Chapter 12AC SourcesSimple AC circuitsPurely Resistive loadPurely Inductive LoadPurely Capacitive LoadThe RLC Series CircuitImpedanceResonancePower in an AC circuitWidth of the PeakTransformerParallel RLC CircuitSummaryProblem-Solving TipsSolved ProblemsRLC Series CircuitResonanceRL High-Pass FilterRLC CircuitRL FilterConceptual QuestionsAdditional ProblemsReactance of a Capacitor and an InductorDriven RLC Circuit Near ResonanceRC CircuitParallel RL CircuitLC CircuitParallel RC CircuitPower DissipationFM AntennaDriven RLC CircuitChapter 12 Alternating-Current Circuits 12.1 AC Sources .............................................................................................................1 12.2 Simple AC circuits..................................................................................................2 12.2.1 Purely Resistive load........................................................................................2 12.2.2 Purely Inductive Load......................................................................................4 12.2.3 Purely Capacitive Load....................................................................................6 12.3 The RLC Series Circuit...........................................................................................8 12.3.1 Impedance......................................................................................................11 12.3.2 Resonance ......................................................................................................12 12.4 Power in an AC circuit..........................................................................................13 12.4.1 Width of the Peak...........................................................................................15 12.5 Transformer ..........................................................................................................16 12.6 Parallel RLC Circuit..............................................................................................18 12.7 Summary...............................................................................................................21 12.8 Problem-Solving Tips...........................................................................................23 12.9 Solved Problems ...................................................................................................25 12.9.1 RLC Series Circuit .........................................................................................25 12.9.2 RLC Series Circuit .........................................................................................26 12.9.3 Resonance ......................................................................................................27 12.9.4 RL High-Pass Filter........................................................................................28 12.9.5 RLC Circuit....................................................................................................29 12.9.6 RL Filter .........................................................................................................32 12.10 Conceptual Questions .........................................................................................34 12.11 Additional Problems ...........................................................................................35 12.11.1 Reactance of a Capacitor and an Inductor ...................................................35 12.11.2 Driven RLC Circuit Near Resonance...........................................................35 12.11.3 RC Circuit ....................................................................................................36 12.11.4 Black Box.....................................................................................................36 12.11.5 Parallel RL Circuit........................................................................................37 12.11.6 LC Circuit.....................................................................................................38 12.11.7 Parallel RC Circuit.......................................................................................38 12.11.8 Power Dissipation........................................................................................39 12.11.9 FM Antenna .................................................................................................39 12.11.10 Driven RLC Circuit....................................................................................40 0Alternating-Current Circuits 12.1 AC Sources In Chapter 10 we learned that changing magnetic flux can induce an emf according to Faraday’s law of induction. In particular, if a coil rotates in the presence of a magnetic field, the induced emf varies sinusoidally with time and leads to an alternating current (AC), and provides a source of AC power. The symbol for an AC voltage source is An example of an AC source is 0() sinVt V tω= (12.1.1) where the maximum valueV is called the amplitude. The voltage varies between and since a sine function varies between +1 and −1. A graph of voltage as a function of time is shown in Figure 12.1.1. 00V0V− Figure 12.1.1 Sinusoidal voltage source The sine function is periodic in time. This means that the value of the voltage at time t will be exactly the same at a later time ttT′=+ where T is the period. The frequency, f, defined as 1/fT=, has the unit of inverse seconds (s−1), or hertz (Hz). The angular frequency is defined to be 2fωπ=. When a voltage source is connected to an RLC circuit, energy is provided to compensate the energy dissipation in the resistor, and the oscillation will no longer damp out. The oscillations of charge, current and potential difference are called driven or forced oscillations. After an initial “transient time,” an AC current will flow in the circuit as a response to the driving voltage source. The current, written as 10( ) sin( )It I tωφ=− (12.1.2) will oscillate with the same frequency as the voltage source, with an amplitude 0I and phase φ that depends on the driving frequency. 12.2 Simple AC circuits Before examining the driven RLC circuit, let’s first consider the simple cases where only one circuit element (a resistor, an inductor or a capacitor) is connected to a sinusoidal voltage source. 12.2.1 Purely Resistive load Consider a purely resistive circuit with a resistor connected to an AC
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