Lesson#28 Topic: AC Circuits 12/7/06 Objectives: (After this class I will be able to) Explain the difference between AC and DC Describe how alternating current is sinusoidal Describe the three ways to measure voltage in an AC circuit Warm Up: What is the difference between AC and DC? Which is more commonly used? Assignment: Packet p675 #11, 13, 14, 15, 17

Alternating Current Many of our most useful electronic devices rely on AC voltage. The current is continually changing direction. The current and voltage change is sinusoidal with time. We use frequency to represent how quickly the voltage oscillates. AC electricity is a form of simple harmonic motion.

Amplitude of Current and Voltage Measuring actual current or voltage is difficult because it is constantly changing in magnitude and direction. AC voltage varies from Vmax to –Vmax You can use Vmax=ImaxR to solve for Imax Can we use Vavg=IavgR to solve for Iavg? What would Vavg equal? Instead of using max values or average values, we use RMS values.

RMS Values Since the peak voltage only lasts for an instant and the average value is zero, we need to use RMS values. RMS = Root mean squared First square all values to make them all positive Take an average of these squared values. Then square root this average. RMS values are the values closest to the actual voltage or current of the circuit.

AC Practice A voltage of peak value 10V oscillates with a period of 1ms. What is the frequency and angular frequency of the signal? A signal generator is set to produce a voltage with a period of 1s. With what frequency does a light bulb wired in this circuit blink? The amplitude of a sinusoidal signal is 3V What is the RMS value?

Lesson#29 Topic: RLC Circuits 12/8/06 Objectives: (After this class I will be able to) Calculate the power dissipated by an AC circuit Define Impedance Describe new elements that may be found in AC circuits Explain how resistors, inductors, and capacitors affect the overall impedance of an AC circuit. Warm Up: What would be the calculated power dissipated by an AC circuit if you used the average voltage times the average current? Assignment: Packet p675 #12, 16, 18, 19, 20

Power in AC circuits To solve for resistance in a circuit you could use Vmax with Imax or Vrms with Irms Either way will work. But when dealing with energy consumption, or average power dissipated, you need to use RMS values. Pavg = VrmsIrms Other previously derived equations for power can also be used with rms voltage and current.

Power in AC circuits The average power dissipated in a stereo speaker is 55W. Assuming that the speaker can be treated as a resistor with 4ohms resistance, find: The RMS value of the AC voltage applied to the speaker and the RMS value of the AC current through the speaker. The peak value of the AC voltage applied to the speaker and the peak value of the AC current through the speaker.

RLC Circuits With DC circuits we saw that greater voltage caused greater current to flow. The same happens in AC circuits. Certain elements in the circuit cause the current to also depend on the frequency of the applied voltage. We can categorize these elements into three types: Resistors, Capacitors, and Inductors. Each element has a different dependence on frequency.

Impedance Impedance is the same thing as resistance. The resistance of a resistor is the resistor’s impedance. Resistors are independent of frequency. Impedance is represented by the symbol Z. ZR = R Capacitors have impedance that is inversely proportional to frequency ZC ~ 1 / f Inductors have impedance that is directly proportional to frequency ZL~ f

AC Practice A resistor has an impedance of 100ohms when Vmax = 10V and f= 100Hz a. What is Z if f = 1000Hz? b. What is Z if f = 0Hz? c. What is Z if f = infinity?

AC Practice 2. When a rms voltage of 15V is applied to a circuit containing only a capacitor, an rms current of 3.7A is produced. a. What is Zc ? b. What is Zc if f is doubled? c. What is the current if f is doubled? d. What is Zc when f = 0Hz? e. What is Zc when f = infinity?

AC Practice 3. When a rms voltage of 15V is applied to a circuit containing only an inductor, an rms current of 3.7A is produced. a. What is ZL ? b. What is ZL if f is doubled? c. What is the current if f is doubled? d. What is ZL when f = 0Hz? e. What is ZL when f = infinity?

Lesson#30 Topic: Capacitors 12/11/06 Objectives: (After this class I will be able to) Describe how a capacitor works Define and explain capacitance Explain how frequency and capacitance affect the impedance of a capacitor Describe how the surface area and distance between plates of a capacitor will affect its capacitance. Warm Up: What would be the current in a DC circuit with a capacitor wired in series with a 12V battery? Should capacitors be used in AC or DC circuits? Assignment: Packet p691 #1, 2, 4 p675 #30, 31

Capacitor Two conducting plates separated by a thin insulating material. The insulator creates a “gap” in the circuit. Current cannot flow through the gap. Current can flow through the wires for a short time while one plate is “sucked” dry of electrons, and the other plate is being saturated with electrons. Positive charge will accumulate on one side and negative charge on the other. The overall capacitor remains neutral.

Capacitor _ + + + _ I N S U L A T O R _ _ + + Electron Flow Current + + Current _ _ _ + + _

Capacitance Capacitance is the quantity of how much charge can be “stored” on each plate. The larger the capacitance of a capacitor, the larger the “capacity” it has to hold charge. The charge found on one of the plates (Q) is directly proportional to the capacitance of a capacitor (C ). Q ~ C

Capacitance q is the charge of a point or particle. Q is the charge of an object (like a capacitor plate) The amount of charge also depends on the amount of voltage that is applied. The larger the voltage, the more charge that will accumulate. Q ~ V

Capacitance Charge will accumulate and current will flow until the voltage across the capacitor is the same as that across the battery. Though these two proportionalities we can use the equation: Q = CV Q = charge stored on one plate of the capacitor V = Voltage across the capacitor C = Capacitance of the capacitor Capacitance has units of Coulomb per Volt or a “Farad” (F).

Impedance of a capacitor In a DC circuit, a capacitor will quickly fill and then no current will flow (infinite impedance). In an AC circuit, it is possible to oscillate the direction of the current fast enough that the capacitor never fills up. At very high frequencies it is as if the capacitor isn’t even there (zero impedance). The larger the Capacitance, the less likely that it will ever fill up and cause impedance. or

Parallel Plate Capacitors The capacitance of a capacitor depends on its structure. If the plates are large in area, then more charge is able to accumulate on the plates. Capacitance is directly proportional to the area of the plates. If the distance between the plates is large, then the electric force causing charges to separate is weak and little charge will accumulate. Capacitance is inversely proportional to the distance between the plates.

Capacitor Practice When a rms voltage of 15V and 10,000Hz is applied to a circuit containing only a capacitor, an rms current of 3.7A is produced. How much charge can be stored on the capacitor when a 15V steady potential difference is applied across it? A capacitor has a capacitance of 6μF. If the width of the gap is doubled, what happens to the capacitance? What if the area of the plates double also?

Lesson#31 Topic: Total Capacitance and Dielectrics 12/12/06 Objectives: (After this class I will be able to) Add capacitors that are wired in series and in parallel Define dielectrics Describe dielectric constant Explain how the material placed between a parallel plate capacitor affects capacitance. Warm Up: A signal generator is wired in series with a capacitor and has a frequency of 2500Hz, a Vrms = 120V and an Irms= 1.5A What is the capacitance of the capacitor? Assignment: Packet

Equivalent Capacitance Wiring multiple capacitors in parallel is similar to adding surface area to one big capacitor. Additional surface area increases capacitance. Capacitors wired in parallel add directly.

Equivalent Capacitance Wiring multiple capacitors in series is similar to increasing the gap between the plates of one large capacitor. Greater distance between plates will decrease capacitance. Capacitors wired in series add inversely.

Dielectrics The type of insulator placed between the gap of a parallel plate capacitor will affect its capacitance. A vacuum would work as an insulator that would provide the lowest capacitance. Any other insulator would make the capacitance increase. We will refer to the capacitance of a vacuum as Co. C ≥ Co

Dielectrics A vacuum allows charges to easily “feel the presence” of the charges on the opposite plate. An insulator will “dull” this feeling. These different insulators are called dielectrics. Capacitance can be found using: C = κCo κ is known as the dielectric constant that is different for each insulator.

Dielectrics Material Dielectric Constant (κ) Vacuum 1 Air 1.000536 Paper 2 Rubber 2.8 Glass 3.8 Water 80.4

Capacitor Practice Is it possible to create a 1.5μF capacitor from two capacitors of capacitance 1μF and 2μF ? Is it possible to create a 1.5μF capacitor from two capacitors of capacitance 2μF each? Two capacitors are used in series. Each capacitor has a C = 3μF when used with a dielectric with κ=2. If one is used with a dielectric with κ=2 and the other is used with a dielectric with κ=4, a) What is the equivalent capacitance of the circuit? b) What is the equivalent impedance of the circuit? c) What is the equivalent capacitance if wired in parallel with each other? d) What is the equivalent impedance if wired in parallel with each other?

Lesson#32 Topic: Inductors and Stored Energy 12/13/06 Objectives: (After this class I will be able to) Describe how inductors affect impedance of AC circuits Define inductance Explain how energy can be stored in a capacitor or an inductor Warm Up: What is the current through a DC circuit that has a 12V battery wired in series with an inductor? Assignment: Packet p691 #7, 8, 11 p676 #37, 40

Inductors An inductor is a long wire wrapped in the form of a coil. For reasons discussed later in the course, this coil resists the change in current. Inductors cause high impedance at high frequencies and have low impedance at low frequency. The proportionality constant between impedance and frequency is called inductance.

Inductance An inductor is a long wire wrapped in the form of a coil. For reasons discussed later in the course, this coil resists the change in current. Inductors cause high impedance at high frequencies and have low impedance at low frequency. The proportionality constant between impedance and frequency is called inductance (L) and has units of Henry’s (H).

Energy stored in a capacitor The amount of energy stored on a capacitor is the same as the amount of work done to fill the capacitor with charge. If the voltage remained constant then E=QV where Q is the final charge and V is the voltage. However V increases as Q increases, so we have to use an average V, so therefore

Energy stored in an inductor Because inductors resist change in current, an inductor will keep current flowing for a brief time after the voltage has been removed. This is a temporary source of energy. Exactly how an inductor works will be explained in the next unit.

Inductor Practice When a rms voltage of 15V is applied to a circuit containing only an inductor, an rms current of 3.7A is produced. If the frequency was originally 10,000Hz, what is the value of the inductors inductance? What is the energy needed to increase the current from zero up to its maximum value?

Inductor Practice 3. When a rms voltage of 15V and 10,000Hz is applied to a circuit containing only a capicitor, an rms current of 3.7A is produced. a) How work is required to charge the capacitor? b) How much energy is release when the capacitor discharges?

Lesson#33 Topic: Phase Shifts and Total Impedance 12/14/06 Objectives: (After this class I will be able to) Explain what it means for two waves to be “in phase” with one another Describe the terms “leading” and “lagging” and how they apply to phase shifts between voltage and current Describe the phase shifts caused by resistors, capacitors, and inductors Use vector diagrams to solve for total impedance and the angle of phase shift Define Resonance Frequency Warm Up: Explain how a radio uses a combination of capacitors and inductors to eliminate all but one set frequency (channel). Assignment: Packet p691 #13, p692 #15, 17, 19 p677 #65

Current in an AC circuit By wiring a capacitor and an inductor in series with a resistor, we can control current. The capacitor will block out low frequency and the inductor will block out high frequency. This will only allow a small range of frequencies. This is very useful for many common devices (like radios and TV’s)

Phase A capacitor wired with an inductor in series can have lower impedance than just a capacitor or inductor wired alone. You cannot just add the impedances as you would with resistors wired in series. One element cancels out the effect of the other.

Phase As we’ve mentioned, in an AC circuit, current will oscillate at the same frequency as voltage. However, the peak voltages don’t always occur at the same time as the peak current. Current doesn’t oscillate with the same phase as the voltage.

Resistors Resistors are independent of frequency Therefore, the voltage peaks occur at the same time as the current peaks through a resistor. Voltage across a resistor is “in phase” with the current through the resistor. We can then compare the voltage graph across the entire circuit to the voltage graph across just the resistor. This will tell us how much the voltage is “out of phase” with the current.

Capacitors Current through a capacitor leads the voltage across the capacitor. When a capacitor is uncharged there is no potential difference across it. Current needs to flow and charge needs to build up on the capacitor before a maximum potential difference is obtained. Max voltage occurs after max current flows. In fact, max voltage always occurs when current is at zero (analyze capacitor diagram). Max I occurs ¼ of a cycle before max V.

Inductors Current through an inductor lags the voltage across the inductor. Inductors resist change in current. If you change an applied voltage, an inductor will resist that change and therefore delay the corresponding change in current. Max voltage occurs before max current flows. In fact, max voltage always occurs when current is at zero. Max I occurs ¼ of a cycle after max V.

Phase Practice An oscillating voltage is applied to an RLC circuit in series. At which of the following frequencies would you expect the current through the circuit to lead the voltage across the circuit? a) At low frequencies b) At high frequencies c) At some frequency in between

Phase Practice 2. When a rms voltage of 15V is applied to a circuit containing only a capacitor, an rms current of 3.7A is produced. a) Which comes first, the max current or the max voltage? b) Suppose the frequency is 10,000Hz what is the time difference between max current and max voltage?

Phase Practice 3. When a rms voltage of 15V is applied to a circuit containing only an inductor, an rms current of 3.7A is produced. a) Which comes first, the max current or the max voltage? b) Suppose the frequency is 10,000Hz what is the time difference between max current and max voltage?

Phase Angles Phase can be represented with angles. A full period is 2π radians. A ¼ period is π/2 radians. The phase shift for the voltage across a resistor is 0 radians. The phase shift for the voltage across a capacitor is π/2 radians. The phase shift for the voltage across an inductor is -π/2 radians. All three waves need to be combined to find the total phase shift across the entire circuit.

Phase Angles Waves can be transformed into vectors and combined exactly like vectors. The phase angles can be used to draw the direction of the voltage vector for each element on a set of x y axis's. VC VR VL

Phase Angles Combine the vectors and then plot the resultant voltage. The angle between the resultant voltage and the voltage across the resistor is the phase shift of the circuit (θ). Vtotal θ

Resonance There are frequencies where the phase shift is zero. This occurs when the effect of the capacitor cancels out the effect of the inductor and vice versa (because they are π radians out of phase with each other). Vc = VL; this is referred to as the resonance frequency. VC VR VR= Vtotal VL

Phase Angle Practice Derive an equation for Vtotal given VL, Vc, and VR. Derive an equation for θ given the same three variables. An 18V alternating voltage is applied to an RLC circuit containing a 8Ω resistor, a 12mH inductor and a 12μF capacitor in series. Is the potential difference across the voltage generator equal to the sum of the individual voltages across each element?

Phase Angle Practice 4. Derive an equation for Ztotal from your previously derived equation for Vtotal and the definition of impedance. Derive an equation for θ given the impedance of each element in an RLC circuit. 6. An 18V, 75Hz alternating voltage is applied to an RLC circuit containing a 8Ω resistor, a 12mH inductor and a 12μF capacitor in series. a) What is the phase shift of the circuit? b) Does the maximum current come before or after the maximum voltage?

Phase Angle Practice 7. An 18V, 75Hz alternating voltage is applied to an RLC circuit containing a 8Ω resistor, a 12mH inductor and a 12μF capacitor in series. a) Which is greater, the impedance of the resistor, capacitor, or inductor? b) What is the impedance of the circuit? c) Is it possible to find a frequency at which the total impedance is less than 8Ω? 8. Derive an equation for resonant angular frequency from ZC = ZL.

Phase Angle Practice A circuit contains a 1.4μH inductor, a 1.82pF capacitor and a 12Ω resistor in series. a) What is the resonance frequency? b) If such a circuit is used in a radio, does the radio pick up AM signals or FM signals? How do you know?