Explain Kirchhoff’s circuit laws

What are Kirchhoff’s circuit laws

Kirchhoff postulated two basic laws in 1845 which are used for writing network equations. Kirchhoff’s laws are basic analytical tools in order to obtain the solutions of currents and voltages for any electric circuit; whether it is supplied from a direct-current system or an alternating current system.These laws concern the algebraic sum of voltages around a loop and currents entering and leaving a node. The word algebraic is used to indicate that summation is carried out taking into account the polarities of voltages and direction of currents. Kirchhoff gave two laws – Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL). 

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Current Law (KCL) or Kirchhoff’s first law or Kirchhoff’s Junction Law states that at any node (junction) in a circuit the algebraic sum of currents entering and leaving a node at any instant of time must be equal to zero. Currents entering the node (Junction) is taken positive and currents leaving the junction is taken negative.

It can be also stated as the sum of currents entering the junction is equal to the sum of currents leaving the junction. This law is also called Kirchhoff’s First Law. this law is based on conservation of charge.

Consider the figure below currents I₁, I_2, I₃  are entering the junction and currents I₄, I₅ leaving the junction. 

According to Kirchhoff’s Current Law\\\;\\\;\;\;\;\;\;\sum_{n=1}^{p}I_{n}=0\;\;\;\;(1)\\\;\\

\;\;\;\;\;\; I₁ + I₂ + I₃ – I₄ – I₅ = 0 \\\;\\\;\;\;\;\;\; I₁ + I₂ + I₃ = I₄ + I₅ \;\;\;\;(2)

where “p” is number of circuit elements.\\Thus as per equation (2) Kirchhoff’s current Law can also be stated as the sum of currents entering the junction is equal to the sum of current leaving the junction 

Now consider the circuit below and apply Kirchhoff”s current  Law at junction “b”\\\;\\\;\;\;\;

Number of circuit elements meeting at the junction are 4. Thus  p = 4 , Then equation (1) becomes\\\;\\\;\;\;\;\;\;\sum_{n=1}^{4}I_{n}=0\\\;\\I₁ + I₂ – I₃ – I₄ = 0\\\;\\I₁ + I₂ = I₃+ I₄

Kirchhoff’s Voltage Law (KVL)

This law is also called Kirchhoff’s second law or Kirchhoff’s Loop law. It states that algebraic sum of voltages in any loop is zero. This Law is based on conservation of energy.\\\;\\Mathematically it can be written as \\\;\\\;\;\;\;\;\;\sum_{n=1}^{P}V_{n}=0\;\;\;\;\;\;(3)\\\;\\where “P” is number of circuit elements in a loop.

While applying Kirchhoff’s Voltage Law there are various sign conventions to be kept in mind.

1. A rise in voltage should be taken positive.
2. A fall in voltage should be taken negative.
3. A voltage drop (IR) should be taken negative if we move in the direction of current through the resistor.
4. A coltage drop (IR) should be taken Positive if we move opposite to the direction of current through the resistor.

consider the circuit diagram below. \\\;\\\;\;\;\;

Lets apply the KVL to the circuit.
In the circuit diagram there are two loops “Loop I” and Loop “II”.
Applying KVL to Loop I.
Starting from  “f” and moving in the direction of arrow.\\ In the Loop I, There are 4 circuit elements ( Two voltage sources and Two Resistances). Thus P = 4, equation (3) becomes\\\;\\\;\;\;\;\;\;\sum_{n=1}^{4}V_{n}=0\\\;\\V₁ – I₁R₁ – V₂ + I₁R₃ = 0     

Similarly Now applying KVL to Loop II, Starting from “d” and moving in the direction of arrow.\\I₂R₂ – V₂ = 0

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