}+G_{2}}i\\i_{2}&=\frac{G_{2}}{G_{1}+G_{2}}i\end{align*}

In general, if a current divider has N conductors in parallel with source current i, the Nth conductor will have current

(13)   \begin{align*}i_{n}=\frac{G_{n}}{G_{1}+G_{2}+...+G_{N}}i\end{align*}

It is very convenient to combine resistors in series and parallel into single equivalent resistance Req.

Such equivalent resistance must have the same values of current and voltage as the original network at the terminal.

Parallel Resistor Examples

Let us review the example below for a better understanding
Find Req for the circuit in Figure.(4)

parallel resistor example
Figure 4. Parallel resistor example

6 Ω and 3 Ω in parallel

    \begin{align*}6\Omega\parallel 3\Omega=\frac{6\times 3}{6+3}=2\Omega\end{align*}

1 Ω and 5 Ω in series

    \begin{align*}1\Omega +5\Omega=6\Omega\end{align*}

2 Ω and 2 Ω in series

parallel resistor example

6 Ω and 4 Ω in parallel

    \begin{align*}4\Omega\parallel 6\Omega=\frac{4\times 6}{4+6}=2.4\Omega\end{align*}

Three resistors in series

parallel resistor example

    \begin{align*}R_{eq}=4\Omega+2.4\Omega+8\Omega=14.4\Omega\end{align*}

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