chapter29

Everyone has power lines coming off the poles that line our streets and that feed their homes with electricity. Instead of direct current (like what is produced from a battery) the power lines carry **Alternating Current **. Alternating current (AC) is produced because of some type of AC generator that operates (usually) by rotating a coil through a magnetic field, thus changing the magnetic flux and generating a changing current as the flux changes. What happens, as a result is that the current produced has variable voltage and current. The electricity produced most commonly follows a sine wave but also can be a square wave or a saw tooth. The most important realization from this is that the voltage and current oscillate between peaks that are positive and negative. The current changes direction on specific intervals. The measure of this interval is **frequency **, which is measured in Hertz (Hz) and measures cycles/second. So if AC is being delivered at 60Hz, the current switches back and forth between its positive peak and its negative peak 60 times in one second. - AC Circuits produce a voltage that alternates in polarity, from positive to negative over time - The current switches direction, i.e. back and forth - AC makes it possible to build electric generators, i.e. motors and power distribution systems, which are far more efficient than direct current (DC)
 * Chapter 29: Alternating Current Circuits **
 * What is Alternating Current? **

Roughly ninety-nine percent (99%) of the electrical energy used today is produced by electrical generators that operate on the basis of alternating currents. Your television, radio, and other household things use alternating current to function. Even those electronics that require direct current must somehow transform the alternating current it is fed to something useable. When we study transformers, we see how that process works. Here’s a way to obtain a deeper and more simplistic understanding of how most appliances and objects work. Alternating Current Presentation and Tutorial [] Alternating Current Java Applet []
 * Why study AC Circuits?! **

** AC Generator **
 AC Generator- a device that //transforms// **mechanical energy** into **electrical energy**. Most AC generators operate by taking advantage of the properties of a magnetic field. By spinning a coil through a constant magnetic field, the magnetic flux changes in turn changing the current that results. Since the coil usually makes complete revolutions, the changing flux and resultant current will change much like the way an object in simple harmonic motion would. The current and voltage will change related to each other and Alternating Current is created.

**Properties of Alternating Current**

We know that when dealing with Alternating Current, the voltage and current oscillate according to some pattern depending on the AC generator and this is shown visually on an oscilloscope and the resulting graph is called the waveform. Just like Simple Harmonic Motion and Sound, the current has a **frequency** (f), cycles per second/Hz, and **wavelength** (lambda), distance per wave/meters, that can be determined from looking at the graph or doing calculations using (lambda = c/f). Later, when we talk about resistor, capacitors, and inductors and how they act in AC circuits, we will see that the voltage and current curves will have a horizontal shift from one another. The horizontal distance (easily measured from the distance between peaks on each graph) is called the phase angle. This is the time that it is said that the voltage lags or leads the current. This will make more sense when we come back to it later. When looking at Alternating Current, time plays a crucial role in determining calculations because the voltage and current are always changing. As is normal, scientists wanted to find a way to compare AC to DC. The way that the two are compared is by their “effective” voltage and current. The **Root Mean Square (RMS)** voltage is considered the effective voltage or the voltage that an equivalent DC power source would supply. RMS Voltage is not the average value (which would actually be zero over an infinite time interval) but rather a value calculated by (square root (integral of V squared from 0 to n divided by n)). This turns out to be that the RMS Voltage is just the Peak Voltage divided by the square root of 2 (or times 0.707). RMS can be useful in calculations within circuits and is necessary to compare AC and DC circuits.

**Ohm’s Law**, I=E/R, still applies in AC circuits. When determining values though, you must keep them all relative to each other. If you are determining RMS voltage, you must use RMS current. For other calculations you might use peak values, average values, or peak-to-peak values.

<span style="font-family: Georgia,serif; font-size: 12pt;">Finally, we should discuss the relationship between reactance and phase in AC circuits. **Reactance** is the opposition to change in current that is present in both capacitors and inductors. Reactance is measured in ohms, and has the variable X**. In capacitors, reactance (XC) increases as the frequency decreases (closer to DC). In inductors, reactance (XL) increases as the frequency increases (very fast changing AC).** These statements will be further explained when we begin to talk about circuits but for reference this is a good place to look back to. Like we said, reactance is the opposition to change in current. Because of this opposition, current and voltage will not peak at the same moment and this is where the concept of **phase** is brought in. In an AC circuit with a resistor, there is no reactance so the peaks will occur at the same time and the two waves (current and voltage) are considered to be **in phase.** When reactance comes into play, the voltage will be affected (either sped up or slowed down) by the reactance. The voltage may lag or lead the current, meaning the two curves are **out of phase**. I**n capacitors the voltage leads the current while in inductors voltage lags the current**. All of this may still seem confusing to you, but don’t worry! We will now move into the behavior of components in AC and finally AC circuits where we will further explain these concepts and they will make more sense in context.


 * Resistors, Capacitors, and Inductors in AC **

<span style="color: black; font-family: Georgia,serif; font-size: 12pt;">When we deal with resistors in AC we can just think of them in the same way as they act in DC when determining basic values using the RMS or effective values. As time changes through, the voltage and current will change as related to trigonometric functions alternating between positive and negative peaks. In essence, the voltage and current switch directions. Because of this, the power generated in the resistor will be constantly changing.
 * <span style="color: blue; font-family: Georgia,serif; font-size: 14pt;">Resistors **



<span style="font-family: Georgia,serif;">- voltage across a resistor is proportional to current(i ) <span style="font-family: Georgia,serif;">through a resistor <span style="font-family: Georgia,serif; font-size: 12pt;">- peak value for voltage is “R” times the peak value of the current <span style="font-family: Georgia,serif; font-size: 12pt;">- voltage and current are in phase- when current is at a maximum so is voltage <span style="background-color: #00ffff; font-family: Georgia,serif; font-size: 12pt;">//Watch this animation on current and voltage as functions of time:// media type="file" key="rc_circuits2.swf" width="468" height="351" <span style="font-family: Georgia,serif; font-size: 12pt;">The vector representation of phase at the right, indicated by rotating is called the phasor diagram. The vectors, or phasors, representing the current and voltage across the resistor rotate with angular velocity ω with respect to the x- and y-axes. The lengths of these vectors represent the peak current Im and voltage V<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">m.

<span style="font-family: Georgia,serif;">The y-components are: current, i(t) = I<span style="font-family: Georgia,serif; vertical-align: sub;">m sin (ωt) and <span style="font-family: Georgia,serif; line-height: 22px;">voltage, v(t) = V<span style="font-family: Georgia,serif; line-height: 22px; vertical-align: sub;">m sin (ωt) <span style="font-family: Georgia,serif; font-size: 12pt;">You can compare i(t) and v(t) in the animation with the x-components of the vectors.

<span style="font-family: Georgia,serif; font-size: 12pt; line-height: 0px; overflow-x: hidden; overflow-y: hidden;">
 * <span style="color: navy; font-family: Georgia,serif; font-size: 14pt;">Capacitors **

<span style="color: black; font-family: Georgia,serif;">As we know, capacitors charge and discharge in AC and create an electric field inside of itself. In DC, a capacitor will charge to its maximum capacity and then the current flow will cease. In AC, since the current is constantly changing (and optimally at high speeds), the capacitor will charge, then discharge and go back and forth as a result of the current. The current won’t stop anymore and there will be a variable electric field created inside the capacitor. This is a better explanation of why a capacitor rejects change in current – also known as Capacitive Reactance X<span style="color: #000000; font-family: Georgia,serif; vertical-align: sub;">c. <span style="font-family: Georgia,serif;">Capacitance reactance (X<span style="font-family: Georgia,serif; vertical-align: sub;">c ) is the ratio of magnitude of the voltage to the magnitude of the current in a capacitor. <span style="color: black; font-family: Georgia,serif;">The following two equations show us what affects X<span style="color: #000000; font-family: Georgia,serif; vertical-align: sub;">c :

<span style="color: black; font-family: Georgia,serif; font-size: 12pt;">Voltage and Current affect reactance. Reactance is very similar to resistance (V=I/R)

<span style="font-family: Georgia,serif; font-size: 12pt;">- Voltage depends on the amount of charge stored on its plates <span style="font-family: Georgia,serif; font-size: 12pt;">- Charge(Q) is determined by the amount of current flowing from the positive plate to the negative plate <span style="font-family: Georgia,serif; font-size: 12pt;">
 * <span style="font-family: Georgia,serif; font-size: 12pt;">Q is equal to the integral of current with respect to time



<span style="color: black; font-family: Georgia,serif; font-size: 12pt;"> <span style="color: black; font-family: Georgia,serif; font-size: 12pt;">Reactance is affected inversely by frequency and capacitance. As frequency increases, reactance decreases. As capacitance increases, reactance decreases.

<span style="color: black; font-family: Georgia,serif; font-size: 12pt;">Because of this reactance, the current will become out of phase with the voltage. We say that the voltage lags the current or the current leads the voltage. This is shown by the following graph:

<span style="color: black; font-family: Georgia,serif; font-size: 12pt;">Finally, we know that electric field is the derivative of voltage so, given the voltage curve from above, we can determine how the electric field inside the capacitor changes.

<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px;">1. There is a difference in the phase <span style="background-color: #00ffff; font-family: Georgia,serif; font-size: 12pt;">//Watch this animation:// <span style="font-family: Georgia,serif; font-size: 12pt;">media type="file" key="cap1.swf" width="371" height="367" <span style="font-family: Georgia,serif; font-size: 12pt;">Look at the relative phase, the voltage across the capacitor is 90° behind the current. You can also look at it as one quarter cycle behind the current. Also notice, the 90° phase difference affects the phasor diagrams at right. The blue arrow represents the voltage and the black arrow represents the blue arrow. The phasors are rotating in the positive direction so the phasor representing V<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">C is 90° //behind// the current.
 * <span style="font-family: Georgia,serif;">Difference between reactance and resistance **

<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px;">2. Reactance is frequency dependent <span style="font-family: Georgia,serif; font-size: 12pt;">Capacitive reactance decreases with frequency. <span style="background-color: #00ffff; font-family: Georgia,serif; font-size: 12pt;">//Watch this animation:// media type="file" key="cap2.swf" width="365" height="365" <span style="font-family: Georgia,serif; font-size: 12pt;">Notice, the frequency is halved but the current amplitude is kept constant. This allows the capacitor to have twice as long to charge up, in turn generating twice the voltage. The blue shading shows charge, the integral under the current curve; the light blue represents positive and the dark blue represents negative. Observe the bigger shaded region before changing sign which shows how the lower frequency leads to a larger charge and therefore a larger voltage.

<span style="color: black; font-family: Georgia,serif; font-size: 12pt;">From previous chapters, we know that inductors create a magnetic field. Inductors, like capacitors, have reactance but here we call it inductive reactance. Inductors resist changes in current more at higher frequencies (AC) and less at lower frequencies (DC). Inductors consist of a coil of wire in which the resistance is insignificant. In inductors we will relate induced voltage and current (remembering that a changing magnetic field creates a voltage). <span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px;">Current sets up a magnetic field which is proportional to the current flowing. The magnetic field’s magnetic flux (φ<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px; vertical-align: sub;">B ) is proportional to the field strengt h. <span style="font-family: Georgia,serif; font-size: 12pt;">Inductive reactance (XL) is the ratio of magnitudes of the voltage to current.
 * <span style="color: #99ccff; font-family: Georgia,serif; font-size: 14pt;">Inductors **

<span style="color: black; font-family: Georgia,serif; font-size: 12pt;">The relationships for calculating inductive reactance XL are:

<span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif; font-size: 12pt;">where "v" is <span style="color: black; font-family: Georgia,serif; font-size: 12pt;">Voltage and current affect reactance. Reactance and resistance are similar (V=I/R)

<span style="color: black; font-family: Georgia,serif; font-size: 12pt;"> <span style="color: black; font-family: Georgia,serif; font-size: 12pt;">Reactance increases at higher frequencies and higher inductance increases reactance

<span style="font-family: Georgia,serif; font-size: 12pt;">Because the magnetic field changes as related as a result of alternating current and higher reactamce, the induced voltage will lag the current and be out of phase. When dealing with the applied voltage though there are new phase angles. Applied voltage is the exact opposite of induced voltage and therefore 180 degrees out of phase. Also, the applied voltage will lead the current.

<span style="background-color: #00ffff; font-family: Georgia,serif; font-size: 12pt;">//Watch the next animation (pay attention to why the voltage and current are out of phase in an inductor) :// <span style="font-family: Georgia,serif; font-size: 12pt; line-height: 0px; overflow-x: hidden; overflow-y: hidden;">﻿media type="file" key="ind1.swf" width="363" height="363" <span style="font-family: Georgia,serif; font-size: 12pt; line-height: 24px; overflow-x: hidden; overflow-y: hidden;">The voltage is 90<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px; vertical-align: super;">o ahead of the current in the phasor diagram and reaches its peak one quarter of a cycle before the current in the graphs.

<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px;">Similar to capacitors, reactance is frequency dependent in inductors <span style="background-color: #00ffff; font-family: Georgia,serif; font-size: 12pt;">//Watch this animation:// <span style="font-family: Georgia,serif; font-size: 12pt;">media type="file" key="ind2.swf" width="387" height="379" <span style="font-family: Georgia,serif; font-size: 12pt;">Notice, the frequency is halved while the current amplitude kept constant. For an inductor, the ratio of voltage to current increases with frequency.

<span style="font-family: Georgia,serif; font-size: 12pt;">Inductive Reactance Java Applet http://www.magnet.fsu.edu/education/tutorials/java/inductivereactance/index.html


 * Summary of Resistors, Capacitors, and Inductors **

Impedance= voltage/current - resistance and reactance are special cases with phase shifts - NOTE: It is easy to remember the voltage is behind the current in a capacitor because the charge does not build up until after the current has been flowing for a while. <span style="font-family: Georgia; font-size: 12pt; line-height: 0px; margin-left: 0.5in; overflow-x: hidden; overflow-y: hidden; text-indent: -0.25in;">

What is a **RC series** combination?! It is simply a resistor and capacitor in series. That is what “R” and “C” stand for. Voltage in an RC series combination is the sum of the voltages on the resistor and capacitor. <span style="font-family: Georgia; line-height: 0px; margin-left: 0.25in; overflow: hidden; text-indent: -0.25in;"> <span style="font-family: Georgia,serif; font-size: 12pt;">This equation looks simple, but due to the fact that resistors and capacitors are not in phase makes it a little more challenging! So keep in mind that the voltage series is //less than// the sum of the resistor and capacitors voltage. <span style="background-color: #00ffff; font-family: Georgia,serif; font-size: 12pt;">//Watch this animation:// <span style="font-family: Georgia,serif; font-size: 12pt;">media type="file" key="rc_circuits2.swf" width="375" height="364" <span style="font-family: Georgia,serif; font-size: 12pt;">Pay particular attention to the phasor diagrams on the right.
 * RC, RL, and RLC Series Combinations **

<span style="font-family: Georgia,serif; font-size: 12pt;">The series impedance ("Z"<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">RC ) is frequency dependent. This is very important in applications of RC Circuits! <span style="font-family: Georgia,serif; font-size: 12pt;"> <span style="font-family: Georgia,serif; font-size: 12pt;">The voltages must add up to give the total voltage of the source; it is in the form of vectors. <span style="font-family: Georgia,serif; font-size: 12pt;">- Filters are used for very important applications of RC Circuits
 * <span style="font-family: Georgia,serif; font-size: 12pt;">RC circuits work as high-pass or low-pass filters for sound waves
 * <span style="font-family: Georgia,serif; font-size: 12pt;">High-pass frequency – // passes high // frequencies and // rejects low // frequencies
 * <span style="font-family: Georgia,serif; font-size: 12pt;">Low-pass frequency- // passes low // frequencies and // rejects high // frequencies

<span style="font-family: Georgia,serif; font-size: 12pt;">What is a **RL series** combination?! <span style="font-family: Georgia,serif; font-size: 12pt;">It is a resistor and inductor in series. <span style="font-family: Georgia,serif; font-size: 12pt;"> <span style="font-family: Georgia,serif; font-size: 12pt;">- The voltage across the inductor is 90<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: super;">o ahead of the current. <span style="font-family: Georgia,serif; font-size: 12pt;">- Inductive Reactance is still: <span style="font-family: Georgia,serif; font-size: 12pt;">- Below are the graphs of voltage and phasor diagram <span style="font-family: Georgia,serif; font-size: 12pt;">

<span style="font-family: Georgia,serif; font-size: 12pt;">What are **RLC series** combinations?! <span style="font-family: Georgia,serif; font-size: 12pt;">A resistor, inductor, and capacitor in series

<span style="font-family: Georgia,serif; font-size: 12pt;">Voltage across the series is the sum of the voltage across the resistor, inductor, and capacitor <span style="font-family: Georgia,serif; font-size: 12pt;"> <span style="font-family: Georgia,serif; font-size: 12pt;">The current is kept sinusoidal <span style="font-family: Georgia,serif; font-size: 12pt;">Voltage across the resistor is in phase with the current <span style="font-family: Georgia,serif; font-size: 12pt;">Voltage across the inductor is 90<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: super;">o //ahead// of the current <span style="font-family: Georgia,serif; font-size: 12pt;">Voltage across the capacitor is 90<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: super;">o //behind// the current

<span style="background-color: #00ffff; font-family: Georgia,serif; font-size: 12pt;">//Watch this RLC series combination animation:// media type="file" key="resonance2.swf" width="360" height="338" <span style="font-family: Georgia,serif; font-size: 12pt;">Notice that the time-dependent voltages always add up, BUT the RMS voltages V do not add up. They can be added by the using phasors, like the ones on the right.

<span style="font-family: Georgia,serif; font-size: 12pt;">Look at this phasor diagram representation:

This diagram helps to show that if you are confused, just think of total voltage as vector addition

<span style="font-family: Georgia,serif; font-size: 12pt;">The voltage across the inductor (v<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">L ) is antiparallel to the voltage across the capacitor (v<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">C ); so the total reactive voltage is v<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">L - v<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">C <span style="font-family: Georgia,serif; font-size: 12pt;">The dependence of Z<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">series and φ on the angular frequency ω is shown in the graphs below. Angular frequency is in terms of a value ω<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">o known to be the **resonant frequency**. <span style="font-family: Georgia,serif; font-size: 12pt;"> <span style="display: block; font-family: Georgia,serif; font-size: 12pt; text-align: left;">Special case where frequency is such that V<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">L =V<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">C <span style="font-family: Georgia,serif; font-size: 12pt;"> <span style="font-family: Georgia,serif; font-size: 12pt;">V<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">L and V<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">C are out of phase by 180<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: super;">o, therefore V<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">L = -V<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">C which means they cancel out as you can see by looking at the phasor diagram. Additionally, the series voltage is equal to the voltage across the resistor. This is known as **series resonance**.

<span style="font-family: Georgia,serif; font-size: 12pt;">A transformer is a device used to raise or lower the voltage in a circuit without an appreciable loss of power or energy. There are two types of transformers, step up and step down. Step up transformers //increase// voltage; conversely, step down transformers //reduce// voltage. <span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px;">It is an iron core on which are wound a primary coil of N<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px; vertical-align: sub;">p (N<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px; vertical-align: sub;">1 ) turns and a secondary coil of N<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px; vertical-align: sub;">s (N<span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px; vertical-align: sub;">2 ) turns. If the primary coil is connected across an alternating-current generator, the primary and secondary voltages are related by <span style="font-family: Georgia,serif; font-size: 12pt;">The currents through the coils are also related by <span style="font-family: Georgia,serif; font-size: 12pt;">The equivalent resistance of the secondary circuit (N<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">s ), as seen by a generator, is
 * Transformers **
 * <span style="font-family: Georgia,serif; font-size: 12pt;">“R” is the restrictive load in the secondary circuit
 * <span style="font-family: Georgia,serif; font-size: 12pt;">The ratio N<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">p /N<span style="font-family: Georgia,serif; font-size: 12pt; vertical-align: sub;">s is called the transformer’s // turns circuit //

<span style="font-family: Georgia,serif; font-size: 16px;">Transformer Java Applet- watch how the primary and secondary winding affect the output volatge <span style="background-color: transparent; color: #000000; display: block; font-family: 'Times New Roman'; font-size: 16px; text-align: left; text-decoration: none;">[] <span style="font-family: Georgia,serif; font-size: 12pt;">**__A real world example of a transformer:__**

<span style="font-family: Georgia,serif; font-size: 12pt;">Transformers are used every day, take on a telephone pole for instance. Power is supplied to houses all over the world through power grids. The voltage can be as high as 765,000V. This power is decreased by a step down transformer to 72,000V at a local substation. From here, the power is additionally stepped down to roughly 220V at a transformer on a telephone pole. The voltage has to be very great in the beginning to withstand traveling long distances. It is stepped down so significantly by a transformer to be used in houses.

<span style="font-family: Georgia,serif; font-size: 12pt;">For a more detailed explanation of a transformer on a telephone pole visit: <span style="font-family: Georgia,serif; font-size: 12pt;">[]

<span style="font-family: Georgia,serif; font-size: 12pt;">If you understand simple harmonic motion, below is an analog between SHM and AC Circuits. <span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px;">RMS Lab <span style="font-family: Georgia,serif; font-size: 12pt;">[]
 * <span style="color: #0f0a85; font-family: Georgia,serif; font-size: 17.5pt;">Still a little confused?! **
 * <span style="color: #5157b8; font-family: Georgia,serif; font-size: 17.5pt;">Lab **

Here are some practice problems from the University of Vermont that we like. Give them a try. See how you do. []
 * <span style="color: #2a2877; font-family: Georgia,serif; font-size: 17.5pt;">Problems/Examples: **

<span style="font-family: Georgia,serif; font-size: 12pt;">Although not common, there is the possibility that the AP E&M exam could ask a question about RLC circuits, like this one from 2011 : []

<span style="font-family: Georgia,serif; font-size: 12pt; margin-top: 0in;">1. As the frequency in the simple AC circuit below increases, the rms current through the resistor… <span style="font-family: Georgia,serif; font-size: 12pt; line-height: 0px; margin-left: 1in; overflow: hidden; text-indent: -0.25in;"> <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;"> <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(a) increases <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(b) does not change <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(c) may increase or decrease depending on the magnitude of the original frequency <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(d) may increase or decrease depending on the magnitude of the resistance <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(e) decreases

<span style="font-family: Georgia,serif; font-size: 12pt; margin-top: 0in;">2. If the frequency in the circuit below is doubled, the inductance of the inductor will <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;"> <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(a) increase by a factor of 2 <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(b) not change <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(c) decrease by a factor of 2 <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(d) increase by a factor of 4 <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(e) decrease by a factor of 4

<span style="font-family: Georgia,serif; font-size: 12pt; margin-top: 0in;">3. If the frequency in the circuit below is doubled, the capacitive reactance of the circuit will <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(a) increase by a factor of 2 <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(b) not change <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(c) decrease by a factor of 2 <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(d) increase by a factor of 4 <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;">(e) decrease by a factor of 4 <span style="font-family: Georgia,serif; font-size: 12pt; margin-left: 1in; text-indent: -0.25in;"> <span style="font-family: Georgia,serif; font-size: 12pt; margin-top: 0in;">4. A 1.0 μF capacitor is connected in series with a 60 Hz AC voltage source. If the Vrms = 120 V, find the peak and rms current. 5. A coil, with an inductance of 0.300 H, is placed in series with a 60 Hz AC, 120 V (rms) voltage source. Determine the rms current <span style="font-family: Georgia,serif; font-size: 12pt;">6. Analyze a series RLC circuit for which R = 250 Ω, L = 0.600 H, C = 3.50 μF, f = 60 Hz, and Vrms = 150 V. (//Hint:// Start by finding XL and XC, than impedance, rms current, phase angle, and voltages) <span style="font-family: Georgia,serif; font-size: 12pt;">

<span style="font-family: Georgia,serif; font-size: 12pt;">Sources: <span style="font-family: Georgia,serif; font-size: 12pt;">Barron's AP Physics C: 2008: 2nd Edition []<span style="background-color: #ffffff; font-family: Georgia,serif; font-size: 12pt;">- majority of the information and all of the animations came from this site! <span style="font-family: Georgia,serif; font-size: 16px; line-height: 24px;">Understanding Basic Electronics: 1st Edition - published by the ARRL // [|www.euclidstube.com/physics/Units/Unit%2022/unit22.pdf] // <span style="font-family: Georgia,serif; font-size: 12pt;">// ﻿ // PHYSICS: For Scientists and Engineers. 5th Edition


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