Introduction to Compound DC Circuits

Welcome to the world of Compound DC Circuits! In previous chapters, we looked at circuits where resistors were either all in a single line (series) or all side-by-side (parallel). In the real world, most electronic devices—like your smartphone or a computer—use a mix of both. These are called compound circuits (or series-parallel circuits).

Don't worry if these look intimidating at first. Think of a compound circuit like a puzzle. By breaking it down into smaller, simpler pieces, you can solve even the most complex-looking diagrams using the same basic rules of physics you already know.

What is a Compound Circuit?

A compound circuit is a circuit that contains combinations of both series and parallel connections. To analyze these circuits, we focus on identifying which parts are in series (sharing the same current) and which are in parallel (sharing the same potential difference).

Analogy: Think of a compound circuit like a river system. Some parts of the river flow through a single narrow channel (series), while other parts split into multiple smaller streams (parallel) before joining back together.

The "Collapse and Expand" Method

The most effective way to solve a compound circuit is a two-phase process. First, we "collapse" the circuit to find the total resistance, then we "expand" it to find individual values.

Phase 1: Collapsing the Circuit (Finding \(R_{eq}\))

To find the equivalent resistance (\(R_{eq}\)) of the entire circuit, follow these steps:

  1. Identify the "Innermost" Groups: Look for a set of resistors that are purely in series or purely in parallel with only each other.
  2. Simplify the Group:
    - For series parts: \(R_s = R_1 + R_2 + ...\)
    - For parallel parts: \(\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + ...\)
  3. Redraw the Circuit: Replace that group with a single equivalent resistor.
  4. Repeat: Continue this process until the entire circuit is simplified into one single resistor and the battery.

Phase 2: Expanding the Circuit (Finding \(I\) and \(V\))

Once you have the total equivalent resistance, you can find the total current (\(I_{total}\)) leaving the battery using Ohm's Law: \(I_{total} = \frac{V_{battery}}{R_{eq}}\).

To find the current or voltage for a specific resistor, work backward from your simplified drawings:

  • If you are "un-collapsing" a series group: The current is the same for every resistor in that group (\(I_{total} = I_1 = I_2\)).
  • If you are "un-collapsing" a parallel group: The voltage (potential difference) is the same across every branch (\(V_{total} = V_1 = V_2\)).

Quick Review: Remember that in Unit 11, we assume all batteries, wires, and meters are ideal. This means wires have zero resistance, and we don't worry about internal battery resistance unless specifically told otherwise.

Step-by-Step Example

Imagine a \(12V\) battery connected to a \(4\Omega\) resistor (\(R_1\)) that is in series with a parallel pair of \(6\Omega\) resistors (\(R_2\) and \(R_3\)).

Step 1: Simplify the Parallel Branch

Find the equivalent resistance of the two \(6\Omega\) resistors in parallel:
\(\frac{1}{R_p} = \frac{1}{6} + \frac{1}{6} = \frac{2}{6}\)
\(R_p = \frac{6}{2} = 3\Omega\)

Step 2: Simplify the Remaining Series Circuit

Now the circuit is just \(R_1\) (\(4\Omega\)) in series with the new \(R_p\) (\(3\Omega\)):
\(R_{eq} = 4\Omega + 3\Omega = 7\Omega\)

Step 3: Calculate Total Current

\(I_{total} = \frac{V}{R_{eq}} = \frac{12V}{7\Omega} \approx 1.71A\)

Step 4: Find Individual Voltages

The current through \(R_1\) is the total current (\(1.71A\)). The voltage drop across \(R_1\) is:
\(V_1 = I \cdot R_1 = 1.71A \cdot 4\Omega = 6.84V\)

Important Conventions & Assumptions

For the AP Physics 2 Exam, keep these standard conventions in mind for compound circuits:

  • Conventional Current: We always model current as flowing out of the positive terminal and into the negative terminal.
  • Ohmic Resistors: Unless a question says otherwise, assume resistors and lightbulbs obey Ohm's Law (\(V = IR\)), meaning their resistance stays constant regardless of current.
  • Ammeters and Voltmeters:
    - An ammeter (measures current) is placed in series and is assumed to have zero resistance.
    - A voltmeter (measures potential difference) is placed in parallel and is assumed to have infinite resistance so no current flows through it.

Common Pitfalls to Avoid

1. Don't add everything at once: You cannot use the series formula for resistors that are separated by a junction. You must resolve the parallel branches first!

2. Watch the Junctions: Remember that current splits at a junction. If \(5A\) enters a parallel split, the sum of the currents in the branches must equal \(5A\).

3. Potential Difference vs. Battery Voltage: In a compound circuit, the voltage across a single resistor is rarely the same as the battery voltage. Always calculate the specific \(V\) for that component using \(V = IR\).

Key Takeaways

- Series Logic: Same current (\(I\)), shared voltage (\(V\)).
- Parallel Logic: Same voltage (\(V\)), shared current (\(I\)).
- Strategy: Simplify (collapse) the circuit from the inside out to find total resistance, then use Ohm's Law to find the specific values you need.
- Power: If you need to find the power dissipated by a component in a compound circuit, use \(P = IV\), \(P = I^2R\), or \(P = \frac{V^2}{R}\) once you have found the specific \(I\) or \(V\) for that component.

Did you know? Lightbulbs in your house are generally wired in parallel so that if one bulb burns out, the rest of the "compound" system of your home stays lit!