Equivalent Circuits for Makers

This is the third article in a series to teach you all the EE you need to know for working with and designing microcontroller based circuits.

Equivalent circuits are a tool for analyzing circuits. You have probably used this tool, possibly without thinking about it. We can use Kirchhoff’s laws to analyze any circuit, but we can sometimes make those equations much more simple by using equivalent circuits.

To create an equivalent circuit we use the equivalent resistance equations. We can combine resistors in two ways:

• Series
• Parallel

Equivalent Series Resistance

Resistors in series are combined by simply adding their values.

$$ R_= = R_1 + R_2 $$ $$ 1k \Omega + 1k \Omega = 2k \Omega $$

By doing the above equivalent circuit, we can easily apply Ohm’s law to find the current through the circuit.

$$ 5V = I \cdot 2k \Omega $$ $$ I = \frac{5V}{2k \Omega} = 0.0025A = 2.5mA $$

Series equivalent resistance is almost trivially easy, but combining resistors in parallel is bit trickier.

Equivalent Parallel Resistance

For two resistors in parallel, the equivalent resistance is:

$$ R_= = (\frac{1}{R_1} + \frac{1}{R_2})^{-1} $$

The same equation can be extended to N resistors in parallel.

$$ R_= = (\frac{1}{R_1} + \frac{1}{R_2} + … + \frac{1}{R_N})^{-1} $$

There is an algebraic short cut for just two resistors that looks like this:

$$ R_= = \frac{R_1 \cdot R_2}{R_1 + R_2} $$

1kΩ1k<b>Ω</b>5V5V1kΩ1k<b>Ω</b>5V5V500Ω500<b>Ω</b>

Like above, using an equivalent circuit makes it much easier to apply both Ohm’s law and Kirchhoff’s laws.

I want to share two important rules of thumb for parallel resistance.

1. The equivalent parallel resistance will always be lower than the lowest value resistor.

2. Adding parallel resistance will always lower the equivalent parallel resistance.

If you think back to the water/pump analogy, adding parallel resistance is like added additional paths for the water to flow. The total resistance will never be higher than the path of least resistance. And, adding paths for the water to flow will always make the total resistance lower.

Equivalent Circuits with KVL and KCL

Doing some simple iterations on an equivalent circuit can make applying KVL and KCL much easier. Consider the parallel combination below of our KCL/KVL circuit.

Now analyzing the circuit is almost trivally simple compared to the equations before.

That’s It

That is how equivalent circuits work. They are powerful but simple. Later when we discuss inductors and capacitors, you will see how they combine in equivalent circuit analysis (spoiler: inductors combine just like resistors but capacitors don’t). You have probably used equivalent circuits plenty without thinking about it. Hopefully this background gives you more understanding of how they are used.