Series Parallel Circuit Examples – Easy Analysis

After learning about series and parallel circuit, we will learn about the series parallel circuit examples, a circuit with combination of series and parallel circuit.

In the actual applications, we will mostly find series parallel circuit instead of series circuit or parallel circuit. We may find a pure series circuit or parallel circuit, but mostly in our life, they will be the combination of series and parallel or we call them series parallel circuits.

Don’t worry, it is not an entirely different thing from series circuit or parallel circuit. We only need to take a different approach to solve this kind of circuit.

We should solve the circuit for each group of parallel resistors to make an entirely series circuit then solve it. Simply like that.

Or we could make it faster if we solve all the series resistors into equivalent series resistances. Then solve all the parallel resistors into equivalent parallel resistances. Last step is to solve those equivalent resistances into an equivalent total resistance.

From the next point until the very end, we will learn everything about how does a series parallel circuit work.

Series Parallel Circuit Analysis

Let’s observe a simple series parallel circuit below.

series parallel circuit examples 1

In the circuit above we can see that we have a voltage source with 4 resistors R1, R2, R3, and R4 in series parallel combination. The resistors R1 and R4 are in series connection, while the resistor R2 and R3 are in parallel connection.

Before calculating the total current flowing in the circuit, we should find the equivalent resistance from those four resistors.

We will use this circuit to solve one of the series and parallel circuits combined examples. We can use two different approaches in order to find the total resistance in that circuit.

First method, we will solve the series resistors to get an equivalent series resistor (RS) and the parallel resistors to get an equivalent parallel resistor (RP) separately. After that, we will solve the equivalent resistor (Req) for RS and RP connected in series.

series parallel circuit examples 2

Find the equivalent series resistor, RS:

    \begin{align*}R_S=R_1+R_4\end{align*}

Find the equivalent parallel resistor, RP:

    \begin{align*}R_P=\frac{R_2\times R_3}{R_2+R_3}\end{align*}

Find the equivalent

    \begin{align*}R_{eq}=R_S+R_P\end{align*}

If the steps above are still too long, we can use another approach.

Second method, we make a single equation consisting of all the resistors at once. We write them in order from positive polarity to negative polarity. This way we will solve the circuit in one go.

series parallel circuit examples 3

    \begin{align*}R_{eq}&=R_1+(R_2||R_3)+R_4\\&=R_1+\frac{R_2\times R_3}{R_2+R_3}+R_4\\&=(R_1+R_4)+\frac{R_2\times R_3}{R_2+R_3}\end{align*}

Finally, we can write the series parallel circuit formula from the circuit above.

How to Analyze a Series Parallel Circuit

Now we will proceed to analyze a series parallel circuit. This will be different from the simple example above because we will analyze the circuit for its equivalent resistance, voltage drop, flowing current, and more.

Observe the circuit below.

series parallel circuit examples 4

Here we have one resistor in series with a voltage source and a pair of parallel resistors.

Series Parallel Equivalent Resistance

First thing first, we have to find the equivalent resistance of the circuit. We will use the second method, one equation consisting of series parallel resistors.

    \begin{align*}R_{eq}&=R_1+(R_2||R_3)\\&=3+\frac{3\times6}{3+6}\\&=3+2\\&=5\end{align*}

Thus the total current flowing in the circuit is

    \begin{align*}I&=\frac{V}{R_{eq}}\\&=\frac{10}{5}\\&=2A\end{align*}

Series Parallel Circuit Voltage Drop

Now we proceed to calculate voltage drop for each resistor.

Keep in mind that:

  • Voltage drop in the series connection has different values across each resistor.
  • Voltage drop in the parallel connection has the same values across each resistor.

The voltage drop across the R1 (VR1) is

    \begin{align*}V_{R1}&=I\times R_1\\&=2\times3\\&=6V\end{align*}

The voltage drop across the R2 and R3 are equivalent to the voltage drop across the equivalent parallel resistor (RP) of R2 and R3.

series parallel circuit examples 5

    \begin{align*}V_{RP}&=I\times R_{P}\\&=2\times2\\&=4V\end{align*}

Thus, we also prove the Kirchhoff’s Voltage Law where the algebraic sum of voltage drop in a closed circuit is zero.

    \begin{align*}V&=V_{R1}+V_{RP}\\10&=6+4\\10&=10\end{align*}

Then the voltage drop across the R2 and R3 are 4 V.

series parallel circuit examples 6

Series Parallel Circuit Current

Next we will calculate the current flowing in the circuit.

Keep in mind that:

  • The current in the series connection will have constant value through every element.
  • The current in the parallel connection will have different values depending on the resistance in each branch.

series parallel circuit examples 7

Since R1 is in series with the voltage source, the current (I1) through it is equal to the total current,

    \begin{align*}I_1=I=2A\end{align*}

Now we need to do some math to determine the current through R2 and R3. From the result before, the voltage drop across the parallel resistor is 4V then we will use it along with Ohm’s law to determine its current.

For I2,

    \begin{align*}I_2&=\frac{V_{RP}}{R_2}\\&=\frac{4}{3}A\end{align*}

For I3,

    \begin{align*}I_3&=\frac{V_{RP}}{R_3}\\&=\frac{4}{6}\\&=\frac{2}{3}A\end{align*}

series parallel circuit examples 8

The total current entering the branches R2 and R3 is I = 2A and it is divided into 4/3 A through the left branch and 2/3A through the right branch.

The current leaving the branch is

    \begin{align*}\frac{4}{3}+\frac{2}{3}=2A\end{align*}

This proves the Kirchhoff’s Current Law which states that the current entering a node is equal to the current leaving a node. Or in other words, the algebraic sum of currents entering and leaving a node is zero.

Series Parallel Open Circuit

Now we will try to make an open connection in a series parallel circuit. We will use the same circuit as above but we give an open terminal in series with R1.

series parallel circuit examples 9

An open terminal has an infinite resistance. Because it is connected in series then the equivalent total resistance in the circuit is infinite. Thus the total current (I) is zero and we won’t get any voltage drop since there is no current flowing in the rest of the circuit.

Next we connect an open terminal in parallel with R2 and R3 just like below.

series parallel circuit examples 10

Since there will be no current flowing through it, you can ignore it. For its voltage drop will be equal to the voltage drop across R2 and R3 since they are in parallel.

Series Parallel Short Circuit

Different with open circuit, most of the time a short circuit can harm the circuit and anything connected into it. First we make a short circuit in parallel with R1.

series parallel circuit examples 11

You may know that an electric current will likely choose the path with the least resistance or impedance. This way, the less resistance the more current flows through it. In an ideal condition, a short circuit has zero resistance, thus the current flowing in the branch with zero resistance is maximum (infinite).

Calculating the equivalent resistance (Rs) for R1 and short circuit in parallel,

    \begin{align*}R_S&=\frac{0\times3}{0+3}\\&=0\end{align*}

We can redraw the circuit into,

series parallel circuit examples 12

Since Rs has zero resistance we can replace it with a single conductor wire. Calculation for the redrawn circuit is simple just like before.

And what if we connect a short circuit in parallel with R2 and R3?

series parallel circuit examples 13

The equivalent resistance for parallel resistors (Rp) is zero.

Short story we can redraw the circuit into below.

series parallel circuit examples 14

Series Parallel Circuit Calculations Examples

Let’s do some exercises to get a better understanding.

1. Observe the circuit below and find the equivalent resistances, voltage drops, and currents.

series parallel circuit examples 15

We can calculate the equivalent resistance in a single equation but we will calculate each parallel connection separately because we need their respective values to calculate voltage drops and current.

The equivalent resistance for the first parallel resistors:

    \begin{align*}R_{P1}&=\frac{2\times3}{2+3}\\&=\frac{6}{5}\\&=1.2\end{align*}

The equivalent resistance for the second parallel resistors:

    \begin{align*}R_{P2}&=\frac{3\times6}{3+6}\\&=\frac{18}{9}\\&=2\end{align*}

The total equivalent resistance is

    \begin{align*}R_{eq}&=R_{P1}+R_{P2}\\&=1.2+2\\&=3.2\end{align*}

The total current in the circuit is:

    \begin{align*}I&=\frac{V}{R}\\&=\frac{12}{3.2}\\&=3.75A\end{align*}

We can redraw the circuit into

series parallel circuit examples 16

The voltage drop across RP1 is

    \begin{align*}V_1&=I\times R_{P1}\\&=3.75\times1.2\\&=4.5V\end{align*}

The voltage drop across RP2 is

    \begin{align*}V_2&=I\times R_{P2}\\&=3.75\times2\\&=7.5V\end{align*}

This proves that

    \begin{align*}V&=V_1+V_2\\12&=4.5+7.5\end{align*}

series parallel circuit examples 17

Next we will calculate currents through each resistor.

For I1,

    \begin{align*}I_1&=\frac{V_1}{R_1}\\&=\frac{4.5}{2}\\&=2.25\end{align*}

For I2,

    \begin{align*}I_2&=\frac{V_1}{R_2}\\&=\frac{4.5}{3}\\&=1.5\end{align*}

For I3,

    \begin{align*}I_3&=\frac{V_2}{R_3}\\&=\frac{7.5}{3}\\&=2.5\end{align*}

For I4,

    \begin{align*}I_4&=\frac{V_2}{R_4}\\&=\frac{7.5}{6}\\&=1.5\end{align*}

This proves that

    \begin{align*}I=I_1+I_2&=I_3+I_4\\3.75=2.25+1.5&=2.5+1.25\end{align*}

2. Observe the circuit below and determine the equivalent resistance, voltage drops, and currents.

series parallel circuit examples 18

The equivalent resistance for the parallel resistors is

    \begin{align*}R_P&=\frac{4\times12}{4+12}\\&=\frac{48}{16}\\&=3\end{align*}

The total equivalent resistance in the circuit is

    \begin{align*}R_{eq}&=R_P+R_3\\&=3+2\\&=5\end{align*}

The total current in the circuit is

    \begin{align*}I&=\frac{V}{R_{eq}}\\&=\frac{10}{5}\\&=2A\end{align*}

series parallel circuit examples 19

The voltage drop across the parallel resistors (VRP) is

    \begin{align*}V_{RP}&=I\times R_P\\&=2\times3\\&=6V\end{align*}

The voltage drop across the R3 is

    \begin{align*}V_{R3}&=I\times R_3\\&=2\times2\\&=4V\end{align*}

series parallel circuit examples 20

The current flowing through R1 is

    \begin{align*}I_1&=\frac{V_{RP}}{R_1}\\&=\frac{6}{4}\\&=1.5A\end{align*}

The current flowing through R2 is

    \begin{align*}I_2&=\frac{V_{RP}}{R_2}\\&=\frac{6}{12}\\&=0.5A\end{align*}

The current flowing through R3 is

    \begin{align*}I_{3}=I=2A\end{align*}

Series Parallel Circuit Examples in Real Life

If you are wondering “where will we find the example of series parallel circuits in our life”, they are already close to you. The most common answer you will find is electrical outlets in your house.

When you are turning off your lamp, disconnecting your electronic device from the outlet, or your outlet is not connected with anything, they will not disturb other outlets. Your electronic device will still run fine and safely.

For the series circuit, one of the examples in real life is Christmas tree lights. This set of light sequences is connected in series. Another series circuit is a connection between your power supply, switch, and light bulb. They are connected in series to make sure you can turn it on or off using a switch.

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