Circuit Basics

What and why?

The basic concepts of circuits revolve around voltage, current, and resistance laws. These concepts can be hard to visualize, so we will use the common analogy of a river flowing down a mountain to explain them.

Principles:

• Node: A node is any point in a circuit where 2 or more components come together.

• Ohm's Law: V = IR, where V is voltage, I is current, and R is resistance

• Kirchhoff's current law (KCL): The sum of the currents entering a node must equal the sum of currents exiting a node.

• Kirchhoff's voltage law (KVL): The sum of voltage drops across any closed loop must be zero.

	Series	Parallel
Resistor	$R_{eq} = R_1 + R_2$	$R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$
Voltage	Splits	Stays the same
Current	Stays the same	Splits

Voltage is the measure of a difference in electrical potential energy. It can be compared to the difference in height from the top of the mountain to the bottom. Just like in physics where we learned that an object in a high place has potential energy, voltage measures the difference in potential energy given relative "heights".

In parallel, voltage is the same across each component. If the river split down the mountain, the difference in heights would remain the same, hence the potential differences would remain the same.

Current is the measure of charge moving in a direction, similar to the flowing water, and the amount of water flowing per second.

In parallel, current is split across each component and is not the same. When a river splits down a mountain, the amount of water stays the same, and each stream will split the water.

Resistance creates a voltage drop across the path of a current to impede the flow of current. Resistance can be compared to the width of the river, which affects the flow of the water through it.

Passive Sign Convention

What is the purpose of sign convention? Passive components (resistors, capacitors, inductors) are used in equations such as Ohm's Law to establish relationships between voltage and current. Voltage polarity and current direction have to be consistent with each other. The universal convention is as follows:

The universal convention for voltage is that current flows from the positive terminal to the negative terminal. Consider the example below:

Now this isn't a typical perspective given such a simple circuit, but this labeling can often happen in more complex circuits before we have completed solving for current values. Remembering Ohm's Law, in this example $I_s = \frac{-V_s}{R_s}$. By the passive sign convention, we would still consider the current to be flowing into the positive terminal of the resistor. In this circuit, our labeled current is negative, which implies that the power, $P = I_sV_s = \frac{-V_s}{R_s}V_s = -\frac{V_s^2}{R_s}$, is negative. This agrees with the passive sign convention! The thing to remember is that when labeling currents, passive convention has a canonical direction and if your labeled current is wrong, it will be a negative value.

KVL and KCL

According to Kirchhoff's Voltage Law, the sum of a voltage drop around any closed loop is zero. This law is crucial to helping us solve complex circuits that you will see in this class.

• Energy is always dissipated by resistors

• Voltage decreases in the direction of positive current flow

• When applying KVL, assume voltage decreases across a resistor in the assumed direction of current flow

According to Kirchhoff's Current Law, the current entering a node is equal to the current exiting the node.

Therefore, by KCL, $I = I_1 + I_2$ in the above circuit.