 Jun 2023

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Consider an isolated circuit consisting of a switch, a resistor, and a voltage source,as shown in Fig. 2.119. If you close the switch, you might predict that the current flowthrough the circuit would jump immediately from zero to V/R, according to Ohm’slaw. However, according to Faraday’s law of electromagnetic induction, this isn’tentirely accurate. Instead, when the switch is initially closed, the current increasesrapidly. As the current increases with time, the magnetic flux through the loop risesrapidly. This increasing magnetic flux then induces an EMF in the circuit that opposesthe current flow, giving rise to an exponentially delayed rise in current. We call theinduced EMF a selfinduced EMF.
When the current flows through the inductor a current that opposes the current flow in the opposite direction is created and this is self induction

EMF , , EMFddt B dA N ddtMMM ∫= − Φ Φ = ⋅ = − Φ (for a coiled wire of N loops) (2.49)where ΦM is the magnetic flux threading through a closed loop circuit (which is equalto the magnetic field B dotted with the direction surface area—both of which are vectors, and summed over the entire surface area—as the integral indicates). Accordingto the law, the EMF can be induced in the circuit in several ways: (1) the magnitudeof B can vary with time; (2) the area of the circuit can change with time; (3) the anglebetween B and the normal of A can change with time; (4) any combination of thesecan occur. See Fig. 2.116.The simple ac generator in Fig. 2.117 shows Faraday’s law in action. A simplerotating loop of wire in a constant magnetic field generates an EMF that can be usedto power a circuit. As the loop rotates, the magnetic flux through it changes withtime, inducing an EMF and a current in an external circuit. The ends of the loop areconnected to slip rings that rotate with the loop, while the external circuit is linked tothe generator by stationary brushes in contact with the slip rings.A simple dc generator is essentially the same as the ac generator, except that thecontacts to the rotating loop are made using a split ring or commutator. The result isa pulsating direct current, resembling the absolute value of a sine wave—there are nopolarity reversals.A motor is essentially a generator operating in reverse. Instead of generating a current by rotating a loop, a current is supplied to the loop by a battery, and the torqueFIGURE 2.116 Illustration of Faraday’s law of induction.
Faraday's law. If a magnetic force passes through the wire loop a EMF is induced. This is increases the more loops that there are

2.24 Inductors
Capacitors store electrical energy in the form of an electrical field. Inductors store energy in the form of a magnetic field
All an inductor is is just wound up coil. A ferrite magnet can be put in the middle like a transformer in order to induce a greater magnetic field effect.
Use right hand rule in order to see the direction of the magnetic field relative to the flow of the current

2.23.15 Quality Factor
Quality factor is used to give us a metric on how much energy is being stored relative to the energy that is lost
 Remember, voltage and therefore energy is lost due to heat when it passes through a resistor. While for inductors and for capacitors the energy is stored

2.23.14 Capacitive Divider
Similar to a regular voltage divider with resistors, a capacitors in series are able to do just that as well

As the frequency goes to infinity, XC goes to 0, and the capacitor acts like a short(wire) at high frequencies; capacitors like to pass current at high frequencies. As thefrequency goes to 0, XC goes to infinity, and the capacitor acts like an open circuit;capacitors do not like to pass low frequencies
Capacitors will allow current to pass only if the frequencies are high. During high frequencies, the reactance is going to 0 which means the current passes through the capacitor like a short circuit.
When the frequency is too low, the reactance becomes infinitiy, and the capacitor acts as a open circuit

Within an ac circuit, the amount of charge movesback and forth in the circuit every cycle, so the rate of movement of charge (current)is proportional to voltage, capacitance, and frequency. When the effect of capacitanceand frequency are considered together, they form a quantity similar to resistance.However, since no actual heat is being generated, the effect is termed capacitive reactance. The unit for reactance is the ohm, just as for resistors, and the formula for calculating the reactance of a capacitor at a particular frequency is given by:121X fC CC = π = ω (2.47) Capacitive reactance
Because capacitors go back and forth charging and recharging, there is no heat that is produced like in a resistor, so thus the "resistance" that forms in a capacitor is a "capacitor reactance". Because it oscillates back and forth it takes into account the angular frequency as well into the equation

2.23.12 Alternating Current in a Capacitor
AC behaves differently when it's connected to a capacitor vs. how DC behaves. This is due to the reactance (which leads to impedance) that forms when AC comes in circuit with capacitance

When two or more capacitors are connected in series, the total capacitance is less thanthat of the smallest capacitor in the group. The equivalent capacitance is similar toresistors in parallel:1 1 1 1tot 1 2C C C CN
When adding capacitors in series, they are added like resistors in parralel
 capacitors in series have a decreased capacitance, but an increase in voltage

When capacitors are placed in parallel, their capacitances add, just like resistors inseries:Ctot = C1 + C2 + ... Cn
Adding capacitors in parralel is like adding resistors in series * Capacitors in parralel increase total capacitance but limited voltage

Intuitively, you can think of capacitors in parallel representing one single capacitor with increased plate surface area. It’s important to note that the largest voltagethat can be applied safely to a group of capacitors in parallel is limited to the voltage rating of the capacitor with the lowest voltage rating.
if you have capacitors in parralel, the voltage will be limited to the capacitor with the lowest voltage rating in parralel

Current and voltage equations for discharging an RC circuit∫= = − = = = − = = = − τ =−−−I VR e tRCIRVV IR V e tRCVVV C Idt V e tRCVVRCs t RCsR s t RC RsC s t RC Cs, ln, ln1 , lntime constant///0(2.44)where I is the current in amps, VS is the source voltage involts, R is the resistance in ohms, C is the capacitance in farads, t is the time in seconds after the source voltage isremoved, e = 2.718, VR is the resistor voltage in volts, and VCis the capacitor voltage in volts. Graph shown to the left inFig. 2.103 is for circuit with R = 3 kΩ and C = 0.1 μF.
RC Discharging current and voltage equations
After 5 time constants, the capacitor is fully discharged

Current and voltage equations for RC charging circuit∫= = − = = = − = = − = − − τ =−−−I VR e tRCIRVV IR V e tRCVVV C Idt V e tRCV VVRCs t RCsR s t RC RsC s t RC s Cslnln1 (1 ) lntime constant///(2.43)where I is the current in amps, VS is the source voltage involts, R is the resistance in ohms, C is the capacitance in farads, t is the time in seconds after the source voltage is applied, e = 2.718, VR is the resistor voltage in volts and VC is thecapacitor voltage in volts. Graph shown to the left is for circuit with R = 10 kΩ, and C = 100 μF. Decreasing the resistancemeans the capacitor charges up more quickly and the voltageacross the capacitor rises more quickly.
These are the equations for taking into account an RC Charging circuit. * As expected, while charging the voltage of the capacitor increases, while the voltage in the resistor decreases (due to voltage drop from resistor).
RC represents the time constant. This just means that it takes R*C time in order to charge the capacitor to 63.6 percent of the voltage sources capacity
Theoretically, the capacitor never fully charges, despite it staying for many time constants, it will never approach 100%. After 5 time constants, the capacitor is considered to be fully charged or fully discharged

E VIdt VC dVdt dt CVdV CV∫ ∫∫= = = = (2.42)
Equation to get energy from capacitor

V C I dtC C∫= (2.41) Voltage across capacitor
Very important equation for getting voltages and current in capacitors

t’s important to note that these equations are representative of what’s definedas an ideal capacitor. Ideal capacitors, as the equation suggests, have several curiousproperties that are misleading if you are dealing with real capacitors. First, if weapply a constant voltage across an ideal capacitor, the capacitor current would bezero, since the voltage isn’t changing (dV/dt = 0). In a dc circuit, a capacitor thus actslike an open circuit. On the other hand, if we try to change the voltage abruptly, from0 to 9 V, the quantity dV/dt = 9 V/0 V = infinity and the capacitor current would haveto be infinite (see Fig. 2.98). Real circuits cannot have infinite currents, due to resistivity, available free electrons, inductance, capacitance, and so on, so the voltage acrossthe capacitor cannot change abruptly. A more accurate representation of a real capacitor, considering construction and materials, looks like the model shown in Fig. 2.93.
Because current in a capacitor varies according to the \(\frac{{dv}}{{dt}}\). If you hookup a constant DC voltage to a capacitor, the change in voltage over time would be 0, which would mean the capacitor current would be 0, so the capacitor acts as a "open circuit"
 Hence the fact why Capacitors are known for being DC blockers, but they enable AC to pass
Additionally, note the changes that occur when a real capacitor is charging. * When a capacitor is charging, the voltage increases over time (to the voltage of the source), and the current experiences an instant spike and decreases over time * When a capacitor is discharging, the voltage decreases over time, and the current spikes and decreases over time as well

2.23.4 Maxwell’s Displacement Current
Maxwell's displacement current about capacitors basically talks about how the charges on opposite ends of the capacitor plates create a sort of electric field due to changing in the magnetic field that gets created when the capacitors are becoming charged.
It's used to demonstrate the current flowing through the dielectric. Of course, in an ideal capacitor, current would NEVER flow through dielectric material

2.23.3 Voltage Rating and Dielectric Breakdown
Because every capacitor has it's own rating in Farads to represent how much charge it can store relative to it's voltage, there is also a limit to how much voltage is allow to run to the capacitor as well.
If the voltage is high enough, the force of the voltage can be enough to puncture and go through the dielectric material. * Remember, electrons are NOT supposed to flow through the dielectric material, the dielectric material is used to keep the charges and their polarities separate. If a dielectric material is puncture, the resistance can go to 0 and it can act as a short which can lead to arcs and etc.

The ratio of charge on one of the plates of a capacitor to the voltage that existsbetween the plates is called capacitance (symbolized C):C QV= (2.32) Capacitance relatedto charge and voltageC is always taken to be positive, and has units of farads (abbreviated F). One farad isequal to one coulomb per volt:1 F = 1 C/1 V
Capacitance equations as well as it's units
 1 Farad = 1C/1V

2.23 Capacitors
Capacitance: Capcitors do one job, that is to store/hold charge
They work by containing 2 conductive plates, separated by a "spacer" called dielectric material. * When you connect a battery to each end of the uncharged capacitor (neutral plates), the electrons from the negative terminal of the battery go to the negative plate of the capacitor. Though, on the other neutral plate, the electrons (that are keeping the other plate neutral) actually flee on move toward the positive terminal of the battery. * This causes one end of the capacitor to be fully positive, and the other end to be fully negative. The capacitor now how a positive charge and a negative charge * To discharge a capacitor, you can connect a wire across it's leads and the electrons will flow to the positive end and the capacitor will be "neutral"

2.22 Mains Power
This section discusses main power and the way that it works in it's entirety.
In the US, three wires run from the transformer, (through meter) and into your "mains" (main circuit breaker for your home). * These 3 wires are Phase A (black wire), Phase B (black wire), and neutral (white wire). * Phase A and B are both connected into the main's main breaker. Neutral is connected to a "neutral bus". Additionally, there is a ground bar in the main panel, this ground bar connects to a rod that can be found lodged in the ground called a "grounding rod"

It's important to note the terms "linetoline" and "linetoneutral". Connected Phase A and Phase B together results in a voltage that is 240V (typically requiring a double pole breaker, and specifically for ovens, dryers, etc.). On the other hand, Phase A or Phase B connected to Neutral would be 120V (lower power appliances like lighting and switches, etc.)

It's also important to note that it's not good practice to just connect all of your breakers on one side of the panel (either Phase A solely, or Phase B solely), as this would lead to an "imbalanced" load. Having your panel imbalanced can lead to voltage imbalances and power loss through the system

Another note: Both neutral bar and ground bar are connected to each other and lead to the ground rod in the ground of your home


Conversion Factors for AC Voltage and CurrentFROM TO MULTIPLY BYPeak Peaktopeak 2Peaktopeak Peak 0.5Peak RMS 1/ 2 or 0.7071RMS Peak 2 or 1.4142Peaktopeak RMS 1/( 2 2 ) or 0.35355RMS Peaktopeak (2 × 2 ) or 2.828Peak Average* 2/o or 0.6366Average* Peak o/2 or 1.5708RMS Average* (2 × 2 )/o or 0.9003Average* RMS o/( 2 × 2 ) or 1.1107* Represents average over half a cycle.
Conversion chart of AC voltage based calculations

Notice that the RMS voltage and current depend only on the peak voltage or current—they are independent of time or frequency. Here are the important relations, withoutthe scary calculus:
Converting from Peak Voltage (Vp) to Root Mean Square Voltage (Vrms) isn't difficult and can be done with the following equations

The measurement that is used instead of the average value is the RMS or rootmean square value, which is found by squaring the instantaneous values of the acvoltage or current, then calculating their mean (i.e., their average), and finally takingthe square root of this—which gives the effective value of the ac voltage or current.These effective, or RMS, values don’t average out to zero and are essentially the acequivalents of dc voltages and currents. The RMS values of ac voltage and currentare based upon equating the values of ac and dc power required to heat a resistiveelement to exactly the same degree
Because a full cycle of a sine (both positive and negative) cancel out and be 0 if you were to take the averages of the waveforms in the sine waves.
Thus, a method known as "Root Mean Square" is used in substitute of "average", and RMS values give us the AC "equivalent" of dc voltages and currents

V(t) = VP sin (2π × f × t) (2.24)where VP is the peak amplitude of the sinusoidal voltage waveform, f is the frequency,and t is the time. Using Ohm’s law and the power law, you get the following:I t V tRVR f tP( ) ( ) sin(2
Defining voltage when working with AC would involve it's peak voltage, along with it's sine wave (due to the voltage, as well as current varying over time).
Typically, for purely resistive loads (only containing resistors), the current and voltage as always IN PHASE (since there is no reactive impedance). * Contrary to purely resistive loads, if you have nonpurely resistive loads (involving both inductance or capacitance), current and voltage will not be in phase

2.20.8 Phase
Though the xaxis refers to time (s), typically degrees are used in order to tell how out of phase AC signals are from one another.
 Lagging and Leading are both keywords that can be used to describe the positioning and how they relate to being in one circuit

2.20.7 Frequency and Period
Frequency and Period go hand in hand. * Frequency tells you "How many full cycles per second" * Period tells you "How many seconds per cycle"

Amplitude
Amplitude can tell us the peaks of voltages or currents based on the degree of the revolution/spin

A complete description of an ac voltage or current involves reference to three properties: amplitude (or magnitude), frequency, and phase.
An AC voltage can be described by: * Amplitude * Frequency * Phase

Alternating current can take on many other useful wave shapes besides sinusoidal.Figure 2.82 shows a few common waveforms used in electronics. The squarewave isvital to digital electronics, where states are either true (on) or false (off). Triangularand ramp waveforms—sometimes called sawtooth waves—are especially useful intiming circuits. As we’ll see later in the book, using Fourier analysis, you can createany desired shape of periodic waveform by adding a collection of sine waves together.
Notice the different types of waveforms that you can be presented with

Besides combining ac and dc voltages and currents, we can also combine separateac voltages and currents. Such combinations will result in complex waveforms.Figure 2.81 shows two ac waveforms fairly close in frequency, and their resultantcombination. The figure also shows two ac waveforms dissimilar in both frequencyand wavelength, along with the resultant combined waveform.FIGURE 2.80FIGURE 2.81 (Left) Two ac waveforms of similar magnitude and close in frequency form a composite wave. Note thepoints where the positive peaks of the two waves combine to create high composite peaks: this is the phenomenon ofbeats. The beat note frequency is f2 − f1 = 500 Hz. (Right) Two ac waveforms of widely different frequencies and amplitudesform a composite wave in which one wave appears to ride upon the other.
Notice what occurs to the output frequency when you have two AC sources that combine that vary in voltage, and frequency

If current and voltage never change direction within a circuit, then from one perspective, we have a dc current, even if the level of the dc constantly changes. Forexample, in Fig. 2.80, the current is always positive with respect to 0, though it variesperiodically in amplitude. Whatever the shape of the variations, the current can bereferred to as “pulsating dc.” If the current periodically reaches 0, it is referred to as“intermittent dc.”
Do not get AC confused with DC. If the voltage or current is above or below the 0 respectively, it MUST cross below the 0 or above the 0 respectively in order to be considered AC
If the positive DC voltage or current stays above 0 but fluctuates, this would be known as "pulsating DC"

The most common way to generate sinusoidal waveforms is by electromagneticinduction, by means of an ac generator (or alternator). For example, the simple acgenerator in Fig. 2.78 consists of a loop of wire that is mechanically rotated aboutan axis while positioned between the north and south poles of a magnet. As theloop rotates in the magnetic field, the magnetic flux through it changes, and chargesare forced through the wire, giving rise to an effective voltage or induced voltage.According to Fig. 2.78, the magnetic flux through the loop is a function of the angle ofthe loop relative to the direction of the magnetic field. The resultant induced voltageis sinusoidal, with angular frequency ω (radians per second).Real ac generators are, of course, more complex than this, but they operate underthe same laws of induction, nevertheless. Other ways of generating ac include usinga transducer (e.g., a microphone) or even using a dcpowered oscillator circuit thatuses special inductive and capacitive effects to cause current to resonate back andforth between an inductor and a capacitor.Why Is AC Important?There are several reasons why sinusoidal waveforms are important in electronics. Thefirst obvious reason has to do with the ease of converting circular mechanical motioninto induced current via an ac generator. However, another very important reason forusing sinusoidal waveforms is that if you differentiate or integrate a sinusoid, you geta sinusoid. Applying sinusoidal voltage to capacitors and inductors leads to sinusoidal current. It also avoids problems on systems, a subject that we’ll cover later. But oneof the most important benefits of ac involves the ability to increase voltage or decreasevoltage (at the expense of current) by using a transformer. In dc, a transformer isuseless, and increasing or decreasing a voltage is a bit tricky, usually involving someFIGURE 2.78 Simple ac generator.
This is the way in which AC is generated.
The mechanics of this align very similarly with bigger systems. Two magents, north and south are positioned in parrelal and a loop of wire is placed in between which spins indefinitely. * This mechanical spin generates electricity and a charge is outputted from the wire. The electricity is created via magnets, hence electromagnetic energy * It spins a certain number of degrees based on time
Angular Frequency just tells us, how many revolutions does it do in a given second. Hence, radians/second

Another type of source that is frequently used in electronics is an alternating source that causes current to periodicallychange direction, resulting in an alternating current (ac). In an ac circuit, not onlydoes the current change directions periodically, the voltage also periodically reverses.
Introduction to AC: * This is where the current switches directions periodically, hence "alternating" * The voltage also changes direction, positive/negative, etc.

Norton’s theorem is another tool for analyzing complex networks. Like Thevenin’stheorem, it takes a complex twoterminal network and replaces it with a simple equivalent circuit. However, instead of a Thevenin voltage source in series with a Theveninresistance, the Norton equivalent circuit consists of a current source in parallel with aresistance—which happens to be the same as the Thevenin resistance.
Norton's theorem is moreso used for current while thevenin is more so used for voltage.
Norton features a resistor in parralel with a current source Thevenin is a resistor in series with a voltage source

Luckily, a man by the name of Thevenin came up with a theorem, or trick, tosimplify the problem and produce an answer—one that does not involve “hairy”mathematics. Using what Thevenin discovered, if only two terminals are of interest,these two terminals can be extracted from the complex circuit, and the rest of thecircuit can be considered a black box. Now the only things left to work with are thesetwo terminals. By applying Thevenin’s tricks (which you will see in a moment),you will discover that this black box, or any linear twoterminal dc network, can berepresented by a voltage source in series with a resistor. (This statement is referredto as Thevenin’s theorem.) The voltage source is called the Thevenin voltage VTHEV, andthe resistance is called the Thevenin resistance RTHEV; together, the two form what iscalled the Thevenin equivalent circuit. From this simple equivalent circuit you caneasily calculate the current flow through a load placed across its terminals by usingOhm’s law: I = VTHEV/(RTHEV + RLOAD)
Thevenin's theorem allows you as the engineer to be able to dumb down and simplify a complex circuit. The methodology in order to do this requires you to treat the problem or variable that you are trying to solve for as a "black box"

The superposition is also an underlying mechanism that makespossible two important circuit theorems: Thevenin’s theorem and Norton’s theorem.These two theorems, which use some fairly ingenious tricks, are much more practicalto use than the superposition. Though you will seldom use the superposition directly,it is important to know that it is the base upon which many other circuit analysistools rest.FIGURE 2.70 The circuit in (a) can be analyzed using the superposition theorem by considering the simpler circuits in(b) and (c).
Superposition is the introductory prerequisite to understanding thevenin's and norton's theorem
Essentially, for superposition, you would make 2 circuits out of the circuit that you are presented with: 1. A circuit where the voltage source is shorted (Voltage is 0) 2. A circuit where the current source is open (open circuit  infinite resistance)

Superposition theorem: The current in a branch of a linear circuit is equal to the sum of thecurrents produced by each source, with the other sources set equal to zero
Superposition theorem explained.
Superposition can only be applied to circuits that are linear, NOT nonlinear. * Linear circuits are simple circuits that are predictable. Such as, doubling the input voltage would double the output voltage. Another example is the input being a sine input and it being outputted as a sine wave but with a different amplitude

Kirchhoff’s Voltage Law (or Loop Rule): The algebraic sum of the voltages aroundany loop of a circuit is zero:V V V Vnclosed path0
Kirchoff's Voltage Law: The sum of the voltages around a closed circuit will be 0, because when the electrons return to negative (or ground), they will go from max DC voltage to 0 volts.
They go from an excited state to a lower state.
Unlike Kirchoff's law about current, voltage experiences drops along the way and at the end of it's loop, it will come down to 0.

Kirchhoff’s Current Law (or Junction Rule): The sum of the currents that enter a junction equals the sum of the currents that leave the junction:I Iin out∑∑ = (2.21)Kirchhoff’s current law is a statement about the conservation of charge flow througha circuit: at no time are charges created or destroyed.
Kirchoff's current law: * The sum of the currents that enter a junction equals the sum that come out of it. *** When it comes to current, it is never lost when it comes out (charges are NEVER lost). What comes in, will come out in one piece**

Often, you will run across a circuit that cannot be analyzed with simple resistor circuit reduction techniques alone. Even if you could find the equivalent resistance byusing circuit reduction, it might not be possible for you to find the individual currents and voltage through and across the components within the network. Likewise,if there are multiple sources, or complex networks of resistors, using Ohm’s law, aswell as the current and voltage divider equations, may not cut it. For this reason, weturn to Kirchhoff’s laws
Kirchoff's Laws are used to analyze more complex circuits. Usually when it comes to smaller circuits, getting the individual currents and voltages are easier, but using kirchoff's law is alot easier to handle more complex circuitry

The most common problems (faults) in circuits are open circuits and shorts. A shortcircuit in all or part of a circuit causes excessive current flow. This may blow a fuse orburn out a component, which may result in an opencircuit condition. An open circuitrepresents a break in the circuit, preventing current from flowing. Short circuits may becaused by a number of things—from wire crossing, insulation failure, or solder splatterinadvertently linking two separate conductors within a circuit board. An open circuitmay result from wire or component lead separation from the circuit, or from a component that has simply burned out, resulting in a huge resistance. Figure 2.59 shows casesof open and shortcircuit conditions. A fuse, symbolized , is used in the circuit andwill blow when the current through it exceeds its current rating, given in amps.
The difference between "short circuit" and "open circuit".
A "short circuit" can lead to a "open circuit" if the short causes the wire to split/burn or open

The two battery networks in Fig. 2.58 show how to increase the supply voltage and/or increase the supply current capacity. To increase the supply voltage, batteries areplaced end to end or in series; the terminal voltages of each battery add togetherto give a final supply voltage equal to the sum from the batteries. To create a supply with added current capacity (increased operating time), batteries can be placed inparallel—positive terminals are joined together, as are negative terminals, as shown inFig. 2.58. The power delivered to the load can be found using Ohm’s power law: P =V2/R, where V is the final supply voltage generated by all batteries within the network. Note that the ground symbol shown here acts simply as a 0V reference, not asan actual physical connection to ground. Rarely would you ever connect a physicalground to a batteryoperated device.
These are a couple ways in which batteries can be connected together in a circuit. They can be connected in series or in parralel.
For batteries connected in series, it would increase the voltage, but for batteries connected in parralel, it increases current capacity
If you are placing batteries in parallel, you SHOULD make them ALL the same voltages and chemistry

The effects of meter internal resistance are shown in Fig. 2.56. In each case, theinternal resistance becomes part of the circuit. The percentages of error in measurements due to the internal resistances become more pronounced when the circuitresistances approach the meters’ internal resistance.
This shows how to take measurements when using a ohmmeter, voltmeter, or ammeter

Intheory, an ideal voltmeter should draw no current as it measures a voltage betweentwo points in a circuit; it has infinite input resistance Rin. Likewise, an ideal ammetershould introduce no voltage drop when it is placed in series within the circuit; it haszero input resistance. An ideal ohmmeter should provide no additional resistancewhen making a resistance measurement.
Measuring instruments and the ideal use cases for them

Now a fishy thing with an ideal voltage source is that if the resistance goes to zero,the current must go to infinity. Well, in the real world, there is no device that can supply an infinite amount of current.
When resistance is at 0, the current is theoretically at infinity

Example 1: You wish to create a multiple voltage divider that powers three loads:load 1 (75 V, 30 mA), load 2 (50 V, 10 mA), and load 3 (25 V, 10 mA). Use the 10 percentrule and Fig. 2.49 to construct the voltage divider.
Great voltage divider problem

Example 1: Find the equivalent resistance of the network attached to the battery byusing circuit reduction. Afterward, find the total current flow supplied by the battery to the network, the voltage drops across all resistors, and the individual currentthrough each resistor.
seriesparralel problem

The 10 Percent Rule: This rule is a standardmethod for selecting R1 and R2 that takes into accountthe load and minimizes unnecessary power losses inthe divider.The first thing you do is select R2 so that I2 is 10percent of the desired load current. This resistance andcurrent are called the bleeder resistance and bleeder current. The bleeder current in our example is:Ibleed = I2 = (0.10)(9.1 mA) = 0.91 mAUsing Ohm’s law, next we calculate the bleederresistance:Rbleed = R2 = 3 V/0.00091 A = 3297 ΩConsidering resistor tolerances and standard resistancevalues, we select a resistor in close vicinity—3300 Ω.Next, we need to select R1, so that the output ismaintained at 3 V.To do this, simply calculate the total currentthrough the resistor and use Ohm’s law:I1 = I2 + Iload = 0.91 mA + 9.1 mA = 10.0 mA = 0.01 A10 V 3 V0.01 A 7001R − = Ω
Bleeder resistance. In this type of application, look at the 10 percent rule!

V IR VR R RV IR VR R R1 1in1 212 2in1 22= = + ×= = + ×(Voltage dividerequations)These equations are called voltage divider equations and are so useful in electronics thatit is worth memorizing them. (See Fig. 2.43.) Often V2 is called the output voltage Vout.
Voltage Divider Equations. Important to memorize

Example 1: If a 1000Ω resistor is connected in parallel with a3000Ω resistor, what is the total or equivalent resistance? Alsocalculate total current and individual currents, as well as thetotal and individual dissipated powers.R R RR R1000 30001000 30003, 000, 0004000 750total1 21 2= ×+ = Ω × ΩΩ + Ω = ΩΩ = ΩTo find how much current flows through each resistor, applyOhm’s law:I VRI VR12 V1000 0.012 A 12 mA12 V3000 0.004 A 4 mA111222= = Ω = == = Ω = =These individual currents add up to the total input current:Iin = I1 + I2 = 12 mA + 4 mA = 16 mAThis statement is referred to as Kirchhoff’s current law. Withthis law, and Ohm’s law, you come up with the current divider
Great intro problem to Kirchhoff's current law
Current divider notes as well!

Rtotal = R1 + R2 + R3 + R4 + ...
Adding together resistors in series

R R R R R1total 11 12 13 14 = + + + +
Calculating resistance total given resistors in parralel

Example: In Fig. 2.39, a 100Ω resistor is placed across a 12V battery. How much current flows through the resistor? How much power does the resistor dissipate?
Look at Power ratings for resisters prior to using them! Always use about 2x the power of what you are going to dissipate in the circuit

The singlepoint ground concept ensures that no ground loops are created. As thename implies, all circuit grounds are returned to a common point. While this approachlooks good on paper, it is usually not practical to implement. Even the simplest circuits can have 10 or more grounds. Connecting all of them to a single point becomesa nightmare. An alternative is to use a ground bus.A ground bus, or bus bar found in breadboards and prototype boards, or whichcan be etched in a custom printed circuit board (PCB), serves as an adequate substitute for a singlepoint ground. A bus bar is simply a heavy copper wire or bar of lowresistance that can carry the sum total of all load currents back to the power supply.This bus can be extended along the length of the circuitry so that convenient connections can be made to various components spaced about the board. Figure 2.35eshows a bus bar return. Most prototype boards come with two or three lines of connected terminals extending along the length of the board. One of these continuousstrips should be dedicated as a circuit ground bus. All circuit grounds should betied directly to this bus. Care must be taken to make sure that all lead and wire connections to the bus are secure. For prototype boards, this means a good solder joint;for wirewrap board, a tight wire wrap; for breadboard, proper gauge wire leads tosecurely fit within sockets. Bad connections lead to intermittent contact that leadsto noise.
Bus bar and common ground design

f several points are used for ground connections, differences in potential betweenpoints caused by inherent impedance in the ground line can cause troublesome groundloops, which will cause errors in voltage readings. This is illustrated in Fig. 2.35c, wheretwo separated chassis grounds are used. VG represents a voltage existing between signal ground and the load ground. If voltage measurements are made between the loadground and the input signal, VS, an erroneous voltage, (VS + VG) is measured. A way tocircumvent this problem is to use a singlepoint ground, as shown in Fig. 2.35d.FIGURE 2.35
Having different grounds can lead to noise in your measurements. The most ideal design is to have all of your loads eventually lead into 1 common ground

For example, in Fig. 2.35a when a load circuit uses a metal enclosure asa chassis ground, resistive leakage paths (unwanted resistive paths) can exist, whichresult in high voltages between the enclosure and earth ground. If, inadvertently, anearthgrounded object, such as a grounded metal pipe, and the circuit chassis aresimultaneously touched, a serious shock will result. To avoid this situation, the chassis is simply wired to an earth ground connection, as shown in Fig. 2.35b. This placesthe metal pipe and the enclosure at the same potential, eliminating the shock hazard.
If the metal that is being used as a ground is not also linked to earth, then it can result in a serious shock

These two terminals are connected together, forming a common return pathfor load current, as shown in Fig. 2.32. The connection between the negative and thepositive terminals of the supplies results in a common or floating return. The floatingcommon may be connected to the earth ground terminal of a supply, if a particularcircuit requests this. Generally, it will neither help nor hinder circuit performance.
This shows the proper way to hookup two DC power supplies and how to incorporate them along with grounding them

As previously mentioned, the ground symbol, in many cases, has been used as ageneric symbol in circuit diagrams to represent the current return path, even thoughno physical earth ground is used. This can be confusing for beginners when theyapproach a threeterminal dc power supply that has a positive (+), negative (−), andground terminal. As we have learned, the ground terminal of the supply is tied to thecase of the instrument, which in turn is wired to the mains earth ground system. Acommon mistake for a novice to make is to attempt to power a load, such as a lamp,using the positive and ground terminals of the supply, as shown in Fig. 2.31a. This,however, doesn’t complete a current return path to the energy source (supply), so nocurrent will flow from the source; hence, the load current will be zero. The correctprocedure, of course, is to either connect the load between the positive and negativeterminals directly, thus creating a floating load, or, using a jumper wire between theground and negative supply, create a grounded load. Obviously, many dc circuits don’tneed to be grounded—it will generally neither help nor hinder performance (e.g.,batterypowered devices need no such connection).Circuits that require both positive and negative voltage require a power supplyto provide each polarity. The supply for the positive voltage will have the negativeFIGURE 2.31
This explains exactly how you would work with a DC power supply
The DC power supply comes with 3 pins (positive, negative, and ground). * The ground can just be connected to the negative terminal of the power supply. Basically, the ground is connected to the case of the power supply which is hooked up to the ground bar in the panel breaker in your house, which is connected to the ground in the earth (in the ground)

The correct definition of an earth ground is usually a connection terminated at a roddriven into the earth to a depth of 8 ft or more. This earth ground rod is wireddirectly to a mains breaker box’s ground bar and sent to the various ac outlets inone’s home via a greencoated or bare copper wire that is housed within the samemains cable as hot and neutral wires. The ground can then be accessed at the outletat the ground socket. Metal piping buried in the earth is often considered an earthground. See Fig. 2.29.
This is what "Earth" ground means

Wiresize is expressed in gauge number—the common standard being the American WireGauge (AWG)—whereby a smaller gauge number corresponds to a largerdiameterwire (high current capacity).
American Wire Gauge (AWG) tells you the size of the gauge of the wire.
The more thinner the wire, the greater the current density since it would involve more collisions. Vice versa for the wire being bigger

For example, Fig. 2.28 shows various ways in which to define voltages by selecting a ground—which in this case is simply a 0V reference marker.
These methods show case how ground can be used in certain examples

Example: Why shouldn’t you connect a wire across a voltage source? For example, ifyou connect a 12gauge wire directly across a 120V source (120V mains outlet), whatdo you think will happen? What will happen when you do this to a 12V dc supply,or to a 1.5V battery?Answer: In the 120V mains case, you will likely cause a huge spark, possibly meltingthe wire and perhaps in the process receiving a nasty shock (if the wire isn’t insulated). But more likely, your circuit breaker in the home will trip, since the wire willdraw a huge current due to its low resistance—breakers trip when they sense a largelevel of current flowing into one of their runs. Some are rated at 10 A, others at 15 A,depending on setup. In a good dc supply, you will probably trip an internal breaker orblow a fuse, or in a bad supply, ruin the inner circuitry. In the case of a battery, there isinternal resistance in the battery, which will result in heating of the battery. There willbe less severe levels of current due to the internal resistance of the battery, but thebattery will soon drain, possibly even destroying the battery, or in an extreme casecausing the battery to rupture.
This explains what would happen if you just connect the positive and negative side of the DC source (battery) or AC Source (wall outlet) together via a wire.
This would mean that the resistance would be 0 ohms which would lead to big damage to the components
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