Math Behind Logic Gates

Understanding the Real Electronics Behind Logic Gates

(The Practical Math Behind Digital Circuits)

When we build circuits with logic chips like the 74LS series, it’s easy to think only in 1s and 0s.
But under the hood, every logic gate follows the same rules as any other circuit, Ohm’s Law, current limits, and voltage thresholds.

So yes, there is real-world math behind those blinking LEDs and logic tables. 

1. Logic Levels — The Digital View

At the logic level, everything is simple:

• 0 V → LOW (logic 0)

• 5 V → HIGH (logic 1)

We don’t care about exact voltages here just whether a signal crosses a defined threshold.
For TTL (Transistor-Transistor Logic) chips like the 74LS86 XOR gate, these thresholds are:

This means a logic gate “decides” based on whether the voltage is above or below that boundary, which is how binary logic becomes real voltage levels.

2. Inside the Circuit — Real Electrical Math

Even though logic circuits process bits, they still obey Ohm’s Law and power equations.
That’s where you can start doing real electronic analysis.

a) Output Current (Sourcing & Sinking)

“Sinking” means the output pulls current down to ground (0 V) when it’s LOW.
“Sourcing” means the output pushes current out toward +5 V when it’s HIGH.

TTL gates can sink more current than they can source.

For the 74LS86:

That’s why we usually wire LEDs from +5 V through a resistor to the gate output, when the gate goes LOW, it sinks current and turns the LED on.

Example LED Calculation

If your LED has a forward voltage (V_f) of 2 V and you use a 330 Ω resistor on a 5 V supply:

$$I = \frac{V_{supply} - V_f}{R} = \frac{5V - 2V}{330Ω} ≈ 9 mA$$

That’s ideal for a TTL output sinking current safely.

3. Power Consumption

Each gate inside a chip draws a small current from the 5 V supply.

For a 74LS86, one gate typically uses about 2 mA.

$$P = V × I = 5 V × 2 mA = 10 mW$$

Since there are four gates per chip, that’s roughly 40mW total, plus a little more when switching rapidly.

4. Fan-Out (How Many Inputs You Can Drive)

If one output drives several inputs, you have to stay within its current limits.
Each TTL input draws a little current:

If an output can sink 8 mA, and each input needs 1.6 mA when LOW:

$$N = \frac{I_{OL(max)}}{I_{IL(max)}} = \frac{8 mA}{1.6 mA} = 5$$

So one output can safely drive up to 5 inputs — that’s the fan-out.

5. Quick Reference Equations

6. Why This Matters

Understanding the “math beneath the logic” helps you design reliable circuits.
You’ll know:

• Why LEDs sometimes don’t light when wired the wrong way.

• Why too many gates on one output can cause logic errors.

• How to predict current draw from your 5 V rail.

Digital circuits look abstract but they’re still analogue at heart.
And once you know both sides, you’re not just wiring gates you’re engineering logic.