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Voltage, Current, and Resistance - Introduction to Circuit Analysis

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In today’s world, technology is seamlessly woven into the very fabric of our lives, transforming how we work, communicate, and interact with the world. Millions of electrical circuits are powering everything from our gadgets and appliances to the infrastructure that drives our daily lives.

These intricate networks are responsible for lighting up our homes, charging our devices, driving our vehicles, and so much more, making the knowledge of electric circuits not just an academic endeavor but a practical necessity.

To understand electric circuits, we need to understand the fundamentals of the driving force behind them, aka electricity. This tutorial series kicks off with a basic understanding of electricity and electric circuits and further delves into the laws governing it and some practical examples. Our aim here isn't to memorize the definitions or derive complex mathematical equations related to these terms but rather to develop an intuitive grasp of electric circuits. Before we begin, we would like to thank our friends at Digilent, whose invaluable support and collaboration have made this series possible.

With that stated, let’s get started with understanding some key terminology and concepts related to electric circuits:

Voltage

On a fundamental level, electricity or electrical energy has to do with the flow of electric charge. Electric charge flows through a circuit to enable us to fulfill a desired objective, even something as basic as lighting up an LED. If you want to learn more about the electric charge at the nitty gritty atomic level as well as the simple yet fascinating math behind it, check out our tutorial - Electric Force and Coulomb's Law.

The movement of electric charge is characterized by two essential parameters: voltage, representing the energy associated with the motion of charges, and current, which defines the rate of that movement. Let us understand both these terms with a simple analogy. A common way to explain an electrical circuit is by drawing a comparison with a hydraulic system, like water in a tank.

Imagine a water tank placed at some height above the ground. The water in the tank has some potential energy due to its elevation. This energy associated with the water is analogous to voltage, which is the energy associated per unit charge present at a point.

Figure 1: Water tank at a height
Figure 1: Water tank at a height

Voltage is a measure of the energy available in a circuit to do work, in this case, moving a charge from one point to another. It is defined as energy per unit charge. We can express this in differential form as:

where v is the voltage difference, w is the energy, Q is the charge, and the differentials dw and dQ are taken as the change in energy and charge between two points in space. The unit for voltage is volts (V).

It is important to note that energy measurements are typically defined in relation to a specific reference point. While dealing with electrical circuits, the primary focus is on the difference in voltage between two points rather than absolute voltage. This difference drives the flow of electric charges in a circuit.

When a voltage is expressed as an absolute value instead of a difference, it is generally assumed to be taken with relation to “ground” or a point of zero voltage potential (0 V). Having a standard reference point provides a common frame of reference, helping us analyze and understand electrical potential at different points in a circuit.

Understanding voltage through the water height analogy helps you see it as the electrical potential energy that drives electric charge flow. Higher voltage is akin to greater potential energy, which provides a greater "push" for electric charges through the circuit.

Current

Returning to our previous water tank analogy, let's imagine the water in the tank is held in place by a valve or gate. As we just discussed, water inherently has its own energy (comparable to voltage) even when it is static. However, when we open the valve at the bottom of the tank, allowing the water to flow, it is analogous to the concept of current.

Figure 2: Water tank at a height with flowing water
Figure 2: Water tank at a height with flowing water

More precisely, the volume of water passing through a specific point, like the valve, corresponds to the current. Current, in an electrical circuit, represents the flow rate of electric charge. Mathematically, we can express this in differential form as:

where i is the current, Q is the charge, and t is the time. The unit for current is amperes (A).

Resistance

Along with voltage and current, another factor plays a role in electric circuits - Resistance. Imagine a water pipe with a section that's narrower than the rest. When water flows through this narrow section, it encounters more resistance than the wider pipe parts. In electrical circuits, resistance works similarly—it's like the narrow part of the pipe, hindering the flow of electric current in a circuit and causing it to slow down.

Figure 3: Explaining resistance with water pipes
Figure 3: Explaining resistance with water pipes

Resistance, measured in ohms (Ω), hinders the flow of electric current in a circuit, causing it to slow down. We will delve deeper into this in further tutorials.

SI Units and Scientific Notation

Now that we've discussed some key terms in electrical circuits, let's talk about units. We'll use the International System of Units or SI units in this context. Here’s a summary of the units we’ve discussed till now.

Figure 4: SI Units
Figure 4: SI Units

Interestingly, the conventional symbol/variable for current is 'I,' not 'A' as you might expect. This choice originates from the French phrase "Intensité du courant," which translates to current intensity.

The International System of Units employs a set of decimal-based multipliers known as SI prefixes. These prefixes allow us to express units in various orders of magnitude, making it easier to work with very large or very small measurements. The table below presents a list of common prefixes and their respective symbols.

Prefix

Symbol

Power of 10

none

-

1

kilo

k

103

mega

M

106

milli

m

10-3

micro

µ

10-6

nano

n

10-9

pico

p

10-12

Passive Sign Convention

In electrical circuits, both current and voltage can assume positive or negative values. Voltage, which signifies the relative energy difference between two points, uses its sign or polarity to indicate which point holds a higher energy level. Likewise, current can also have either positive or negative signs, helping us determine the direction of its flow within a circuit. This positive and negative distinction often serves as a way to interpret the behavior of both voltage and current in circuits.

Passive Sign Convention is a set of rules used in circuit analysis to consistently define the direction and polarity of current and voltage in passive components in electrical circuits. Since current and voltage are related to each other, their direction and polarity must remain consistent. Learn more about the relationship between current and voltage in our tutorial - Ohm’s Law Tutorial with Easy Practice Problems.

Now, with this foundational knowledge in place, let's explore a typical electrical circuit. In this context, we will focus on passive circuit elements, which are components capable of storing energy but not generating it.

Figure 5: Typical electrical circuit
Figure 5: Typical electrical circuit

A general two-terminal electrical circuit element is shown in the above figure with current, i, flowing through the element and some voltage difference, v, across its terminals.

The universal sign convention for passive components assumes the sign of the current relative to the voltage. In this convention, a positive current always enters the circuit element at a higher voltage. Therefore, a circuit element following the passive sign convention would look something like this:

Figure 6: Electrical circuit with the passive sign convention
Figure 6: Electrical circuit with the passive sign convention

While a circuit element that does not adhere to the passive sign convention would look something like this:

Figure 7: Electrical circuit not following passive sign convention
Figure 7: Electrical circuit not following passive sign convention

In circuit analysis, you can arbitrarily assume a direction for either voltage or current as a reference. If your assumption is correct, you'll get a positive value. You'll get a negative value if it's opposite to the actual direction.

Double Subscript Notation

So far, we have seen that circuit diagrams often represent voltage polarities using the plus sign to denote positive voltage and the minus sign to denote negative voltage. Another way to represent voltage polarities is using what’s known as the double subscript notation. To understand this better, let's revisit our previous example of a typical two-terminal electrical circuit element, but this time, let's label the terminals 'A' and 'B'.

Figure 8: Two terminal circuit element
Figure 8: Two terminal circuit element

Let’s assume that the voltage at A is higher than that at B and that the current is flowing from A to B using the double subscript notation; we can denote the voltage across the element as VAB.

Figure 9: Double subscript notation
Figure 9: Double subscript notation

In other words, VAB is the voltage at point A with respect to point B, with point A being at a higher potential. However, this holds true only if the sign of the voltage is positive. In our example, if VAB has a negative value, say VAB is equal to -6 Volts.

Figure 10: Negative voltage in double subscript notation
Figure 10: Negative voltage in double subscript notation

This simply means that our initial assumption was incorrect and that B is at a higher potential. So, our circuit element can be accurately represented as VBA with point B at a higher potential.

Figure 11: Corrected double subscript notation
Figure 11: Corrected double subscript notation

You may observe that the potential difference observed from point A to B (or top to bottom in this case) is simply the negative of the potential difference from point B to A. Hence, we can say that

Power and Sign Conventions

If you take a look at gadgets or appliances, you'll often see that they have a certain "wattage" or power rating. But what does this mean? Power, or electric power, is defined as the rate at which work is done or energy is transferred in an electric circuit.

Mathematically, electrical power associated with a certain circuit element is written as the product of the current passing through the element and the voltage difference across the element.

This is written as a function of time because power is calculated at a specific instant. The unit of power is the watt (W).

Power in a circuit can either be absorbed by a circuit element or generated by it. This is determined based on the passive sign convention. Here's how it works:

  • Absorbing Power: If the signs of voltage and current align with the passive sign convention, the element absorbs power. In practical terms, this means that the element is using electrical energy like a resistor heating up when current flows through it.
  • Generating Power: Conversely, if the signs of voltage and current do not align with the passive sign convention, the element is generating power. In this scenario, the element provides electrical energy to the circuit, like a solar panel producing electricity when exposed to light.

Let's look at this with an example. Consider a circuit element with a voltage difference, v, across points A and B and a current, i, passing through it.

Figure 12: VAB following sign convention with current directions reversed
Figure 12: VAB following sign convention with current directions reversed

In the above scenario, the signs for current and voltage align with the passive sign convention, indicating that the circuit element is absorbing power.

Figure 13: VAB not following sign convention with current directions reversed
Figure 13: VAB not following sign convention with current directions reversed

In the above scenario, the signs for current and voltage do not align with the passive sign convention, and as a result, power is being generated. This is because the positive current is not entering from the direction of higher voltage or positive polarity.

Summary

So far, we've covered a lot about electrical circuits. We began by defining voltage and current in terms of electrical charge, familiarising ourselves with common prefixes and scientific notation symbols. We delved into the concept of the passive sign convention, exploring its significance in understanding power within circuit elements.

Additionally, we briefly introduced the essential relationship between current and voltage, recognizing the need for consistent sign conventions. Now, we're ready to explore Ohm's Law, which describes how current and voltage relate to each other, adding another familiar variable into the mix: resistance.

Ohm's Law stands as both the simplest and most fundamental principle in the world of electrical circuits, making it a cornerstone of electrical engineering and circuit analysis. Stay tuned for our next tutorial, where we explore Ohm's Law and show you how to put the concepts you've learned into action, allowing you to analyze and solve real electrical circuits.

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