Grade 10

Advanced

10 min

Lesson 7 of 12

Complex and Combination Circuits

Not Started

Integrates series and parallel rules to solve sophisticated circuit networks.

Why does a single blown fuse in your car stop the radio but keep the headlights shining? The secret lies in how engineers weave series and parallel paths together to control the flow of energy.

Learning Objectives

1.Simplify 'nested' circuit diagrams into a single equivalent resistance
2.Apply Kirchhoff’s Current Law (KCL) to track how electricity splits and merges at junctions
3.Apply Kirchhoff’s Voltage Law (KVL) to ensure energy is conserved across any closed loop

The Strategy of Simplification

When you look at a complex circuit, don't panic! Think of it like a set of Russian nesting dolls. To find the total resistance (

R_{total}

), you must work from the 'inside out.' Identify the smallest groups of resistors that are purely series or purely parallel. Replace them with their equivalent resistance (

R_{eq}

), and redraw the circuit. Repeat this until only one resistor remains. For series, we use

R_s = R_1 + R_2

. For parallel, we use

\begin{gathered}\\frac{1}{R_p} = \\frac{1}{R_1} + \\frac{1}{R_2}\end{gathered}

. This process of 'collapsing' the circuit allows us to use Ohm's Law (

V = IR

) on the entire system to find the total current provided by the battery.

A $12V$ battery is connected to a $10\\Omega$ resistor ( $R_1$ ) in series with a parallel pair of two $20\\Omega$ resistors ( $R_2$ and $R_3$ ).

1. Simplify the parallel pair: $R_p = \\frac{20 \\times 20}{20 + 20} = 10\\Omega$ . 2. Add the series resistor: $R_{total} = R_1 + R_p = 10 + 10 = 20\\Omega$ . 3. Find total current: $I = \\frac{V}{R} = \\frac{12}{20} = 0.6A$ .

Quick Check

If you have a parallel branch inside a series circuit, which part should you simplify first?

Work from the most 'nested' part of the circuit outward.

Answer

The parallel branch.

Kirchhoff’s Junction Rule (KCL)

Electricity behaves like water flowing through pipes. At any junction (a point where three or more wires meet), the total current entering must equal the total current leaving. This is Kirchhoff’s Current Law (KCL), and it is a direct result of the Conservation of Charge. Charge cannot be created or destroyed; it just takes different paths. Mathematically:

\begin{gathered}\\sum I_{in} = \\sum I_{out}\end{gathered}

. If you know the total current entering a house and the current going to the kitchen, you can calculate exactly how much is left for the rest of the rooms!

A main wire carries $15A$ into a junction. The junction splits into three paths. Path A carries $5A$ and Path B carries $7A$ . How much current is in Path C?

1. Identify total current in: $15A$ . 2. Identify known current out: $5A + 7A = 12A$ . 3. Solve for unknown: $15A - 12A = 3A$ .

Quick Check

KCL is based on the conservation of which physical property?

Answer

Electric charge.

The Energy Balance: Kirchhoff’s Voltage Law (KVL)

While KCL tracks charge, Kirchhoff’s Voltage Law (KVL) tracks energy. Think of a battery as a ski lift that gives charges 'potential energy' (voltage). As charges move through resistors, they 'drop' that energy. KVL states that the sum of all potential differences (voltages) around any closed loop must be zero:

\begin{gathered}\\sum V = 0\end{gathered}

. This means the energy supplied by the source is exactly equal to the energy used by the components in that loop. It’s like a rollercoaster: no matter how many hills and loops you go through, if you end up back at the start, your total change in height is zero.

A $24V$ battery is connected to a $4\\Omega$ resistor in series with a parallel pair ( $6\\Omega$ and $12\\Omega$ ). Find the voltage drop across the $4\\Omega$ resistor.

1. Find parallel resistance: $R_p = \\frac{6 \\times 12}{6 + 12} = 4\\Omega$ . 2. Find total resistance: $R_{total} = 4 + 4 = 8\\Omega$ . 3. Find total current: $I = \\frac{24}{8} = 3A$ . 4. Find voltage drop across the $4\\Omega$ resistor: $V = I \\times R = 3A \\times 4\\Omega = 12V$ .

Key Takeaways

1Simplify complex circuits by replacing parallel and series groups with equivalent resistors ( $R_{eq}$ ) from the inside out.
2Kirchhoff’s Current Law (KCL) states that current entering a junction equals current leaving it (Charge Conservation).
3Kirchhoff’s Voltage Law (KVL) states that the sum of voltages around any closed loop is zero (Energy Conservation).

Quick Check

Multiple Choice

Kirchhoff's Current Law (KCL) is a statement of which conservation law?

Multiple Choice

If a closed loop contains a $9V$ battery and two resistors, and the first resistor has a $3V$ drop, what is the voltage drop across the second resistor?

True/False

The equivalent resistance of two resistors in parallel is always smaller than either individual resistor.

What's Next?

Review Tomorrow

In 24 hours, try to recall the difference between Kirchhoff's Current Law (junctions) and Voltage Law (loops) without looking at your notes.

Practice Activity

Sketch a circuit with a $12V$ battery and three $6\\Omega$ resistors (two in parallel, one in series) and calculate the total current.

Browse

Browse

Complex and Combination Circuits

Learning Objectives

The Strategy of Simplification

Kirchhoff’s Junction Rule (KCL)

The Energy Balance: Kirchhoff’s Voltage Law (KVL)

Key Takeaways

Quick Check

What's Next?

Complex and Combination Circuits

Learning Objectives

The Strategy of Simplification

Kirchhoff’s Junction Rule (KCL)

The Energy Balance: Kirchhoff’s Voltage Law (KVL)

Key Takeaways

Quick Check

What's Next?