# Difference between revisions of "Circuit Lab"

Circuit Lab is a Division C and Division B event about the study of circuitry, electricity, and magnetism. It was previously an event in 2013, 2014, 2019, and 2020, when it was called Shock Value in Division B. Circuit Lab is a laboratory event which deals with the various components and properties of direct current (DC) circuits. Historically, the fields which have been tested in this event are DC circuit concepts and DC circuit analysis (both theory and practice).

The event consists of two parts: the written test and the hands-on component. The written test consists of questions spanning a list of topics and can include a variety of question types. The hands-on component can test a variety of circuit elements in order to complete a task (such as constructing a magnet or determining the value of a resistor). Scoring consists of summing the points earned on the test and the hands-on component. The written test will account for 50-75% of the total score, with the hands-on task consisting of the remaining 25-50%. Ties are broken based on questions on the written test.

The event rotates with other physics events, including Optics and It's About Time.

## What is a Circuit?

Let's take an example of a battery, for now. The battery has a positive (+) end, and a minus ( - ) end. When you touch a wire onto both ends of the battery at the same time, you have created a circuit. (It is generally ill advised to attempt this experiment. Not only will there be nothing to see, but short-circuiting a battery is potentially dangerous). What just happened? Current flowed from one end of the battery to the other through the wire. Therefore, the definition of a circuit can simply be a never-ending looped pathway for electrons (the battery counts as a pathway!).

The Requirement of a Closed Conducting Path

There are two requirements which must be met to establish an electric circuit. The first is clearly demonstrated by the above activity. There must be a closed conducting path which extends from the positive terminal to the negative terminal. It is not enough that there is a closed connecting loop; the loop itself must extend from the positive terminal to the negative terminal of the electrochemical cell. An electric circuit is like a water circuit at a water park. The flow of charge through the wires is similar to the flow of water through the pipes and along the slides of the water park. If a pipe gets plugged or broken such that water cannot make a complete path through the circuit, then the flow of water will soon cease. In an electric circuit, all connections must be made and made by conducting materials capable of carrying a charge. Metallic materials are conductors and can be inserted into the circuit to successfully light the bulb. There must be a closed conducting loop from the positive to the negative terminal in order to establish a circuit and to have a current.

## Basic Electrical DC Circuit Theory

### Current

Current is the rate of flow of charge past a particular point or region. In the context of electric circuits, the charge is usually carried by electrons, so we consider current to be the rate of flow of electrons past a certain point in the circuit. In most conductors, the atoms are bonded in such a way so that the electrons can move around the material without being localized to a particular atom. This allows the electrons to "flow."

As it turns out, the movement of individual electrons within a conductor (known as the electron's drift velocity) is relatively slow (around several micrometers/second for 1 A in a 2mm diameter copper wire.) However, electricity (or electric current) moves at the speed of light. Imagine a long, thin tube (representing the wire) filled with a single-file row of ball bearings (representing the electrons) to rationalize this. Pushing a ball bearing into the tube on one end causes another bearing to fall out the other end almost immediately. Thus the "effect" of the current transfers almost immediately through the tube, but each bearing travels much slower.

This atomic view of current also helps explain the need for a closed-loop to create a circuit. This means that you cannot simply attach a wire to one end of a battery and expect electrons to flow through it. There must exist a closed loop. It is not possible on the atomic scale (within reasonable conditions) for electrons to flow off a wire into free space. Furthermore, electrons cannot accumulate within the circuit because a massive buildup of negative charge would quickly cause any current flow to cease. Therefore, for every electron put into a circuit by a particular device (like a battery), it must accept an electron if any flow is achieved.

### Current Flow and Direction

When Benjamin Franklin was first investigating the properties of electricity, he observed positive and negative charges at play in this phenomenon and that there was some flow occurring. However, he had no way of knowing if the positive charges or the negative charges were flowing. He arbitrarily selected positive charges as the mobile charge carrier. With more advanced technology, we now know that the negatively charged particles (i.e., electrons) that flow in a circuit are negatively charged particles. The problem is that by that time, positive charge flow had already become the convention. This became known as conventional flow notation while the opposite direction became known as electron flow notation. This distinction does not fundamentally change any circuits or much of the underlying circuit theory. The only thing that changes is the direction the arrow is drawn when we draw the current flow. In other words, are the charge carriers flowing out of the positive end of a battery or the negative end?

Just as where in mathematics subtracting a negative is equivalent to adding a positive, a flow of positive charges in one direction is the same current as a flow of negative charges in the opposite direction. As such, in most applications, the choice of current direction is an arbitrary convention.

In nearly every context you will encounter (including this wiki page), circuits will be discussed in conventional flow notation. To convert to electron flow notation, you can reverse the current directions in all branches.

### Voltage

Electrons do not simply move on their own. They move only in response to a force. This notion of the ability to move a charged particle describes voltage. If an electric field exists in space (from any source), a point charge in this space will feel a force from this field. The strength of this force will depend on the strength of the field at the charge's location and the magnitude of the charge. Voltage describes the force caused by this field for a unit charge. In other words, it describes the potential of the electric field to move a unit charge at a particular point in space. In this sense, voltage describes the "push" a circuit gives to the electrons. An equivalent analogy is a hilly piece of land. The gravitational potential would be the amount of potential energy a ball has at various elevations per kilogram. Similarly, the electric potential (voltage) describes the amount of potential energy a charged particle has per unit charge.

The important take-aways from this analogy are that voltage, in general, describes an electric field's ability to move a particle and that voltage as a single quantity is irrelevant. It must be given to a reference point. For example, a table on the second floor of a building could be four feet high relative to the floor but could be 20 feet high relative to the ground floor. Similarly, the voltage at a particular point in the circuit must be given with a particular reference point. Normally, we choose one point in the circuit as the ground from which we reference all other voltages. However, we will sometimes refer to the voltage across a component. This means we choose one side of the component as a reference and see the potential on the other side of the component.

How do voltages fit into the context of a circuit? A battery is a common type of voltage source. This means that some mechanism inside the battery pushes negatively charged electrons out of the negative terminal and into the wire. This "push" provided by the battery is the voltage. In the gravity analogy, the battery's negative side is the "uphill" side, and the positive terminal is the "downhill" side for the electrons. These electrons then bump electrons in the atoms of the wire repeatedly until finally, electrons arrive back at the positive end of the battery. Batteries accomplish this with a chemical reaction, but there are several ways to generate a voltage. In conventional current form, the battery pushes positive charges out the positive side of the battery, through the circuit, and back into the negative side. This is the way circuits will generally be notated, so you will see the current flowing out of the battery's positive side. We will say that the positive terminal is at a higher voltage or potential than the negative terminal.

## Voltage, Resistance, and Amperes

For more in-depth information, see Circuit Lab/Episodes, written by a user for the old wiki (pre-2009 season)

### Amperes

To understand Amps, the coach of a baseball team can be used as an analogy. A coach wants to make his/her team the best that it can be. There are two ways they can do this: making the team score as much as possible and making the opposing team score as little as possible. Focusing on both would be impossible so naturally, the coach is going to have to choose one area to focus on. Say the coach wants to score more runs; this circumstance can be related to the concept of "amperes." The number of runs the team makes is the score - the more they get, the better their chance of winning. Similarly, amperes measure the amount of current flowing per second through an area.

On the other hand, if the coach wants to win the game, he/she doesn't necessarily have to have the team score a whole lot of runs, the team just needs to score more than the opponent. So, maybe the coach's resistance to their scoring of runs will be high, which means that the number of runs needed to achieve the same goal is less.Resistance to current flowing is also one of the important terms.

Now, how do these concepts of amperes and resistance relate? If resistance is multiplied by the amperes, the product would be the voltage of a circuit. This relationship was discovered by Georg Simon Ohm, and it says, simply, that:

$\displaystyle{ V = I \times R }$

Or

Voltage = Current times Resistance

• Sometimes E is used in place of V, for electromotive force (EMF)

### Voltage

Imagine a battery as a super-soaker, and the water that comes out of it as voltage. The harder someone pumps that super-soaker, the harder the stream is going to be when it comes out of the gun. Voltage is the potential for that water exit the gun quickly: the more the gun is pumped, more "voltage" is added in, the faster that water will go.

But sometimes, there will be a "multi-functioning" nozzle which even allows for adjustments of the water speed even further. For a "wider" and "bigger" stream of water, the nozzle may have to be changed to one with a bigger opening. A nozzle with a bigger opening increases the amount of space that the water is allowed to go through. The water still has a high "voltage", or potential for speed and force but the overall pressure of the water is decreased due to the stream's increased spread. The bigger the nozzle gets (think of it like the resistance), the smaller the hitting power (current, which is a speed in electricity too) is.

Voltage is technically electrical potential. While in many cases it is treated as an absolute, it is important to remember that in circuits mostly the difference in voltage is discussed, a potential difference, and that things like Ohm's laws only apply to potential differences, not just electrical potential. However, in the context of circuits, Voltage is often used in reference to potential difference.

### Resistance (Ω)

A resistor is just a piece of metal, and the piece in the center is what provides the resistance. A resistor limits the flow of electrical current.

And as for what resistance is itself - it is the force against the flow of the electrons. They transform the electrical energy they absorb into heat energy.

Imagine electrons - flowing along the wire, pushing new electrons to flow on, and so on. This wire is not very hard to flow in - it's made of a material that's very conductive. But what would happen if something was placed in the middle of the wire that was harder for the electrons to flow through? They're going to be bumping into all the atoms in the material, which will cause the atoms to vibrate. This, in turn, will cause nearby air molecules to take some energy. That energy is in the form of heat. Thus, heat would be created from the electrons bumping into atoms inside the resistor.

### Other Analogy

The other way that these three are explained is using water as an example. Imagine the basic components of a circuit, a battery, wire, and a resistor. In the water analogy, this translates to a pump (because the battery pushes electrons around the circuit), some large pipe (wire), and a section of much smaller pipe (resistor). In the water analogy, the flow rate of the pump is the same as the voltage of the battery, and the pressure in the tubing if the same as the current in the circuit. This is a pretty simple way to explain voltage, current, and resistance. If the voltage is increased, but the resistance (pipe size) remains the same, it logically takes more pressure. However, if the flow rate the remains same and put in large pipes, it takes a lot less pressure to so the same job. Conversely, if the pressure is dropped, but the pipe size remains the same, the flow rate goes down, and if a constant pressure is maintained, but the pipe size increases, the flow rate goes up. And that's all there is to it. Thus one can easily comprehend the relationships in Ohm's law. It may help to read the derived units section to understand the units used on the water side.

### Application of Ohm's Law

This section doesn't teach any theory behind Ohm's law, but this is one of the easiest ways to apply the law (or the power law, P=IV, or any similar law). Basically, take a circle and divide into half, then divide one of the halves in half again, so there is half a circle at the top, and two quarters at the bottom.

Then an equation would be added (any equation in the form a=bc). In the case of the power law, P would go into the half circle, and I and V would go into quarter circles. Now a certain value can be covered up to determine how to solve for the covered value. For example, for finding I using P and V, P and V are uncovered since P is on top of V, it can be determined that $\displaystyle{ I=P/V }$. If the letters are next to each other (i.e. finding P from I and V) then simply multiply. Sure, the math behind it is very simple, but in a competition, this method goes a lot quicker than rearranging equations.

### Base and Derived Units

SI base units are the base quantities that are independent. There is a total of seven units, but the ones important to this event are meters (m, length), kilograms (kg, mass), amperes (A, electric current), and seconds (s, time). Derived units are units that come from a combination of the base units. The ones important to this event are newtons, joules, watts, coulombs, volts, farads, siemens, and ohms. The table below shows how each of the units is related.

Derived Units
Quantity measured Unit name Unit symbol Expression in other SI Units Base SI Units
Electrostatic Force Newton N - kg*m*s-2
Energy, work Joule J N*m m2*kg*s-2
Power Watt W J/s m2*kg*s-3
Electric Charge Coulomb C - s*A
Electric Potential Difference Volt V W/A m2*kg*s-3*A-1
Electric Resistance Ohm Ω V/A m2*kg*s-3*A-2
Electric Conductance Siemens S A/V s3*A2*m-2*kg-1

Another important derived quantity that does not have a special unit name is the electric field strength, measured in V/m.
One coulomb is also equal to the charge of 6.24 x 1018 electrons.

## Electric Charges and Fields

All matter is made up of tiny particles called atoms. These atoms contain three basic subatomic particles which are known as protons (+), neutrons and electrons (-). Note that neutrons have no charge. Every atom either has an overall net positive, negative or neutral charge. The number of protons relative to the number of electrons (or the number of electrons relative to the number of protons) determines the charge of the atom. The charge of these atoms then determines the charge of the matter it makes up. Charged atoms are commonly reffered to as charges.

### Coulomb's Law

Coulomb's Law is a formula formulated by Charles Coulomb that expresses the electrostatic force between two charges. This formula includes an electrostatic constant, also known as Coulomb's constant, equal to roughly 8.988 × 109 N ⋅ m2 ⋅ C−2.

$\displaystyle{ F = \frac{kq_1q_2}{r^2} }$

F = electrostatic force in Newtons

k = Coulomb's constant (8.988 × 109 N ⋅ m2 ⋅ C−2)

q1 = charge of one of the charges in Coulombs

q2 = charge of the other charge in Coulombs

r = distance between the charges in meters

## Symbols

An important part of Circuit Lab is being able to read schematics, which are visual representations of electronic systems. Electronic components like those described below are represented using drawings, and determining what something represents may not always be intuitive. Listed below are some of the most common visual symbols for electronic devices.

Component Basic Description Typical Symbol Additional Symbol(s)
Trace Junction Shows where two wires (or traces) connect Resistor Reduces current flow  Battery Provides power to a circuit  Capacitor Stores electrical energy  Diode Conducts current in one direction Switch Opens and closes a circuit  Lamp Produces light when it receives current  ## Circuit Elements

### Conductors and Insulators

A pre-requisite to understanding what conductors and insulators are is understanding the phrase "electron conductivity." This phrase is essentially the measure of a substance to transport electrons. Conductors have a high electron conductivity while insulators have a low electron conductivity. Conductors are substances which transmit electrons (electricity) relatively easily with little resistance. On the other hand, insulators do not transmit electrons easily and if they do, they have a lot of resistance. Bear in mind, not all conductors have the same electron conductivity and not all insulators have the same electron conductivity. Some common examples of conductors and insulators are below.

### Sources

• Voltage Source: a theoretical component which outputs a precise, constant voltage regardless of current. Their primary usage is in modeling real components. For example, a battery can be modeled as a voltage source in series with a resistor equal to its internal resistance.
• Current Source: a theoretical component which outputs a precise, constant current, regardless of the voltage.

### Resistors This is a basic ¼ watt resistor, the actual resistor is the part in between the two silver leads

The color bands around the resistor signify what the resistance is, and what the tolerance is (how accurate it is). The color codes are:

Resistor Color Codes
Color Value
Black 0
Brown 1
Red 2
Orange 3
Yellow 4
Green 5
Blue 6
Purple (Indigo) 7
Gray 8
White 9
Gold .1
Silver .01
Common Tolerance Codes
Color Percent
Silver 10%
Gold 5%
Red 2%
Brown 1%

The most common tolerance is Gold, followed by Brown, but this doesn't rule out the other possibilities. To convert the color codes into resistance values on a resistor with 3 bands and a tolerance band, read the first two bands off in order, and then multiply that by 10^(color of third band). In the picture it would be green, then blue, thus 56, and then multiply it by 10^0 which is 56 x 1, or 56 ohms. If the resistor has 4 or more bands, read the first however many necessary, usually 3, until you only have one color (not tolerance) left, and multiply by 10^(color of last band).

### Resistor Networks

Networks of resistors between two points can be simplified into an equivalent single resistor, for which the resistance can be calculated according to the configuration and values of the resistors within the network.

#### Series Resistance

The resistance of a resistor is directly proportional to the length of the resistive material. As such, because placing resistors in series effectively adds the lengths, resistances add in series. Therefore, for a chain of resistors, the equivalent resistance is equal to the sum of individual resistors.

#### Parallel Resistance

In parallel, it is not the resistances that add, but the conductances. An analogy for this is to imagine a crowd of people trying to get through a door. A single door will allow so many people per minute, but if a second, adjacent, identical door is opened, the same number of people per minute will simultaneously move through that door. Therefore, twice the number of people will move through the doors per minute. Similarly, two identical resistors in parallel will conduct twice the current as a single one. Therefore the total conductance is equal to the sum of individual conductances in parallel. As conductance is the reciprocal of resistance, the usual formula is that $\displaystyle{ \frac{1}{R_t}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n} }$ for n resistors in parallel.

#### Networks Containing Both Series and Parallel

Many real circuits will contain a combination of both series and parallel components. To simplify these networks, one must find parts of the networks that are purely one or the other and simplify them according to the formulas above. One can repeat this process until the network is simplified into a single equivalent resistor.

### Wheatstone Bridge

A wheatstone bridge is used to measure an unknown resistance value to a high degree of accuracy. It uses 4 resistors set up in a diamond fashion (shown below) and a voltmeter. In the schematic below, Rx is the unknown resistance, R1 and R3 are fixed resistance values (generally the same, but they don't have to be the same, also generally >1% tolerance, but again, not always) and R2 is a variable resistor (potentiometer, this is not always the case, see below). By adjusting R2 until the voltmeter reads 0 volts, you know that the ratio between the R1/R2 and R3/Rx is equal.

To understand this, think of a circuit with two resistors of equal value in series, connected to a +5v source, because the resistances are equal, the voltage drop is equal, this kind of circuit is called a voltage divider, because the voltage in between the two resistors is 1/2 the input voltage. Again, imagine a circuit with 2 resistors in series connected to a +5v source, however this time, the resistors are 50 ohms and 25 ohms, because the total resistance is 75 ohms, at 5v, we can calculate the current, and from there calculate the voltage drop from each resistor, you should have gotten 3.33 volts across the first, and 1.66 for the second one; the voltage happens to be in the same ratio as the resistance values; now that we've proved that, we can apply it to the wheatstone bridge.

With that in mind, we now know that the ratio of the resistors is what controls the voltage at the midpoint, so if two sets of resistors have the same ratio, then they would have the same voltage. When the voltage across the bridge is 0, the sets of resistors (R1/R2 and R3/Rx) have the same voltage, and thus the same ratio of resistance values! Since we know the ratio of the first leg (R1/R2, remember we set R2 to a known value to balance the bridge) and we know R3, it's fairly simple to solve for Rx.

What if you don't want to have to change R2? Then, using the same principle, you can take the voltage across the bridge, and calculate Rx from that. Basically, by applying all the concepts discussed here (Kirchhoff's laws, Ohm's law, etc) you end up at the equation

$\displaystyle{ V_G = ({R_x \over {R_3 + R_x}} - {R_2 \over {R_1 + R_2}}) }$

### Capacitors

Capacitors are, in DC at least, a device that stores a charge. When capacitors are in a circuit, they are said to resist change in voltage (i.e. if the voltage in a circuit goes up, the capacitor charges, taking away the excess voltage. If the voltage drops, the capacitor discharges, adding back to the circuit to make up the difference. There are many types of capacitors (Mylar, polystyrene, electrolytic, etc), but they all do the same basic job. At the most basic level, a capacitor is comprised of two plates separated by a dielectric (insulating material) that stores a charge, there are a few basic concepts it may be helpful to know. First off, look at the charging circuit below, the capacitor is uncharged in the beginning, but when the switch is closed, it begins to charge, as it starts to charge, the resistance across it is small (thus a current flows through the circuit, charging the capacitor), however as the voltage of the capacitor reaches $\displaystyle{ V_0 }$, the current decays exponentially, because the voltage is smaller, less current flows (remember?). This can also be shown by trying to measure the resistance of a capacitor (see below, because the meter puts out a small current, that charges the capacitor). It's useful to in some cases calculate the voltage for a capacitor as it is charging or discharging, for which 2 formulas are incredibly helpful. For a charging capacitor in an RC circuit:

$\displaystyle{ V_c = V_o(1-e^{-t/(RC)}) }$

For a discharging one:

$\displaystyle{ V_c = V_o(e^{-t/(RC)}). }$

Other Analogy
In the water analogy, a capacitor is simulated as a piece of rubber blocking the pipe. Using this example, we can see that a DC current would flow for a time, but when the rubber reached its elastic limit, it would stop, the same as the capacitor charge curve discussed above. However, an AC current (imagine the water moving back and forth very fast) would simply move the rubber back and forth, never stopping the rubber on the other side from flowing (this is true for electricity, but in an effort to not drone on too long, it's not in there).

## Kirchhoff's Laws

Kirchhoff has two well-known laws of circuits: Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL). They are simplifications of Maxwell's Laws of Electromagnetism that are valid for most practical circuits.

### Kirchhoff's Current Law

A node is a junction in a circuit where two or more electrical components meet. Kirchhoff's Current Law states that sum of currents entering a node is equal to the sum of currents leaving the node. This is based on the assumption that charges cannot accumulate in a node of the circuit. Equivalently, if one designates the direction of the currents with a sign (eg all currents leaving the node are negative), the sum of currents at each node equals zero.

#### Node Method

The Node Method is a powerful tool of circuit analysis that is based upon Kirchhoff's Current Law. Basically, one writes the equation for every node in the circuit based upon unknown variables. Then from the resultant system of equations, one can calculate all the unknown variables and solve the circuit. It is often unnecessary for simple circuits but becomes quite convenient for large circuits.

Detailed method:
1. Select a node to be your ground and assign it a voltage of zero. (N.B. The term "ground" in the context of circuit analysis does not necessarily mean that it is connected to the ground. Instead, it is a node designated at zero electrical potential from which all other voltages are measured.)
2. Assign every other node in the circuit a variable voltage. You may in certain cases be able to calculate a voltage for a few nodes (e.g. if the negative terminal of the battery is connected to the ground, the node connected to the positive terminal will have a known, positive voltage).
3. Write the KCL equation for every node in the equation. [Example soon].
4. Solve the resulting system of equations for all the unknown variables. At this point, you know the voltage for every node in the circuit and should be able to easily calculate anything else.

 Nodal Analysis Example 2021 Rickards Invitational, Question 41 What is the node voltage at the point directly above the 15 mA source? (Answer to two decimals) Solution: First, to write a nodal equation, we need to find the current entering the node. Analyzing the circuit, we can see that only a 15 mA current source enters the node, which can simply just be written as +15 mA. Now we need to find the current exiting the node. Looking at the circuit again, there is a 20 mA current source exiting the node and current on the right branch exiting the circuit, which can be expressed as: $\displaystyle{ \frac{V_{node} + 12 \textrm{V}}{(450 + 200 Ω)} }$ Now, we can plug this in our final equation, which is: $\displaystyle{ 15 \textrm{mA} − 20 \textrm{mA} − \frac{V_{node} + 12 \textrm{V}}{(450 + 200 Ω)}= 0 }$ Using math, we can simplify this equation and solve for Vnode, the voltage on the upper node. $\displaystyle{ V_{node} = -\textbf{15.25 V} }$ *Credit: Jaggie34

### Kirchhoff's Voltage Law

Kirchhoff's Voltage Law (KVL) states that for any closed loop in a circuit, the sum of voltages will be zero. One must be very careful with sign convention for this to work. For example, in a simple series circuit of resistors and voltage sources, one must choose either the voltage sources or the resistors to have negative voltages. This is based on the assumption that there is no changing magnetic field.

#### Mesh Method

The Mesh Method is another technique of circuit analysis based upon Kirchhoff's Voltage Law. Essentially, one designates a variable for a current circulating through every loop in the circuit, and then writes an equation for each of these in terms of KVL.

 Mesh Analysis Example 2021 Rickards Invitational, Question 46 What is the value of ix? (Answer to the nearest whole number in μA) Solution: We can create two loops of current to represent the current sourced from batteries V1 and V2. In this scenario, we can express the loops of current i1 and i2 for batteries V1 and V2, respectively, as shown in the circuit example. Current i1 is flowing counterclockwise while ix is flowing clockwise. First, we can observe that ix is equivalent to -i2. To write a mesh equation for loop i1, we firstly know that its source is from battery V1. V1 is the source of the voltage across R1, so we subtract this from 10 V, the voltage of the voltage source of this current loop. Dependent battery source 100ix is only in the way of the current produced by V1, so we subtract that from 10 V as well, along with the voltage across the R2 resistor, where V1 partially sources its voltage along with V2. Now we have this equation for current loop i1: $\displaystyle{ 10 \textrm{V} − 1000i_1 − 100i_x − 3000(i_1+i_2) = 0 }$ Knowing that ix is equal to -i2, we can replace ix in our equation with -i2. 3000(i1 + i2) can be distributed and like terms can be combined to create our final simplified version of the equation: $\displaystyle{ −4000i_1 − 2900i_2 = −10 \textrm{V} }$ Now that we have the mesh equation for i1, we have to write the mesh equation for loop i2. We understand from that V2 is the source of current loop i2. V2 is also responsible for the voltage drop across R3 and R4, so we subtract the expression for both of those voltages from 8 V. From the mesh equation for i1,we know that V1 is partially responsible for the voltage drop across R2. Likewise, V2 is also partially responsible for the voltage drop across R2, so we include it in our equation. Subtracting the expression for the voltage across R2 from 8 V also in our equation, we get the equation: $\displaystyle{ 8 \textrm{V} − 2000i_2 − 3000(i_1 + i_2) − 2500i_2 = 0 }$ This equation can be simplified to the following through distributing and combining like terms: $\displaystyle{ −3000i_1 − 7500i_2 = −8 \textrm{V} }$ Using a bit of math, we can solve for the current of loops i1 and i2. $\displaystyle{ i_1= 0.0024 \textrm{A} }$ $\displaystyle{ i_2= -9.4 \times 10^{−5} \textrm{A} = -94 μ\textrm{A} }$ $\displaystyle{ i_x= -i_2 = \textbf{94 μA} }$ *Credit: Jaggie34

### Superposition Analysis

Another common method of circuit analysis is Superposition Analysis. It uses both of Kirchhoff's Circuit Laws and doesn't focus on a particular one. The purpose of this method of circuit analysis is to find the input or effect of each power source of the circuit by focusing on each individual source and replacing all other sources with a short or removal; current sources are removed and voltage sources are replaced with a short (wire). After this, the corresponding algebraic sum of voltages and currents of each individual component is found with polarity taken into account of course.

 Superposition Analysis Example 2021 November SMEC, Question 39 What is the power consumed by V1, R1, and R2 respectively? (Answer to the nearest whole number) Solution: For this circuit example, we will use superposition because there are only two sources. Normally, with a circuit with 3 sources or fewer, superposition can be used. In superposition, each individual source is focused on one by one to measure its impact on the circuit. All the other sources except for the one focused on are replaced with a wire or removed from the circuit. Voltage sources are replaced with wires and current sources are removed. First, we’ll focus on the battery and remove the current source. By removing the current source, we have created an open circuit. This means that no current can be supplied by the battery to the two parallel resistors, and therefore no power. Now, we can move on to the current source. To focus on the current source, the current source will be put back into the circuit and now the battery will be replaced with a wire. Looking at the circuit right now, we can use current division to solve for the current through each resistor, which we can use to solve for the power consumed by each resistor. $\displaystyle{ I_{R1} = \frac{R_{eq}}{R_1} \times I_1(\frac{66.667 Ω}{200 Ω} \times 3 A) = 1 \textrm{A} }$ $\displaystyle{ I_{R2} = \frac{R_{eq}}{R_2} \times I_1(\frac{66.667 Ω}{100 Ω} \times 3 A) = 2 \textrm{A} }$ Now in this scenario, you don’t necessarily have to do this and use plain logic that double the current will go through R2 than R1 and solve for the current that way, but for demonstration purposes, the current division formula is used here. Now that we have the current through each resistor, we just find the power using P = I2R. $\displaystyle{ P_{R1} = (1 \textrm{A})^2 \times 200 Ω = \textbf{200 W} }$ $\displaystyle{ P_{R2} = (2 \textrm{A})^2 \times 100 Ω = \textbf{400 W} }$ Since we have found the power consumed by resistors R1 and R2, all that’s left to do is find the powerconsumed by the battery using the simple formula P = IV by putting the battery back in the circuit. The battery consumes power because the current source is external, so the battery acts like a resistor by consuming power. $\displaystyle{ P_{V1} = 3 \textrm{A} \times 5 \textrm{V} = \textbf{15 W} }$ *Credit: Jaggie34

## Equivalent circuits

Just as networks of individual resistors can be simplified into a single equivalent resistor, so also can more complicated networks. Any network containing resistors, current sources, and voltage sources can be transformed into a Thevenin or Norton equivalent. This is especially useful when analyzing circuits containing other components, as the entire rest of the circuit around the component may be a network which has an equivalent.

### Thevenin equivalent

The Thevenin equivalent circuit between two points consists of a voltage source in series with a resistor. In order to find the Thevenin voltage, you must find the open-circuit voltage across the two points (ie when it is broken open). The resistance is found by removing all the power sources (replacing current sources with shorts and voltage sources with breaks) and finding the equivalent resistance of the resultant resistor network.

### Norton equivalent

The network can also be represented by a Norton equivalent. It consists of a resistor in parallel with a current source. The Norton resistance is equal to the Thevenin resistance. The Norton equivalent current is equal to the current that passes between the two points if you short circuit them.

## Digital Logic

In digital logic in circuits, a current corresponds to a "true" or "1", and no or very little current corresponds to a "false" or "0". A logic gate will take 1 or more of these signals, perform a logical operation on it, and then either send a true or a false on its way.

A simple example of a logic gate is a transistor. If it receives very little current (false/0), then it does not allow current to pass through it (false/0). If it receives a current (true/1), then it allows current to pass through it (true/1).

Gate Description Boolean Operator Distinctive Shape Rectangular Shape Truth Table
NOT Turns true into false and false into true.

Commonly called an inverter.

$\displaystyle{ \overline A }$ or ~$\displaystyle{ A }$  INPUT OUTPUT A NOT A 0 1 1 0
AND Outputs true only when all inputs are true. $\displaystyle{ A \cdot B }$ or $\displaystyle{ A }$ & $\displaystyle{ B }$  INPUT OUTPUT A B A AND B 0 0 0 0 1 0 1 0 0 1 1 1
OR Outputs true if any inputs are true. $\displaystyle{ A+B }$  INPUT OUTPUT A B A OR B 0 0 0 0 1 1 1 0 1 1 1 1
NAND Outputs false when all inputs are true. $\displaystyle{ \overline {A \cdot B} }$ or $\displaystyle{ A \vert B }$  INPUT OUTPUT A B A NAND B 0 0 1 0 1 1 1 0 1 1 1 0
NOR Outputs false when any inputs are true. $\displaystyle{ \overline {A + B} }$ or $\displaystyle{ A - B }$  INPUT OUTPUT A B A NOR B 0 0 1 0 1 0 1 0 0 1 1 0
XOR Outputs true when exactly one input is true. $\displaystyle{ A \oplus B }$  INPUT OUTPUT A B A XOR B 0 0 0 0 1 1 1 0 1 1 1 0
XNOR Outputs true when input signals are the same. $\displaystyle{ \overline {A \oplus B} }$ or $\displaystyle{ A \odot B }$  INPUT OUTPUT A B A XNOR B 0 0 1 0 1 0 1 0 0 1 1 1

## Alternating Current (AC)

As of the 2013 season, Alternating Current is a disallowed topic under the rules. However, as of the 2018 season, Alternating Current is a allowed topic under the rules to a certain extent.""

Up until now, the entire discussion (minus a mention when talking about capacitors) has dealt with DC, or direct current. In a DC circuit, current flows from the positive to the negative terminals of a battery or other source, that it (electrons flow the opposite, see above). However, the power in your house is AC, not DC. AC, or alternating current, is much more complicated. Basically, the direction of the current flow change, if voltage vs time is plotted, instead of a line (for DC), a sine wave is graphed. This yields many advantages, namely, the use of transformers for voltage step-up/step-down.

### Single Phase

Single phase AC (only one wire that's 'hot' or has voltage in it) is most of what's in your home. It has a hot and neutral line. The hotline varies between the minimum and maximum voltage, and neutral stays around 0v. The switch direction does not matter much for simple devices like light bulbs, but more complicated devices generally convert AC into DC before use through a rectifier (usually a diode bridge). As a side note, an inverter is used to convert DC into AC. The benefit of this is that AC enables the use of transformers to easily step-up and step-down the voltage.

### Transformers

The diagram of a transformer is as shown below:

On the left hand side of the diagram is the primary circuit, where the the initial AC voltage source provides power to the circuit. The primary coil (on the left hand side of the transformer) and secondary coil (on the right hand side of the transformer) are wrapped around the iron core. These coils are both considered to be electromagnetic inductors. The changing magnetic field in primary coil then induces a voltage in the secondary coil according to the formula

$\displaystyle{ \frac{V_p}{N_p}=\frac{V_s}{N_s} }$,

where $\displaystyle{ V_p }$ and $\displaystyle{ V_s }$ are the voltages of the primary and secondary coils respectively, and $\displaystyle{ N_p }$ and $\displaystyle{ N_s }$ are the number of turns on the primary and secondary coils respectively. Thus, it can be seen that the voltage of the each coil is directly proportional to the number of turns in that coil.

It isn't possible to build a DC transformer because the magnetic field of the primary coil would be constant. Remember that, according to Faraday's law, constant magnetic fields can't induce the voltage needed to create a current in the secondary coil, only changing magnetic fields can.

### Phase Shift

Imagine a sine wave. Now, put another sine wave on top of it, only flipped. These waves are said to be 180 degrees out of phase - either wave could be moved 180 degrees along the x-axis and the waves would line up. This is phase shift.

### Power Factor Power Triangle, the power factor is the angle between apparent power and real power, from wikipedia

In a normal AC waveform graph, there is both a voltage vs. time waveform and a current vs. time one. The two are normally in sync and always have positive power (signs switch as the zero-crossing, so the product is always positive). This occurs in a purely resistive load like a perfect lightbulb, and when the two are in sync, the power factor is always 1, a perfect power factor. However, this never happens. On the other hand, when the voltage and current are 90 degrees out of phase, the power transfer is half negative and half positive, so there is no net power transfer. This occurs in a purely inductive or capacitive load (i.e. a motor), and the power factor is 0. This also never happens. In the real world, every wire has resistance, inductance, and capacitance (they are very negligible, small wire is in the 1-10 milliohms/ft), which causes the power factor to never be perfect.

Although there are many units that go along with AC, the most important are: watts, which are real power, the work that can be done through a motor; volt-amps (VA) are apparent power, the power that wires and cables must handle; volt-amps-reactive (VAR) are the power the wires must carry, but cannot do real work. These three main units are related in the power triangle shown to the right. Most power companies do not want reactive power, charging more for businesses with low power factor loads (bigger wires are needed), and inductors or capacitors can be placed to correct for the bad power factor. However, there are some uses for reactive power. For example, in drilling rigs, when a rig holds the top drive, most rigs using AC motors don't actually use any power. They only use enough to overcome the resistive losses in the wire/control system (VFD) and motor, approximately 50A at 60V, or 30KVA. This is a tiny amount of power to hold a block weighing a little over a million pounds. This occurs because about 1200A (720KVA) flows between the motor and the VFD, but the motor is almost a purely inductive load, so no real power is used.

### Polyphase

Although a complex-sounding word, polyphase has a simple meaning. Whereas a single phase system has one wire with changing voltage, a polyphase (many-phase) system has multiple wires carrying current at a time, shifted (time delayed) by a certain amount. The most common is 3-phase power, which is used in many factories and industrial places, anywhere where large motors are involved. In this system, there are 3 wires carrying power, each of them shifted 120 degrees from the other. In this case, there is always a voltage between the phases (due to the shift) and a neutral is not required. The benefit to this is that the power through the system is constant, instead of varying like single phase (add up the magnitude of the 3 waves at any point and it will always be the same number). Polyphase is also useful in motors because the motors become self-starting; in an AC motor, the magnetic field must come from two points, and there must be a phase shift between the two so that the field 'rotates' around the stator (motor shaft). In a single phase system, this isn't possible, so most motors need to be moving to start (so the stator is already moving through the field), in three phase, the phase shift is already present, so the motor can start itself, it can also apply full torque without any speed, because the field can 'hold' the stator in place. This is useful in cranes and oil derricks, removing the need for a mechanical break. Systems that use more than 3 phases exist, but almost solely for motor applications where higher speeds are required.

## Other Topics

### Inductors

An inductor is essentially an electromagnet that exhibits special characteristics in a circuit. It can be described as the opposite of a capacitor, but this is slightly misleading. An inductor can store a charge in a magnetic field (capacitors store it in an electric field) and can maintain a constant current in a circuit (capacitors maintain a constant voltage). An inductor easily conducts DC (capacitors easily conduct AC), but it AC is put through an inductor, the magnetic field grows and collapses with the rise and fall of the current, which tends to opposes the flow of AC through an inductor.

In the water analogy from above, an inductor is a waterwheel, with a constant flow of water through it. The waterwheel spins normally. However, with an alternating flow, the water wheel continuously tries to turn back and forth, limiting the flow of water.

### Diodes

The defining feature of a diode is that it primarily allows current to flow through it in one direction. Applying voltage in the direction that causes current to flow is known as forward-biasing the diode. Applying a voltage in the other direction across the terminals is known as reverse-biasing. An ideal diode has zero resistance in the forwards direction and infinite resistance in the backwards direction.

Most diodes today are formed from the junction of P-type and N-type silicon. Other materials such as Gallium Arsenide (GaAs) and Germanium are also used to make diodes.

The schematic symbol for a P-N diode is an arrow with a line across the tip. This indicates that conventional current can flow in the direction of the arrow but is blocked in the reverse direction.

In the water analogy, the diode is simply a check-valve (one-way valve).

#### Characteristics

Forward Voltage

All practical (non-ideal) diodes exhibit a forward voltage drop. In other words, when a diode is forwards biased, it will exhibit a voltage drop across it. For a typical P-N junction diode, this forward voltage drop will be around 0.7 V. This also means that it requires more than 0.7 V across the terminals of the diode to actually get any significant current flow through the diode (i.e. the diode "turns on" at 0.7 V). An ideal diode has zero forward voltage drop.

Reverse Recovery Time

Reverse recovery time describes the amount of time it takes for the diode to stop current flow in the reverse direction. In other words, when switching from forward to reverse biasing, this is the amount of time it takes for the diode to switch from conducting to non-conducting or to "turn off". For P-N diodes, the reverse recovery time will be around 1 microsecond.

Breakdown (Avalanche or Zener) Voltage

The last major characteristic of diodes is the breakdown voltage, the maximum amount of voltage the diode can withstand without breaking down and conducting when reverse biased. An ideal diode has an infinite breakdown voltage, but practical diodes will break down when a sufficiently high voltage is used to reverse-bias the diode. For P-N diodes, this can be anywhere from around 50 V to hundreds, if not thousands, of volts. While there are subtle differences, generally both "avalanche" and "zener" will refer to diode breakdown. These are simply two different mechanisms responsible for the breakdown of the junction.

#### Types

P-N Diode

These diodes are common, general-purpose diodes that are formed from the junction of P-type and N-type semiconductors.

Schottky Diode

Schottky diodes are formed from the junction of a semiconductor with a metal. They tend to have a very fast reverse recovery time (often around 0.1 - 10 ns) and lower forward voltage drop than P-N diodes. However, they are often limited by a relatively low reverse breakdown voltage.

Light Emitting Diode (LED)

Light Emitting Diodes (LEDs) are diodes designed to produce light at the junction. Generally, LEDs producing shorter wavelength colors have higher forward voltage drops.

Zener Diode

While most diodes should not be operated near their breakdown voltage, zener diodes are designed to break down repeatedly at a precisely set voltage. So a 1.2 V zener diode will breakdown when a reverse voltage exceeding 1.2 V is applied to it. They are often used to clamp or limit voltages to a certain amount since the reverse voltage across a zener diode will not exceed its rated reverse voltage.

#### Circuit Analysis with Ideal Diodes

Ideal diodes can generally be approximated by either a short or a break in the circuit. To analyze a circuit with an ideal diode, and educated guess must be made on whether or not the diode conducts. Once a solution is found for this circuit, one much check if it makes sense. If the diode is assumed to conduct, one must make sure it is conducting current in the correct direction. If the diode is assumed not to conduct, one must check to make sure the voltage across the non-conducting diode is such that it would not conduct. If this is not the case, the assumptions must be switched and the circuit recalculated.

For first-order diode approximations, diodes may be assumed to have a 0.7 V forward voltage. If this is specified in the problem, you will need to take this voltage drop into account when writing out the equations for KVL and KCL. As before, make an educated guess about the biasing of the diode and check these assumptions afterwards.

## Lab Component

Breadboards are a common tool used when prototyping electronics. They are a plastic boards with an array of sockets that electronic components can plug into. Underneath the sockets are copper clips that serve to hold any inserted components in place as well as make electrical connections between components.

Breadboards facilitate prototyping by making it easy to make temporary connections between components. Along each long edge of the board are two columns of sockets, often indicated by the colored lines and shown by the red arrows in the image. These are known as the power rails or as bus strips. Each column in the power rails is fully connected by the copper clips underneath the board. This makes them easily accessible from anywhere on the board and hence makes them useful for supplying power to components. Some larger breadboards will have a split in the middle of the power rail. This can be checked quickly with a multimeter. All the other sockets on the board are connected in groups horizontally, in 1x5 rows. This space is generally used to connect components together.

### Multimeters Fluke 287 multimeter, note the separate jacks for measuring current.

Multimeters are devices that can measure various electrical properties of circuits and electronic components. The most common measurements are voltage, resistance, and current. Multimeters are a combination of voltmeters, ohmmeters, and ammeters. Modern multimeters include the ability to measure many other properties such as capacitance, temperature, and frequency of AC signals.

#### Basic Usage

First, identify the desired quantity to be measured (e.g. volts, amps, ohms). Most meters will have 3-4 sockets along the bottom to plug in the probes. Plug the black probe into the COM socket. This is COMmon ground. Plug the red probe into the socket corresponding to the property you are measuring. On most meters, there will be two options for current: mA and 10A. These ports are normally protected by a fuse. The limit of the fuse should be indicated next to the socket. Generally, the mA socket will be fused to around 200 mA. If expected measurement exceeds the rating of the fuse, use the 10A socket. If the expected measurement exceeds 10A (and the 10A socket is fused for 10A), a different device will be required to perform the measurement. (If this wiki is your primary source of information, then you should definitely not be measuring anything in the range of 10A as this is an extremely dangerous amount of current.)

Once the probes have been plugged in, turn the dial on the meter to the corresponding symbol of the property being measured. Some multimeters will have various units for each type of measurement (e.g. uA, mA, A). Estimate the magnitude of the quantity you are measuring and pick the correct range. If you have any doubts, choose the highest value. Note that on most meters, the high-current setting will only read from the high-current socket.

When reading the meter, it is important to keep track of units. Misjudging the units being measured can cause calculations to be off by several orders of magnitude. For example, if you are on the mV setting, then the number displayed will be in mV. Also, if using a meter that has different precision levels, make sure to use the level of precision that provides the most unique digits; a reading of 1 V is less helpful than a reading of 1123 mV. Finally, as with all pieces of equipment, typical safety procedures and measures should be taken when using meters. Generally, most multimeters are easy and safe to use, but common sense still applies.

Measuring Voltage

The Voltmeter measures the voltage between two points in a circuit. As a result, the circuit should be powered on when measuring voltage (otherwise all points on the circuit will have a voltage of 0V). With the circuit on, touch the black lead to one point and the red lead to the other point. This is called placing the meter in 'parallel' to the circuit. The number displayed is the voltage at the red probe relative to the voltage at the black probe. Remember that voltage refers to a potential difference between two points.

When the voltmeter is placed in parallel to the circuit, a small amount of current will be siphoned off the circuit and into the meter to actually make the measurement. However, the Voltmeter has a very high internal resistance, around $\displaystyle{ 10\text{M}\Omega }$, so it has minimal impact on the circuit (i.e. siphons off a minimal amount of current) when placed in parallel.

Measuring Current

The Ammeter measures the current through a section of a circuit. To use an Ammeter, find the segment of the circuit being measured and break the circuit apart at that location. Place one lead on one end of the break and the other lead on the other end. This is referred to as placing the meter in 'series' with the circuit.

It is necessary for the current to pass through the meter in order to measure it. To prevent the meter from dropping a large amount of voltage and impacting the rest of the circuit, the Ammeter has a very low internal resistance. Because of this fact, it is essential that the multimeter NOT be placed in parallel with the circuit. Doing so is equivalent to placing a low-resistance wire in parallel with the component. In some situations, this can cause large amounts of current to flow through the multimeter and damage it. As a preventative measure, multimeter designers put fuses on these sockets which will blow when too much current is passed through. If this occurs, find the rating of the fuse on the face of the meter and replace the fuse with one of the same size and rating.

Measuring Resistance

The Ohmmeter measures the resistance of a circuit element. It can be placed either in series or parallel to the element. If an Ohmmeter is not provided, one can also place a Voltmeter parallel to the element and an Ammeter in series with the element and apply Ohm's Law to calculate the resistance.