Circuit Theory Laws (Ohm's Law, KVL, KCL)

Before we can analyze circuits, we must understand the three fundamental electrical quantities: voltage, current, and resistance.

Electricity is invisible, which makes it hard to picture. The water analogy helps us visualize how charge, voltage, current, and resistance behave.

There is an important historical distinction in how we describe the direction of current in a circuit.

Conventional Current (Engineering Convention)

• Current flows from the positive terminal (+) to the negative terminal (-) of a power source.
• This convention was established before scientists discovered the electron and understood that negative charges are the ones physically moving.
• This is the standard convention used in most electrical engineering and circuit analysis.

Electron Flow (Physics Reality)

• Electrons (which carry negative charge) actually flow from the negative terminal (-) to the positive terminal (+).
• This is the physically accurate direction of charge carrier movement in metallic conductors.

Ohm's Law defines the mathematical relationship between voltage, current, and resistance in any electrical circuit.

The Ohm's Law Triangle

A helpful visual aid to solve for any of the three variables: cover up the value you want to find, and the remaining values show the formula.

       -----
      |  V  |
      |-----|
      | I | R |
       \-----/
          

Worked Example

Given: V = 6 V, R = 150 Ω
Find: I

Using Ohm's Law:
I = V / R
I = 6V / 150 Ω
I = 0.04 A = 40 mA
          

In a series circuit, components are connected end-to-end in a single loop. Current has only one path to flow, meaning it must flow through every component in sequence.

Kirchhoff's Voltage Law (KVL)

KVL states: The algebraic sum of all voltages around any closed loop in a circuit must equal zero. In practice, this means the sum of all individual voltage drops across resistors in a loop must equal the total voltage supplied by the source.

Worked Example: Series Analysis

Step 1: Total Resistance (R_T)
R_T = R1 + R2 + R3 = 200 + 400 + 400 = 1000 Ω = 1 kΩ

Step 2: Circuit Current (I)
I = V_T / R_T = 12V / 1000 Ω = 0.012 A = 12 mA

Step 3: Voltage Drops (using Ohm's Law on each resistor)
V1 = I × R1 = 12mA × 200 Ω = 2.4 V
V2 = I × R2 = 12mA × 400 Ω = 4.8 V
V3 = I × R3 = 12mA × 400 Ω = 4.8 V

Step 4: Verify KVL
V_T = V1 + V2 + V3 → 2.4V + 4.8V + 4.8V = 12V  [KVL Satisfied!]
          

In a parallel circuit, components are connected across the same two nodes, creating multiple branches for current. Voltage is identical across each branch, but the total current divides between them.

Kirchhoff's Current Law (KCL)

KCL states: The algebraic sum of currents entering a junction (node) must equal the sum of currents leaving that junction. Current does not get consumed; it simply splits when paths divide and merges when paths rejoin.

Parallel Formulas

For multiple resistors, use the reciprocal formula: R_T = 1 / (1/R1 + 1/R2 + 1/R3 + ...)

For exactly two parallel resistors, you can use the simplified product-over-sum rule: R_T = (R1 × R2) / (R1 + R2)

Worked Example: Parallel Analysis

Step 1: Total Resistance (Product-over-Sum)
R_T = (100 × 200) / (100 + 200) = 20000 / 300 = 66.67 Ω

Step 2: Branch Currents (Voltage is 12V across both)
I1 = Vs / R1 = 12V / 100 Ω = 0.12 A = 120 mA
I2 = Vs / R2 = 12V / 200 Ω = 0.06 A = 60 mA

Step 3: Total Current (using KCL)
I_T = I1 + I2 = 120 mA + 60 mA = 180 mA
Verification: I_T = Vs / R_T = 12V / 66.67 Ω = 0.18 A = 180 mA  [Matches!]
          

Most practical electrical devices contain combination circuits, mixing series and parallel elements together in a network. To solve these, reduce the circuit step-by-step.

Combination Reduction Strategy

1. Identify series and parallel branches. Look for points where current divides (parallel nodes) and paths where current has only one way to flow (series branches).
2. Simplify the circuit by calculating the equivalent resistance of parallel groups first, turning them into a single equivalent resistor.
3. Add series components together. Repeat the simplification process until the entire circuit is reduced to a single equivalent resistance.
4. Solve for total current using Ohm's Law with the total voltage and equivalent resistance.
5. Expand back outward, using branch current division and KVL/KCL rules to find the individual voltages and currents for every component.

Worked Example: Combination Circuit

        R1 = 100 Ω
    +--------/\/\/\/--------+
    |                       |
    |      R2 = 200 Ω        |  R3 = 300 Ω
    +--------/\/\/\/--------+---/\/\/\/---+
    |                                     |
    +=============== 12V =================+
          

Step 1: R1 and R2 are in parallel
R_12 = (R1 × R2) / (R1 + R2) = (100 × 200) / (100 + 200)
R_12 = 20000 / 300 = 66.67 Ω

Step 2: R_12 is in series with R3
R_T = R_12 + R3 = 66.67 + 300 = 366.67 Ω

Step 3: Calculate total current (I_T)
I_T = Vs / R_T = 12V / 366.67 Ω = 0.0327 A = 32.7 mA

Step 4: Calculate voltage drop across R3
V3 = I_T × R3 = 32.7mA × 300 Ω = 9.81 V

Step 5: Find the voltage drop across the parallel bank (R1 and R2)
Using KVL: V_bank = Vs - V3 = 12V - 9.81V = 2.19 V

Step 6: Find individual currents in R1 and R2
I1 = V_bank / R1 = 2.19V / 100 Ω = 21.9 mA
I2 = V_bank / R2 = 2.19V / 200 Ω = 10.9 mA
Verify KCL: I1 + I2 = 21.9mA + 10.9mA = 32.8 mA ≈ I_T  [Satisfied!]
          

Practice applying circuit laws. Click "Show Answer" on each problem below to verify your logic and view the worked solution.

I = V / R
I = 9V / 470 Ω
I = 0.01915 A = 19.15 mA