**In this section Kirchhoff’s Law will help us understand how to analyse a DC circuit. DC means Direct Current where the current travels in one direction, from the +ve terminal of the voltage source (battery) to the -ve terminal.**

♦

**Why to analyse a DC circuit?**

**And what does it mean?**

Well, it means we take the components that **we know the value of in a circuit and we calculate what we don’t know. **For example a value of a resistor or the flow of current.

Circuits get biiig. Really big. With DC circuit analysis we could **simplify a large circuit**, so a few components would represent the large circuit. In theory of course, so example we can substitute 100 resistors with 1: which will be the sum of them all. But we will get to this shortly.

**Let’s focus on Mr. Kirchhoff. He has 2 important laws:**

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## 1. Kirchhoff’s Current Law

Let’s take this junction for an example. It is part of a random circuit.

- There are
**4 wires**joined together at a**node**

- Each wire has a
**different current**in it

**I1**and**I2**are flowing__toward__the node- Since current flows from +ve to -ve terminal, the node has a lower potential than whatever we have on the other side of the I1 and I2 wires.

**I3**and**I4**are flowing__away__from the node- for them the node is at hire potential than wherever I3 and I4 currents are flowing to.

**We can express the currents the following way:**

From the equation, it is straightforward why **I3** and **I4** are negative. But think of it this way: They are in the opposite direction as **I1** and **I2**, in such way that they don’t go towards the node, they go away from it.

**The sum of the total current entering a junction is equal to the sum of total current exiting the junction. **

In other words, the sum of all the currents is equal to zero. The sigma sign is a mathematical representation for ‘sum’:

♦

## 2. Kirchhoff’s Voltage Law

**To understand this, let’s start with a small circuit:**

- We have
**1 voltage**source:**E**and two**resistors R1**and**R2.**

- The red arrows symbolise the voltage drop
- The
**arrow**points towards the__higher potential.__

- The

- There is a current
**I**in the circuit.

- The voltage drop across
**R1:****V(R1) = IR1**and across**R2:****V(R2) = IR2**.

- The current is the same across both resistors.

- The voltage drop across
**R1**plus the voltage drop across**R2**is equal to the voltage source E.

**E**is the**EMF**.

**We can express the above as:**

**Kirchhoff’s 2nd law states that the sum of voltages around a closed circuit is equal to zero. **

**So, the sum of all EMFs (multiple batteries) minus the voltage drops across all elements (resistors) is equal to 0:**

**Solved Example:**

**Find the current flowing across R1.**

#### Solution:

**1. Use Kirchhoff’s Current Rule**

- We have
**3 currents**of which we only know the value of**I**. Node**X**joins them together.

**I1**and the current source**I**flow__towards__**node X**while**I2**flows__away__.

- from this we can write the following equations. Note,
**I1**and**I2**are negative as they flow towards the node.

** **

**2. Substitute equation 2 and 3 to equation 2**

- Our aim is to find the
**voltage at node X.**

- By knowing
**Vx**we will know the voltage drop across**R1.**The value of the resistor**R1**is given, thus, by using Ohm’s law we can calculate**I1**.

** **

**3. Substituting to the equation:**

** **

**4. Use equation 2 to find I1 and equation 3 to find I2**

**Solved Example 2:**

**Find the current flowing in the circuit as well as the individual voltage drops across each resistor.**

**Solution:**

**1. Use Kirchhoff’s Voltage Rule**

- There are
**4 resistors**in the circuit and**2 voltage sources**

- All resistors are
**in series**. thus, the**same current**flows through them

- By Kirchhoff’s Voltage Law the sum of the voltage drops across each resistor is equal to the sum of all the voltage sources.

**VR1**below simple means the voltage drop across**R1**. Voltage by Ohm’s Law is the resistance x current:**V = RI**.

- By substitution we can easily find the value of
**I**.

**2. Use Ohm’s Law to find the voltage across each resistor**

** **

**optional reading:** Success in Electronics book by Tom Duncan

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