Series and Parallel Resistor Networks

1. Resistors in Series

When components are connected end-to-end, creating a single path for the current, they are said to be in series. The defining characteristic of a series circuit is that the current is the same through every component.

Let's derive the formula for the equivalent resistance of a series circuit using Kirchhoff's Voltage Law (KVL), which we covered in the previous lesson.

As the text demonstrated, by applying KVL ($\sum V=0$) to a simple series circuit with a voltage source $V$ and N resistors, we get:
$V-V_1-V_2-\cdots -V_N=0⟹V=V_1+V_2+\cdots +V_N$
Since the current $I$ is the same through all resistors, we can substitute Ohm's Law ($V_i=I{R}_i$) for each voltage drop:
$I{R}_S=I{R}_1+I{R}_2+\cdots +I{R}_N$
Where $R_S$ is the equivalent series resistance. Dividing by the common current $I$ gives us the simple and elegant result:

Equivalent Resistance in Series:
$R_S=R_1+R_2+R_3+\cdots +R_N={\sum }_{i=1}^N{R}_i$
The total resistance of resistors in series is simply the sum of their individual resistances.

2. Resistors in Parallel

When components are connected across the same two points (or nodes) in a circuit, they are said to be in parallel. The defining characteristic of a parallel circuit is that the voltage drop is the same across every component.

This time, we'll use Kirchhoff's Current Law (KCL) to derive the formula.

From the reading, we see that KCL ($\sum I_{in}=\sum I_{out}$) applied to a junction where the current splits tells us:
$I=I_1+I_2+\cdots +I_N$
Since the voltage $V$ is the same across all parallel resistors, we can substitute $I_i=V/R_i$:
$\frac{V}{R_P}=\frac{V}{R_1}+\frac{V}{R_2}+\cdots +\frac{V}{R_N}$
Where $R_P$ is the equivalent parallel resistance. Dividing by the common voltage $V$ gives us the formula:

Equivalent Resistance in Parallel:
$\frac{1}{R_P}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\cdots +\frac{1}{R_N}={\sum }_{i=1}^N\frac{1}{R_i}$
The reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances. An important consequence is that the equivalent resistance in a parallel combination is always less than the smallest individual resistance.

A Very Common Special Case: Two Resistors in Parallel

For the common scenario of exactly two resistors in parallel, we can derive a simpler-to-use formula. Your interest in algebraic manipulation will make this derivation straightforward.

As derived in the video, for two resistors $R_1$ and $R_2$ in parallel:
$R_P=\frac{R_1{R}_2}{R_1+R_2}$
This is often remembered as the "product over sum" rule. It is a very handy shortcut.

3. Analyzing Combined Series-Parallel Circuits

Most real-world circuits are not purely series or purely parallel. They are combinations of both. The strategy to analyze them is to reduce the circuit in steps:

1. Identify a simple series or parallel group of resistors within the larger circuit.
2. Calculate the equivalent resistance for that group.
3. Redraw the circuit, replacing the group with its single equivalent resistor.
4. Repeat steps 1-3 until the entire circuit is reduced to a single equivalent resistance.

Let's watch a complete, detailed example of this reduction process.

The process you just saw is the standard method for analyzing any reducible resistor network. The key is to be methodical: identify, calculate, and redraw.

Conclusion

This lesson completes our foundational module on DC circuits. We've established a powerful technique for simplifying complex resistor networks, which is a crucial step in many circuit analysis problems.

Key Takeaways:

• Resistors in Series: The current is the same through all. Equivalent resistance is the sum of individual resistances: $R_S=\sum R_i$.
• Resistors in Parallel: The voltage is the same across all. The reciprocal of the equivalent resistance is the sum of the reciprocals: $\frac{1}{R_P}=\sum \frac{1}{R_i}$. The equivalent resistance is always smaller than the smallest resistor in the combination.
• Combined Networks: These are simplified by methodically identifying and reducing series and parallel sub-circuits, redrawing the circuit at each step, until a single equivalent resistance is found.

With the completion of this module, you have a solid theoretical understanding of the physics governing DC circuits. You can now analyze voltage, current, resistance, and power using Ohm's Law, Kirchhoff's Laws, and series-parallel reduction.

Preview of the Next Lesson:
We are now ready to bridge theory and practice. In our next module, we will finally unbox your Arduino! We will begin by exploring the Arduino board itself and writing your first simple program. You will learn how to use the Arduino to act as your measurement tool, verifying the electrical principles we've just studied and bringing the physics to life on your breadboard.