Alternating Current Circuits: Phasor Diagrams & Resonance
Learn about AC circuits, phasor diagrams, resistance, reactance, resonant circuits, power factor, and practical applications in speaker systems. Includes worked examples and comparison of capacitors and inductors.
Alternating Current Circuits: Phasor Diagrams & Resonance
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Presentation Transcript
28.1. Alternating Current Reminder: All waves can be analyzed in terms of sinusoidal waves (Fourier analysis Chap 14). Sinusoidal wave (Chap 13) : Vp sin Angular frequency : [] = rad/s = / 6 = phase
Example 28.1. Characterizing Household Voltage Standard household wiring supplies 110 V rms at 60 Hz. Express this mathematically, assuming the voltage is rising through 0 at t = 0.
28.2. Current Elements in AC Circuits • Resistors • Capacitors • Inductors • Phasor Diagrams • Capacitors & Inductors: A Comparison
Displacing Functions g g is moved to the right (forward) by to give f. f x x cos Displacement: sin is cos moved forward by /2. Phase: sin lags cos by /2. sin Derivative: moves sinusoidal functions backward by /2. phase is increased by /2. Integral: moves sinusoidal functions forward by /2. phase is decreased by /2.
Resistors I + VR + I & V in phase
Capacitors When V(t) > 0 : I + + VC I leads V by 90 I peaks ¼ cycle before V Capacitive reactance DC: open ckt. HF: short ckt.
Inductors When V(t) > 0 : I L + + I trails V by 90 I peaks ¼ cycle after V Inductive reactance DC: short ckt. HF: open ckt.
Table 28.1. Amplitude & Phase in Circuit Elements Resistor V & I in phase Capacitor V lags I 90 V leads I 90 Inductor
Example 28.2. Equal Currents? • A capacitor is connected across a 60-Hz, 120-V rms power line, • and an rms current of 200 mA flows. • Find the capacitance. • What inductance, connected across the same powerline, • would result in the same current? • (c) How would the phases of the inductor & capacitor currents compare? (a) (b) Capacitor: ICleads V by 90. Inductor: V leads ILby 90. (c) ICleads ILby 180.
Phasor Diagrams Phasor = Arrow (vector) in complex plane. Length = mag. Angle = phase. V leads I by 0. ( same phase ) V leads I by 90. V leads I by 90. ( V lags I by 90 )
Capacitors Revisited I + VC Vp e i t I leads V by 90 Taking the real part as physical Taking the imaginary part as physical Impedance
Inductors Revisited I L + Vp e i t I lags V by 90 Taking the real part as physical Taking the imaginary part as physical
Capacitors & Inductors: A Comparison C L translator: E B q B V I Z Y
Table 28.2. Capacitors & Inductors Defining relation Defining relation;differential form Opposes change in V I Energy storage Open circuit Short circuit Behavior in low freq limit Short circuit Open circuit Behavior in high freq limit Reactance Admittance / Impedance V leads by 90 Phase I leads by 90
Application: Loudspeaker Systems Loudspeaker C passes High freq Loudspeaker system with high & low frequency filters. L passes low freq
28.3. LC Circuits I V +
Analyzing the LC Circuit I V +
Resistance in LC Circuits – Damping + VR I L + VC + (see next page)
Resistance in LC Circuits – Damping + VR I VC + L + (see next page)
Solutions to Damped Oscillator Ansatz:
28.4. Driven RLC Circuits & Resonance + VR I + L + VC + Driven damped oscillator : Long time: oscillates with frequency d. Resonance if d =0.
Resonance in the RLC Circuit VC& VL are 180 out of phase. i.e., if
Frequency Response of the RLC Circuit Series circuit same I phasor for all. VR in phase with I. VC lags I by 90. VL leads I by 90. High Q Low Q See Prob 71 for definition of Q. At resonance, = 0.
Example 28.4. Designing a Loud Speaker System • Current flows to the midrange speaker in a loudspeaker system through a 2.2-mH inductor in series with a capacitor. • What should the capacitance be so that a given voltage produces the greatest current at 1 kHz ? • If the same voltage produces half this current at 618 Hz, • what is the speaker’s resistance ? • If the peak output voltage of the amplifier is 20 V, • what will be the peak capacitor voltage be at 1 kHz ? (a) Greatest I is at resonance:
(b) If the same voltage produces half this current at 618 Hz, what is the speaker’s resistance ? At resonance: • If the peak output voltage of the amplifier is 20 V, • what will be the peak capacitor voltage be at 1 kHz ? Peak voltage is at resonance (1 kHz).
28.5. Power in AC Circuits Capacitor: I leads V by 90 , P = 0 Resistor: I & V in phase , P > 0 I & V out of phase , P Power factor Dissipative power = I2 R large power factor reduces I & hence heat loss.
Conceptual Example 28.1. Managing Power Factor You’re chief engineer of a power company. Should you strive for a high or a low power factor on your lines? Power factor Generator : fixed Vrms . To maintain fixed <P>, Irms cos = const. Smaller power factor higher Irms. higher power loss. Ans.: keep power factor close to 1.
Making the Connection Transmission losses on a well-managed electric grid average about 8% of the total power delivered. How does this figure change if the power factor drops from 1 to 0.71? To deliver the same power Transmission losses: ( doubles to 16% )
28.6. Transformers & Power Supplies Transformer: pair of coils wound on the same (iron) core. Works only for AC.
Direct-Current Power Supplies Diode passes + half of each cycle Diode Diode cuts off half of each cycle RC (low freq) filter
