#### Transcript AC Circuits - GTU e

AC CIRCUITS Enrollment no.: 130010111001- Abhi P. Choksi 130010111051-Anuj Watal 130010111023- Esha N. Patel Guidied by: M. K. Joshi, P.R.Modha A.D.PATEL.INSTITUTE OF TECHNOLOGY NEW V.V.NAGAR AC Circuits Capacitive Reactance Phasor Diagrams Inductive Reactance RCL Circuits Resonance Resistive Loads in AC Circuits Ohm’s Law: V Vrms I I rms R R Vt V0 sin 2ft Vt V0 I t sin 2ft I 0 sin 2ft R R R is constant – does not depend on frequency No phase difference between V and I Capacitive Reactance At the moment a capacitor is connected to a voltage source: I0 V + - Current is at its maximum Voltage across capacitor is zero + C V=0 - Capacitive Reactance After a long time, the capacitor is charged: V + - + C V - Current is zero Voltage is at its maximum (= supply voltage) Capacitive Reactance Now, we reverse the polarity of the applied voltage: I V + + - + C Current is at its maximum (but reversed) Voltage hasn’t changed yet V - Capacitive Reactance Apply an AC voltage source: I(t) = I0sin(2ft+/2) V(t) = V0sin(2ft) + an AC current is present in the circuit a 90° phase difference is found between the voltage and the current C Capacitive Reactance We want to find a relationship between the voltage and the current that we can use like Ohm’s Law for an AC circuit with a capacitive load: Vrms I rms X C We call XC the capacitive reactance, and calculate it as: 1 XC 2fC units of capacitive reactance: ohms (W) Capacitive Reactance A particular example: V0 = 50 V f = 100 Hz - C = 750 mF + 1 1 XC 2.12 W -4 2fC 2 100 Hz 7.5 10 F Capacitive Reactance Power is zero each time either the voltage or current is zero Power is positive whenever V and I have the same sign Power is negative whenever V and I have opposite signs Power spends equal amounts of time being negative and positive Average power over time: zero Capacitive Reactance X C 1 2fC The larger the capacitance, the smaller the capacitive reactance As frequency increases, reactance decreases DC: capacitor is an “open circuit” and XC high frequency: capacitor is a “short circuit” andX 0 C Phasor Diagrams Consider a vector which rotates counterclockwise withan speed angular 2f : V V0 sin(2ft) This vector is called a “phasor.” It is a visualization tool. V0 t=2ft Phasor Diagrams For a resistive load: the current is always proportional to the voltage V, I V0 sin(2ft) I0 sin(2ft) I0 = V0 / R V0 voltage phasor t=2ft current phasor Phasor Diagrams For a capacitive load: the current “leads” the voltage by /2 (or 90°) V, I current phasor V0 sin(2ft) I0 sin(2ft + /2) voltage phasor t=2ft Inductive Reactance A coil or inductor also acts as a reactive load in an AC circuit. AC Inductive Reactance As the current increases through zero, its time rate of change is a maximum – and so is the induced EMF I EMF L t AC Inductive Reactance As the current reaches its maximum value, its rate of change decreases to zero – and so does the induced EMF I EMF L t AC Inductive Reactance The inductive reactance is the Ohm’s Law constant of proportionality: Vrms I rms X L AC X L 2fL units of inductive reactance: ohms (W) Inductive Reactance The voltage-current relationship in an inductive load in an AC circuit can be represented by a phasor diagram: V, I V0 sin(2ft) voltage phasor t=2ft current phasor I0 sin(2ft - /2) Inductive Reactance X L 2fL Larger inductance: larger reactance (more induced EMF to oppose the applied AC voltage) Higher frequency: larger impedance (higher frequency means higher time rate of change of current, which means more induced EMF to oppose the applied AC voltage) RLC Circuit Here is an AC circuit containing series-connected resistive, capacitive, and inductive loads: R C AC L The voltages across the loads at any instant are different, but a common current is present. RCL Circuit The current is in phase with voltage in the resistor. VL I The capacitor voltage trails the current; the inductor voltage leads it. We want to calculate the entire applied voltage from the generator. VC VR RCL Circuit We will add the voltage phasors as vectors (which is what they are.) We start out by adding the reactive voltages (across the capacitor and the inductor). This is easy because those phasors are opposite in direction. The resultant’s magnitude is the difference of the two, and its direction is that of the larger one. VR VL - V C I RLC Circuit Now we use Pythagoras’ Theorem to add the VL – VC phasor to the VR phasor. V = VR2 + (VL - VC)2 VR VL - V C I RCL Circuit The current phasor is unaffected by our addition of the voltage phasors. It now makes an angle f with the overall applied voltage phasor. V = VR2 + (VL - VC)2 f I RLC Circuit We can make Ohm’s Law substitutions for the voltages: VC IX C VL IX L VR IR V VR VL VC I R IX L IX C 2 2 2 V I R I X L X C 2 2 2 2 V I R X L X C 2 2 2 2 RCL Circuit Our result: V I R 2 2 X L XC suggests an Ohm’s Law relationship for the combined loads in the series RCL circuit: V IZ I R X L X C 2 2 Z R2 X L X C 2 Z is called the impedance of the RCL circuit. SI units: ohms (W) RCL Circuit -- Power If the load is purely resistive, the 2 average power dissipated is P I rms R V = VR2 + (VL - VC)2 We can use the phasor diagram to relate R to Z trigonometrically: VR I rms R R cos f V I rms Z Z f VL - VC R Z cos f “power factor” P I rms R I rms Z cos f 2 2 I VR RLC Circuit -- Resonance Series-connected inductor and capacitor: Capacitor is initially charged. When connection is made, + discharge current I flows through inductor. Induced EMF opposes and limits discharge current. C L I RCL Circuit -- Resonance When capacitor is discharged, current I slows and stops. Decrease of magnetic flux in inductor induces EMF that opposes the decrease (and continues the current I). + C L I - RCL Circuit -- Resonance Induced current charges C with the opposite polarity to its original state. When the capacitor is charged, the current I is stopped. + C L RCL Circuit -- Resonance Now the capacitor begins to discharge through the inductor again – this time in the opposite direction (new discharge current = -I). + Opposite induced EMF across inductor again limits this new discharge current. The cycle continues. C -I L RLC Circuit -- Resonance Energy is alternately stored in the capacitor (in the form of the electrical potential energy of separated charges) and in the inductor (in its magnetic field). When the magnetic field collapses, it charges the capacitor; when the capacitor discharges, it builds the magnetic field in the inductor. C L RCL Circuit -- Resonance This “LC oscillator” or “tuned tank circuit” oscillates at a natural or resonant frequency of f res 1 2 LC C L RLC Circuit -- Resonance At the resonant frequency, how are the inductive and capacitive reactances related? f res XC 1 2 LC 1 2f resC 1 LC 2C C 2 LC The reactances are equal to each other. X L 2f res L 2L 1 2 LC L L LC LC XC LC C LC RCL Circuit -- Resonance At the resonant frequency, when the inductive and capacitive reactances are equal, what is the situation in the circuit? VL I VC VR I VR RLC Circuit –SERIES Resonance At the resonant frequency, when the inductive and capacitive reactances are equal, what is the impedance of the circuit? Z R2 X L X C R2 R 2 At resonance, the circuit’s impedance is simply equal to its resistance, and its voltage and current are in phase. If the resistance is small, the current may be quite large.