# Kirchoff's Voltage Law

Kirchoff's Voltage Law (KVL) is a circuit law related to voltages in a closed loop.

It states that the the sum of voltages traversed in a closed loop is equal to zero. This is also known as the conservation of energy. This is mathematically described as:

Expressed another way, the total voltage gains are equal to the total voltage drops in a closed loop:

Voltage polarities must be assigned to each circuit element in accordance with the direction of the electric current, which can be made arbitrarily, and will be taken into account when using KVL. A negative voltage result means that the assigned voltage is of the opposite polarity to the actual voltage and that the current travels in the other direction.

We can use the circuit below to illustrate how KVL is used:

To apply KVL we draw closed loops around the circuit and assign polarities to each circuit element through the loop in the direction of the current. We have chosen a clockwise KVL loop around the circuit where the voltage source provides a voltage gain (negative to positive) and each resistor drops a voltage as indicated by the positive and negative signs.

We can derive the following equation using KVL going clockwise around the loop from the voltage source:

We can then replace each unknown voltage drop in terms of the source current and resistance via Ohm's Law:

Which we can then rearrange and solve for the source current Is:

Which is the result we would expect for the above circuit by Ohm's Law as the total effective resistance is the sum of the resistances for a series circuit.

## Gustav Kirchhoff

Gustav Robert Kirchhoff was born on March 12 1824 in Königsberg which was in Prussia back then, now known as Kalingrad, Russia. He was a German physicist who not only contributed to the fundamental laws of electronics but also spectroscopy and he also coined the term black body radiation in 1862.

Kirchhoff formulated his circuit laws which are ubiquitous in electronics, in 1845, while still a student. I want to emphasize this extremely important detail, age is NOT a barrier to achieve greatness and discover amazing things that can and will CHANGE THE WORLD. Einstein was only 27 when he published Zur Elektrodynamik bewegter Körper on Special Relativity in 1905 so let's instead be inspired by these incredible humans and continue pushing boundaries AT ALL AGES.

### Example:

1. Use KVL to determine the current supplied by the voltage sources in the circuit below:

The first step is to draw a KVL loop around the circuit and assign polarities in accordance with the direction of the current: We can then traverse the loop clockwise from the negative terminal of the voltage source and apply KVL:

### Questions:

1. a) Determine the current It in the circuit below using KVL

2.262mA

b) What is the voltage across R1?

1.063V

c) What is the voltage across R2?

2.262V

d) What is the voltage across R3?

4.976V

2. a) Determine the total current flowing through the circuit below using KVL

1.176mA

b) What is the voltage across R3?

1.176V

c) What is the voltage across R1?

2.824V

d) What is the current flowing through R1?

705.882uA

e) What is the current flowing through R2?

470.588uA

3. a) Determine the voltage provided by the current source in the circuit below using KVL

5V

b) What is the voltage across R1?

0.5V

c) What is the voltage across R2?

2.5V

d) What is the voltage across R4?

2V

e) What is the current flowing through R2?