Analyzing Simple Circuits with Ohm's and Kirchhoff's Laws

Hello! Welcome to the first lesson in our course on electricity. Your goal is to build a practical and theoretical understanding of electricity, from fundamental laws to real-world applications like mains power and generators.

This first module lays the foundation. In this lesson, we will focus on the three most fundamental laws of circuit analysis: Ohm's Law and Kirchhoff's two laws. By the end of our 60 minutes, you'll be able to combine these laws to systematically analyze simple DC circuits and find the voltages and currents within them. These skills are the essential building blocks for understanding all the more complex topics we'll cover later.

Let's begin.

The Basic Quantities: Voltage, Current, and Resistance

Before we dive into the laws, let's quickly re-acquaint ourselves with the three key players in any circuit:

• Voltage (V): Measured in Volts (V). This is the electric potential difference between two points. You can think of it as the "pressure" or "push" that causes charge to move. A battery or a generator provides a voltage.
• Current (I): Measured in Amperes, or Amps (A). This is the rate of flow of electric charge. It's the amount of charge passing a point per unit of time.
• Resistance (R): Measured in Ohms (Ω). This is a measure of the opposition to the flow of current. A light bulb filament, for example, has resistance.

These concepts are often explained using a water analogy: voltage is like the water pressure from a pump, current is the flow rate of the water, and resistance is like a narrow section of pipe that restricts the flow.

Ohm's Law: The Fundamental Relationship

Georg Ohm discovered a simple and crucial relationship between these three quantities for many materials. This is Ohm's Law. It states that the voltage across a resistor is directly proportional to the current flowing through it.

The formula is:
$V=I\times R$

This simple equation is the cornerstone of circuit analysis. It allows you to find any one of the three values if you know the other two.

A critical aspect of applying Ohm's law, and all circuit laws, is getting the signs right. A convention called the passive sign convention is used. It states that for a passive component like a resistor, positive current flows from a higher potential (+) to a lower potential (-). This means that as current flows through a resistor, there is a voltage drop across it.

To see a clear explanation of Ohm's Law and the importance of defining current direction and voltage polarity consistently, please watch the first two minutes of the following video.

Kirchhoff's Laws: Analyzing Complex Circuits

Ohm's law is perfect for a simple circuit with one resistor. But what about more complex circuits with multiple paths and power sources? For these, we need two more tools, developed by Gustav Kirchhoff in the 1840s.

Kirchhoff's laws are not just mathematical tricks; they are direct consequences of fundamental physical principles: the conservation of charge and the conservation of energy. Your background in statistics and physical systems might give you an appreciation for how these fundamental conservation laws provide the constraints we need to solve the system.

• Kirchhoff's Current Law (KCL): Based on the conservation of charge.
• Kirchhoff's Voltage Law (KVL): Based on the conservation of energy.

The following video provides an excellent explanation of these two laws, the physical principles behind them, and the terminology used in circuit analysis.

A Summary of Kirchhoff's Laws

Let's formalize what you just saw:

1. Kirchhoff's Current Law (KCL) - The Junction Rule

KCL states that the algebraic sum of currents entering any node (or junction) is zero. More intuitively:

The total current flowing into a node must equal the total current flowing out of it.
$\sum I_{in}=\sum I_{out}$

Charge can't be created or destroyed, so any charge that flows into a junction must immediately flow out.

2. Kirchhoff's Voltage Law (KVL) - The Loop Rule

KVL states that the algebraic sum of the potential differences (voltages) around any closed loop is zero. More intuitively:

The sum of voltage rises must equal the sum of voltage drops in any closed loop.
$\sum V_{rises}=\sum V_{drops}\text{or}{\sum }_{\text{loop}}V=0$

This comes from the conservation of energy. If you start at any point in a loop and trace a path back to your starting point, your net change in potential must be zero. Any potential gained from batteries must be lost as voltage drops across resistors.

A Systematic Approach to Circuit Analysis

With Ohm's Law, KCL, and KVL, you can solve for the currents and voltages in any linear DC circuit. Given your background in mathematics and econometrics, you'll recognize the process as setting up a system of linear equations to find a set of unknown variables.

Here is a general procedure:

1. Label Everything:

   • Identify and label all the nodes.
   • Assign an unknown current variable to each branch of the circuit (e.g., $I_1,I_2,I_3$). Don't worry about the direction; if your final answer is negative, it just means the current flows opposite to your assumed direction.
   • Using the passive sign convention, mark the voltage drops (+ to -) across each resistor based on your assumed current direction.

2. Formulate Equations:

   • Apply KCL at one or more nodes. You'll need as many independent equations as you have unknowns. A common strategy is to apply KCL at all nodes with 3 or more branches, minus one. For a circuit with two such nodes, KCL at one gives all the information.
   • Apply KVL for the independent loops in the circuit. Traverse each loop (clockwise or counter-clockwise) and sum the voltage rises and drops. Voltage sources are rises if you go from - to +, and drops if you go from + to -. Resistors are always voltage drops in the direction of the current.

3. Solve the System:

   • You now have a system of linear equations. Use substitution, elimination, or matrix methods (like Cramer's rule or matrix inversion, which you've likely encountered in linear algebra and econometrics) to solve for the unknown currents.
   • Once you have the currents, you can use Ohm's Law ($V=IR$) to find the voltage drop across any resistor.

Putting Theory into Practice

Let's watch a final video segment where this method is applied to a more complex circuit. The presenter will set up the necessary KCL and KVL equations but won't solve them. That part will be up to you.

Test your understanding!

In the video you just watched, the following circuit was analyzed:

A two-loop DC circuit with two voltage sources and three resistors.

The analysis resulted in this system of three equations with three unknowns ($I_1$, $I_2$, $I_3$), with resistances in kΩ:

1. $20-1\cdot I_1-3\cdot I_3=0$ (KVL for left loop)
2. $-10-2\cdot I_2-3\cdot I_3=0$ (KVL for right loop)
3. $I_1+I_2=I_3$ (KCL at the top node)

Your task: Solve this system of linear equations to find the values for $I_1$, $I_2$, and $I_3$. Since the resistances are in kΩ and voltages are in Volts, your answers for current will be in milliamperes (mA).

Show answer

This is a classic system of linear equations, solvable by substitution.

First, substitute equation (3) into (1) and (2) to eliminate $I_3$:

Sub into (1):
$20-I_1-3(I_1+I_2)=0$
$20-4I_1-3I_2=0$
$4I_1+3I_2=20\text{(Eq. A)}$

Sub into (2):
$-10-2I_2-3(I_1+I_2)=0$
$-10-3I_1-5I_2=0$
$3I_1+5I_2=-10\text{(Eq. B)}$

Now we have a 2x2 system. Let's solve for $I_1$ in Eq. A:
$I_1=\frac{20-3I_2}{4}$

Substitute this into Eq. B:
$3\left(\frac{20-3I_2}{4}\right)+5I_2=-10$

Multiply by 4 to clear the fraction:
$3(20-3I_2)+20I_2=-40$
$60-9I_2+20I_2=-40$
$11I_2=-100$
$I_2=-\frac{100}{11}\approx -9.09\text{mA}$

Now, back-substitute to find $I_1$:
$I_1=\frac{20-3(-100/11)}{4}=\frac{20+300/11}{4}=\frac{(220+300)/11}{4}=\frac{520}{11\cdot 4}=\frac{130}{11}\approx 11.82\text{mA}$

Finally, find $I_3$ using Eq. (3):
$I_3=I_1+I_2=\frac{130}{11}-\frac{100}{11}=\frac{30}{11}\approx 2.73\text{mA}$

Interpretation:

• $I_1$ and $I_3$ are positive, so their actual direction of flow matches the direction assumed in the video.
• $I_2$ is negative, meaning it actually flows in the opposite direction to the one assumed (i.e., it flows up through the 2kΩ resistor, not down). This is a common and perfectly correct result.

For a more comprehensive text-based walkthrough, including guidelines on when to favor KVL (mesh analysis) versus KCL (nodal analysis), the following document is an excellent resource.

Conclusion

In this lesson, we have established the foundational toolkit for DC circuit analysis.

Key Takeaways:

• Ohm's Law ($V=IR$) defines the relationship between voltage, current, and resistance for a single component.
• Kirchhoff's Current Law (KCL) states that the sum of currents at a node is zero, a consequence of the conservation of charge.
• Kirchhoff's Voltage Law (KVL) states that the sum of voltages in a closed loop is zero, a consequence of the conservation of energy.
• By systematically applying these three laws, we can set up a system of linear equations to solve for all unknown currents and voltages in a complex circuit.

Preview of the Next Lesson:
We've focused on the "how" of charge movement. In the next lesson, we'll address the "what for" by exploring electrical power and energy. We'll see how voltage and current relate to the rate at which energy is delivered and consumed in a circuit, connecting these abstract quantities to the practical concept of work done, which aligns directly with your goals.