### Unit 3: Electronic Circuit Theory

Although the study of electric and magnetic fields is interesting in and of itself, it may not seem directly useful in the real world. However, the interplay between these phenomena is responsible for much of the technology you see in your everyday life. For example, all electronics apply various features of electromagnetism, so that computers, HDTV, iMacs and iPads, smartphones, motors, fans, lights, and so on are applied electromagnetic devices. In this unit, we will take a quick look at the foundations of electronics while at the same time adding to our understanding of electromagnetism.

**Completing this unit should take you approximately 8 hours.**

Upon successful completion of this unit, you will be able to:

- state Ohm's Law in words;
- apply Ohm's Law to simple circuits;
- calculate effective resistance of a network of resistors;
- determine the resistance of a cylindrical wire;
- compare and contrast voltage and current;
- use the junction and loop rules to analyze basic circuits;
- explain how a battery works;
- calculate the power in a DC circuit; and
- calculate the power in an AC circuit.

### 3.1: Electric Current, Voltage and Resistance

An

**electric current**refers to the flow of electric charge through a circuit. We measure an electric current as the net rate of flow of electric charge across an imaginary fixed surface that cuts through the region where the charge is moving. The charge can be negatively charged electrons or positive charge carriers, such as protons or positive ions.**Voltage**, as defined earlier, refers to the electric potential difference between two points, and we will see that in the context of electric circuits, and can be thought of as a kind of pressure that causes the current to flow.As you read this text, pay attention to the formula for the current, which will also define its standard unit, the ampere. One ampere corresponds to one Coulomb per second, crossing a given surface.

If you look at our imaginary reference surface sideways, it is clear that it has two sides that we could call "

**left**" and "**right**". This means the current also has a**direction**because a charge can cross the surface from right to left or left to right. Just as with velocity for larger objects, we can introduce a "**coordinate**" choice so that one of these directions is labeled as the positive direction and the other as negative.A convenient choice is to imagine that positive charges are crossing the surface and choose their direction as positive. With that choice, we also call the

**current**positive because a positive number of Coulombs is moving in with a velocity that we labeled as positive. This is also called the conventional current direction.For example, let's say positive charges are crossing from left to right, then we say the conventional current direction is from left to right. Because charge is conserved, the total amount of charge on the right of the surface must increase as we are adding new positive charge through the current.

Now, imagine instead that

**negatively-**charged electrons flow through the same surface, but from right to left (opposite to what we assumed at the outset). What does that do to the total charge on the right? If it loses negative charge through the current flow, then the right-hand side will end up with more and more**positive charge**that is no longer balanced by negative electrons! So the effect of negative electrons flowing from right to left is the same as the effect of positive charges flowing from left to right: the total charge on the right increases (and by the same argument, the total charge on the left decreases).This means that we will always be allowed to think of a current as being made up either of positive charge carriers flowing in one direction or equivalently consisting of negative charge carriers flowing in the opposite direction. This is the explanation for figure 24(b) in the text, where the arrows indicating the velocities of the electrons point in the opposite direction to the arrow indicating the conventional current direction.

This is important to understand because it is actually the negative electrons that transport charge inside a metallic conductor, whereas the positive atomic nuclei stay fixed in the crystalline structure of the metal.

The next video will review current and introduce the concepts of resistance and

**drift velocity**.Make sure you understand the idea behind drift velocity before reading about Ohm's Law. Drift velocity is an average of the velocities that the electrons in a conductor move at a microscopic level. We observe drift velocity in conductors that prevent the flowing electrons from accelerating freely due to an effect we call electrical

**resistance**.You can think of the drift velocity as analogous to the terminal velocity of a falling skydiver: when

**terminal velocity**is reached, the downward pull of gravity is balanced by the upward push of air resistance, so the skydiver no longer accelerates as you would expect for a falling object without air resistance.As the fall continues at constant speed, the gravitational potential energy keeps decreasing, but it is no longer converted to mechanical kinetic energy. Instead, the air resistance drains away the potential energy in the form of heat.

In this analogy, you can compare the average behavior of the electrons in a conductor to that of the skydiver: the pull of the electric force tries to accelerate the electrons toward the positive end of the conductor, just like gravity tries to pull the skydiver down. However, the electrons in the conductor experience collisions that have the same effect as the force of air resistance on the skydiver. When these two opposing effects are exactly balanced, the electrons end up moving at the drift velocity calculated at the end of section 20.1 in the text.

We learn in this video that the important thing to remember is that the drift velocity stays

**constant**during the entire trip of the electrons through the wire – they do not accelerate, even though they are continuously exposed to the electric field inside the conductor. The flowing electrons lose electrical potential energy, too. This energy is converted to heat and sometimes even to light. Incandescent light bulbs are essentially resistors that use this effect.- Watch this video.

### 3.2: Ohm's Law

**Ohm's Law**states that the current through a conductor between two points is directly proportional to the voltage across the two points. According to Ohm's Law, the voltage drop, , across a resistor when a current flows through it is calculated using the equation , where equals the current in amps (A) and is the resistance in ohms (Ω). Another way to think of this is that is the voltage necessary to make a current flow through a resistance .Read this text, which introduces Ohm's Law and simple circuits.

While the lecture you watched earlier has already covered this material, watch this brief video to review how we define the

**unit ohm**.

### 3.3: Resistance and Resistivity

Ohm's Law is an empirical relationship that applies to many different types of conductors, but not to all (e.g., incandescent light bulbs show a more complicated behavior while they heat up or cool down). This law can also be explained in more detail by making use of the microscopic picture that we introduced in the context of the drift velocity: The current through a wire is proportional to the drift velocity, so that any effect that manages to increase the drift velocity will also increase the current. One way to increase the drift velocity is to reduce the rate at which electrons undergo collisions in the material.

Read this section which defines the concepts that quantify the factors that control the current. The

**electrical resistance**of an object measures its opposition to the flow of electric current.**Electrical conductance**is the reciprocal quantity – it describes how easy it is for an electric current to pass. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The**SI unit**of electrical resistance is the**ohm (Ω)**, while electrical conductance is measured in**siemens (S)**(formerly called "mho"s and represented by**℧**).Get a feeling for the interplay of resistivity and geometry in determining the resistance of a wire by manipulating the interactive simulation at the bottom of the page linked above.

Review the lecture on currents to see how resistance and resistivity are related. Then watch this video, which explains the difference between

**resistance**, which is the ratio of voltage to current,**resistor**, a device that creates resistance, and**resistivity**, which is a material-dependent property of the device.

### 3.4: Power in Electric Circuits

When a current flows through a resistor, electrical potential energy of the moving charges is converted into thermal energy (as in a toaster oven) or light (as in a light bulb). This is the reason why the drift velocity stays constant in the resistor even though the potential energy of the charges does not. If you look at the label on a light bulb or most electrical appliances, you will in fact see a number that quantifies how much electrical energy they convert to other forms of energy (such as heat) every second, under normal operating conditions. That is the power rating, measured in watts.

As the following section in the text explains, watt is the standard unit of power – and power is the rate at which energy is transferred.

The text also combines the definition of electrical power with Ohm's Law to get some alternative, but equivalent ways of relating power to the other quantities we have already encountered. Watch this video, which discusses these relationships further.

### 3.5: Alternating Current vs. Direct Current

**Direct current (DC)**is the flow of electric charge in only one direction. It is the steady state of a constant-voltage circuit. Most well-known applications, however, use a time-varying voltage source.**Alternating current (AC)**is the flow of electric charge that periodically reverses direction. If the source varies periodically, particularly sinusoidally, the circuit is known as an alternating current circuit. Examples include the commercial and residential power that serves so many of our needs.Watch this video, which leads us from the discussion of electrical power to the definition and properties of AC circuits.

Read this text, which explains that, because the voltage in an AC circuit changes all the time, there is some ambiguity as to what "the" voltage of an AC outlet in your home actually means. Two different ways to characterize the AC voltage are mentioned in the video: the peak voltage and the rms voltage – an averaged value that accounts for the fact that the voltage in an AC circuit spends most of the time at levels that are smaller than the peak value.

As we learned in the text, electricity is delivered to U.S. households at a frequency of 60 hertz. In Europe, that frequency is 50 hertz. Watch this video if you are not sure how frequency is related to the concept of alternating current.

Different countries also employ different voltages in the electricity being delivered to homes. This is described in the video below.

### 3.6: Electric Hazards and the Human Body

The reason electric shock is so dangerous to humans is due to a trait we share with all other animals which plants do not possess: our bodies use electricity to transmit information along our nerve cells. Electricity allows us to transmit messages over long distances inside our body, much faster than would be possible if we relied on the diffusion of chemical molecules, such as hormones (a method we also use). It becomes hazardous when an electric shock interferes with these nerve signals.

Most of the electrical appliances you use in your home draw their energy from AC outlets. In all likelihood, you have also encountered the problem that too many devices were plugged in at the same time and tripped the circuit breaker. This may seem like a nuisance, but it is designed to save lives. To appreciate this, it is important to have a basic understanding of electrical hazards, as discussed below.

Watch this video for a brief overview of why household wiring includes a third pole, called a ground.

Read this text to further explore this aspect of electricity in the field of biology.

### 3.7: Resistors in Circuits

Most circuits have more than one component (a

**resistor**) that limits the flow of charge in a circuit. A measure of this limit on charge flow is called**resistance**. The total resistance of a combination of resistors depends on both their individual values and how they are connected.Before we look at different ways to combine resistors, watch this video, which summarizes the graphical symbols used to draw circuit diagrams. We represent capacitors, batteries, and resistors with abstract symbols that are meant to remind you of their basic design.

We have already covered combinations of capacitors in series and parallel. We can make the same two basic types of arrangement with resistors. Just as with capacitors, we can simplify the analysis of a

**resistor circuit**step-by-step, by imagining replacing a combination of resistors by a single ("**effective**") resistor. We can do this because the internal details of a resistor are irrelevant to its function in a circuit. All we really care about is the value of the**resistance**that needs to be plugged into Ohm's Law, as applied to the current and voltage found at the two terminals leading into and out of the resistor.You could say the idea behind analyzing circuits is to get as far as possible by treating parts of the circuit as a

**black box**. And we can indeed treat complicated arrangements of resistors as a black box, so long as there are just two terminals connecting the black box to the outside (because resistors must have two terminals).It does not matter whether several individual resistors exist between the two terminals in question, or just a single wire, as long as we know the resistance of our black box. Read this text, which implements this strategy, with the aid of two basic rules: one for series and one for parallel resistors.

Watch this video for a brief review of series resistors connected to a battery.

Next, watch this video to see how to deal with resistors connected to a battery in parallel.

To really appreciate the power of the "black-box" thinking that underlies circuit analysis, we have to go one step further by combining three resistors in series

**and**in parallel at the same time. Jennifer Cash discusses this in her next video.Our textbook also presented an example with three resistors (example 21.3), which is part of this video.

Read this text, which provides an in-depth discussion of batteries and how they act in electric circuits.

Note that you may be confused by the term "electromotive force" (emf) for the voltage the battery creates – and you are right to be confused. It is not a force, but a term we have used historically for so long that it is no longer practical to modernize all the textbooks by inventing a different name. Just think of emf as "internally generated voltage". In a battery, it is caused by the chemical process that separates positive and negative charges to keep individual electrons on the two battery terminals at a constant potential-energy difference.

The reading above addresses issues that arise when you are trying to characterize the performance of a battery, or increase the voltage or current the battery can deliver. This involves stacking batteries to form a series configuration, or putting them in parallel. Put batteries in series to increase the battery voltage. Increasing the current involves overcoming the internal resistance which is ultimately a by-product of the battery's chemistry. Use a parallel configuration to get more current without degrading the battery voltage.

To illustrate the

**internal resistance**of a battery, we can look for the heating that should occur when a large current flows through any resistor. In this video, a large car battery is made to produce a large current by creating a short circuit (that is a circuit between its terminals containing nothing but a good conductor). Indeed, the battery not only produces heat, it sparks!

### 3.8: Kirchhoff's Rules

The "

**black-box**" approach of the previous section has its limitations: you cannot reduce any**arbitrary**combination of batteries and resistors to a single battery and a single resistor. If you could, circuit analysis would not only be boring, but it would not be able to produce all of the functionality that electronics are able to deliver in everyday life.We need additional tools to analyze even more complicated circuits. Fortunately, the beauty of physics is that it manages to reduce the complexities of everyday life to a relatively small number of fundamental principles. In this section, we deal with two additional principles that apply to all circuits (except when they operate at the kind of frequencies that generate radio waves – more on that later).

**Kirchhoff's rules for circuit analysis**are basically applications of the conservation laws to circuits. The first rule applies the conservation of**charge**, while the second rule applies the conservation of**energy**. Conservation laws, even used in a specific application, such as circuit analysis, are so basic as to form the foundation of that application.**Kirchhoff's first rule**, also called**the junction rule**, states that the sum of all currents entering a junction must equal the sum of all currents leaving the junction.**Kirchhoff's second rule**, also called**the loop rule**, states that the algebraic sum of the changes in potential around any closed circuit path (loop) must be zero.Watch this video, which explores how to apply Kirchhoff's rules to generate equations to find the unknowns in circuits. These unknowns may be currents, voltages (called

**emfs**when relating to batteries), or resistances.Read the accompanying textbook so you can carefully go through an example (21.5).

The main purpose of Kirchhoff's Laws is to help

**predict**the values for quantities, such as current and voltage, in a circuit. We would not be able to design electronic circuits, if we could not make these predictions! Anyone working in electronics needs to have a firm grasp of these laws. Watch these videos to get additional practice.Note that when people talk about junctions in circuits, they typically refer to the points where three or more wires come together. But in principle, even the connection between only two wires is a simple junction. It is just that we do not need any additional laws to figure out what the current and voltage on both sides of a two-wire junction must be: they are simply equal. You only face the difficulty of not being able to directly see what those quantities must be – without having to look to see how the junction is connected to the rest of the circuit – when you have junctions with more than two wires.

In Kirchhoff's second (loop) rule, we deal with voltages. Watch this video for a reminder that voltages are

**differences**in electric potential.When applying Kirchhoff's second rule, it is important to know when to count a voltage as positive, and when to count it as negative. This depends on the coordinate direction that you have chosen in each branch of the circuit, which also fixes what you call the positive current direction. Watch this video, which shows the specific case of the resistor one more time.

Watch this video for a discussion on how to treat batteries according to Kirchhoff's second rule. The direction in which the electric potential changes is opposite to how it behaves in a resistor. This is why people use a different name for the voltage across a battery: electromotive force. It is not a force, but it is "

**reminiscent**" of a force in that the battery does work on the charges that flow through it, elevating them to a higher potential.As we mentioned earlier, we need Kirchhoff's rules for circuits that are more complex than simple combinations of series and parallel resistors. It can be instructive to connect these new rules to what you have already learned about series circuits. Watch this video to see how to apply the loop rule to a circuit that you could understand without using this rule.

A potential point of confusion exists when you try to apply Kirchhoff's first rule for the current: if you do not know all the currents in a circuit, how do you choose the direction of the currents, as Kirchhoff's first rule dictates?

The short answer is: do not worry about it. It will work out fine

**no matter**which direction you draw the arrows labeled by the currents , , etc. in a circuit diagram like figure 21.25.There is a long answer for those who want to understand

**why**it works. It is okay to skip ahead to the next section if you are not interested in the details!The confusion about current directions in Kirchhoff's first law is removed if you think about the blue arrows in figure 21.25 as coordinate axes, not as currents. Remember that current through any imaginary surface in the circuit has a direction. This is still true here. And just as for velocities, we can specify the direction of a current by using plus or minus signs relative to a coordinate axis that we can choose arbitrarily. We make this choice of

**coordinate axis**once and for all in each segment of the circuit that goes from one junction to another. That is what the blue arrows in figure 21.25**really**mean. If a current of positive charge in a given segment is moving in the direction of the coordinate arrow for that segment, then that current counts as positive, and otherwise it counts as negative.To be specific, let's look at the junction labeled "a" in figure 21.25. Kirchhoff's rule now tells us to sum up all the currents in the wires whose arrows point into the junction, and set that total equal to the sum of all the currents in the wires whose arrows point out of the junction. At junction "a", there is only one arrow pointing toward it, and that is labeled by . The other two arrows are pointing away from junction "a", and we decide to label them by the variables I2 and I3. Then their corresponding currents satisfy the equation .

The variable is the current in the horizontal wire, counted as positive if the flow of positive charge is in the same direction as the arrow we drew there – pointing toward "a". Now we could have equally well chosen to draw all arrows so that they point away from junction "a" – but then the same current that was previously described by the value (positive charges flowing toward "a") would have to have its sign reversed and be called . This sign change is caused only by the change in the reference direction represented by the coordinate arrow, but the actual current it describes is the same as before.

In Kirchhoff's first law, we now have only arrows pointing out of the junction, so we must total up all the currents on one side of the equation, and set the result equal to zero – because none of the currents belong to arrows pointing into the junction. The total corresponding to the outgoing arrows again contains as before, but it also includes the value labeling the arrow we just reversed. We include the minus sign so that we are still describing the same physical current as before the arrow was reversed. This makes the equation for Kirchhoff's Law . But this equation is mathematically equivalent to the equation we got with the original choice of arrow direction, as you can see by bringing to the other side.

This is why the set of simultaneous equations that you obtain by applying Kirchhoff's first law is not substantially affected by the choice of coordinate arrow directions in the individual branches of the circuit. You can make that choice completely at random, provided you remember that some of the currents that are found as solutions to the equations can turn out negative, which means they describe flow of positive charge opposite to the chosen arrow direction.

### 3.9: DC Voltmeters and Ammeters

Now that we have spent some time performing the theoretical calculations for designing an electronic circuit, we should go back and ask how to

**measure**the currents and voltage we just calculated. Most discoveries in electromagnetism would have been impossible without measurement devices. For example, although Coulomb may have figured out the forces between charged objects, how can you determine whether those charges are sitting still or in motion?In a circuit, voltages can exist with or without any moving charges. For example, a battery with open terminals shows a voltage even if no current is able to flow. On the other hand, electrical current is, by definition, the directed flow of charge.

In principle, it is possible to detect a voltage in the same way you detect the presence of charge: by measuring repulsive or attractive electrostatic forces between two movable objects, connected to the points whose potential difference you are trying to find. But that is difficult to do in practice, because static charges are easily lost by unintended routes, for example in humid air. Instead, the measurement of voltage and current often relies on a different effect that can only be produced by moving charges:

**magnetism**.**Voltmeters**measure voltage;**ammeters**measure current.**Voltmeters**are connected in parallel with the device whose voltage is being measured. A**parallel connection**is used because objects in parallel experience the same potential difference.**Ammeters**are connected in series with the current of the device that is being measured. A**series connection**is used because objects in series have the same current passing through them.We will discuss the magnetism on which both of these devices rely in more detail in later parts of this course. Read this text, which introduces voltmeters and ammeters from a practical perspective first, without going into the physics of magnetism. For now, you should just be aware that the force which deflects the needle in a galvanometer is in fact

**magnetic**, and this force obeys a**different law**than the one discovered by Coulomb for electrostatic forces.Watch this video to learn about the differences in how to hook up voltmeters versus ammeters.

The problem with the kind of voltage measurements we have described so far is that they are really based on magnetism. It turns out that we can only generate magnetic forces if there is at least some current flow. Consequently, for a voltmeter to function, it must allow some current to flow through it instead of flowing through the intended path in the circuit you are measuring. To minimize this unintentional rerouting of current, one tries to make the internal resistance of a voltmeter as large as possible.

We can use a

**galvanometer**(which is always based on magnetism and hence current flow) to push this idea to the extreme with a special configuration that involves a**variable**resistor, called a**potentiometer**. The basic insight behind this setup is that there is one particular scenario in which a voltmeter gives a perfectly accurate reading without drawing any current whatsoever: that is when the reading is precisely zero volts. A vanishing voltage means vanishing current by Ohm's Law. Read this text to learn how we can use a technique called a**null measurement**to measure an unknown emf (voltage).The role of the adjustable resistor is crucial because it needs to be "

**tuned**" just right to make the galvanometer show a null reading (zero voltage). Watch this video, which recaps this information. The discussion also includes the**Wheatstone bridge**, a setup that is used to measure resistances instead of voltages, using the same idea of making a null measurement.A detailed discussion of how to make precise measurements is not only important for practical applications. It is also a good way to get acquainted with the tremendous predictive power that our physical laws provide. This understanding helps us design devices to reveal what is going on inside a circuit even though it is all happening at a microscopic level, invisible to the naked eye. Measurement devices are the bridge between this micro-world and the macroscopic world of pointer arrows and displays that we can perceive and interpret.

The range of validity of Kirchhoff's rules is broad. Simply put, Kirchhoff's rules are valid whenever the conservation laws on which they are based are valid. The following three energy forms are accounted for in Kirchhoff's rules: electrical potential energy, the energy generating an electromotive force, and the energy lost to heat in a resistor. An energy form that is not included in Kirchoff's rules is electromagnetic radiation.

With this in mind, we can apply Kirchhoff's Laws to many other types of circuits, containing not just batteries and resistors. In this we add capacitors to the mix. As you recall from our discussion of capacitance, a capacitor is able to store electric charge in an amount proportional to the voltage between its plates.

Read this text to explore a topic we have not yet discussed: the details of the

**charging**and**discharging**process. Since this process involves current flow, we have to look at capacitors in a complete circuit.Watch this video to learn about a new mathematical concept that appears in this context: the

**exponential function**. It is a function that appears frequently in physics when growth or decay processes are described. Here, the growth and decay refers to the amount of charge stored on the capacitor. The time it takes to charge and discharge a capacitor is characterized by a time that is directly proportional to the capacitance.