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Physics 2 Voronkov Vladimir Vasilyevich

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Lecture 3 Alternating Current (AC) Inductors in AC Circuits Capacitors in AC Circuits Series RLC Circuit Impedance

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Alternating Current (AC) The voltage supplied by an AC source is harmonic (sinusoidal) with a period T. AC source is designated by

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Applying Kirchhoff’s loop, at any instant: Applying Kirchhoff’s loop, at any instant: The instantaneous current in the resistor is:

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Where Imax is the maximum current: And eventually:

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Phasor Diagrams A phasor is a vector whose length is proportional to the maximum value of the variable it represents (Vmax for voltage and Imax for current in the present discussion) and which rotates counterclockwise at an angular speed equal to the angular frequency associated with the variable. The projection of the phasor onto the vertical axis represents the instantaneous value of the quantity it represents.

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Phasor diagram for a circuit with a resistor is: Phasor diagram for a circuit with a resistor is: The phasor diagram for the resistive circuit shows that the current is in phase with the voltage.

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The projections of the phasor arrows onto the vertical axis are determined by a sine function of the angle of the phasor with respect to the horizontal axis. We can use the projections of phasors to represent values of current or voltage that vary sinusoidally in time. The projections of the phasor arrows onto the vertical axis are determined by a sine function of the angle of the phasor with respect to the horizontal axis. We can use the projections of phasors to represent values of current or voltage that vary sinusoidally in time.

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RMS The average value of the current over one cycle is zero. What is of importance in an AC circuit is an average value of current, referred to as the rms current.The notation rms stands for root-mean-square, which in this case means the square root of the mean (average) value of the square of the current:

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Because I2 varies as sin2 t and because the average value of I2 is Imax/2, the rms current is Because I2 varies as sin2 t and because the average value of I2 is Imax/2, the rms current is Thus, the average power delivered to a resistor that carries an alternating current is Alternating voltage is also best discussed in terms of rms voltage, and the relationship is identical to that for current:

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One reason we use rms values when discussing alternating currents and voltages in this chapter is that AC ammeters and voltmeters are designed to read rms values. Furthermore, with rms values, many of the equations we use have the same form as their direct current counterparts. One reason we use rms values when discussing alternating currents and voltages in this chapter is that AC ammeters and voltmeters are designed to read rms values. Furthermore, with rms values, many of the equations we use have the same form as their direct current counterparts.

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Inductors in AC Circuits Kirchhoff’s rule for AC circuit with an inductor is: After derivation we get:

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The maximal current in the inductor is The maximal current in the inductor is We can define the inductive reactance as resistance of an inductor to the harmonic current: The instantaneous voltage across the inductor is:

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Plot of the instantaneous current iL and instantaneous voltage vL across an inductor as functions of time. The current lags behind the voltage by 90°.

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Capacitors in AC The current is /2 rad = 90° out of phase with the voltage across the capacitor:

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The maximal current is: The maximal current is: The capacitive reactance of the capacitor to the sinusoidal current is: Then the instantaneous voltage across the capacitor is:

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Plot of the instantaneous current iC and instantaneous voltage VC across a capacitor as functions of time. The voltage lags behind the current by 90°. Plot of the instantaneous current iC and instantaneous voltage VC across a capacitor as functions of time. The voltage lags behind the current by 90°.

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The RLC Series Circuit For convenience, and not losing generalization, we can assume that the applied voltage is and the current is

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The voltage across each element has a different amplitude and phase: The voltage across each element has a different amplitude and phase:

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Impedance Using the previous calculations we can define a new parameter impedance: So, the amplitudes of voltage and current are related as Using the phasor diagram:

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Power in AC Circuit The average power delivered by the source is converted to internal energy in the resistor. No power losses are associated with pure capacitors and pure inductors in an AC circuit.

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Series RLC Circuit Resonance A series RLC circuit is in resonance when the current has its maximum value. So resonance is at XL=XC, the frequency 0 when XL=XC is called the resonance frequency: This frequency corresponds to the natural frequency of oscillation of an LC circuit

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The average power dissipating in the resistor is Then at resonance the average power is a maximum and equals .

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Units in Si voltage (potential difference) V V (Volt) current (electric current) I A (Ampere) inductance L H (Henry) inductive reactance XL Ohm) capacitive reactance XC Ohm) Impedance Z Ohm) Power P W (Watt)