capacitor ac circuits 2, Exercises of Circuit Theory

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Alternating Current Circuits

Alternating Current Circuits

Electrical appliances in the house use alternating current (AC) circuits. If an AC source applies an alternating voltage to a series circuit containing resistor, inductor, and capacitor, what are the amplitude and time characteristics of the alternating current. Other devices will be discussed  (^) Transformers  (^) Power transmission  (^) Electrical filters Introduction

AC Voltage

The output of an AC power source is sinusoidal and varies with time according to the following equation:  (^) Δ v = Δ Vmax sin ωt  (^) Δ v is the instantaneous voltage.  (^) Δ Vmax is the maximum output voltage of the source.  (^) Also called the voltage amplitude  (^) ω is the angular frequency of the AC voltage.

AC Voltage, cont.

The angular frequency is  (^) ƒ is the frequency of the source.  (^) T is the period of the source. The voltage is positive during one half of the cycle and negative during the other half.

2 ƒ π ω π T

Resistors in an AC Circuit

Consider a circuit consisting of an AC source and a resistor. The AC source is symbolized by

Δ vR =  Vmax= Vmax sin  t

Δ vR is the instantaneous voltage across the resistor.

Resistors in an AC Circuit, cont.

The instantaneous current in the resistor is The instantaneous voltage across the resistor is also given as ΔvR = Imax R sin ωt

sin sin

max

I max

R R

v V

i ωt ωt

R R

Phasor Diagram

To simplify the analysis of AC circuits, a graphical constructor called a phasor diagram can be used. A phasor is a vector whose length is proportional to the maximum value of the variable it represents. The vector 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.

A Phasor is Like a Graph

An alternating voltage can be presented in different representations. One graphical representation is using rectangular coordinates.  (^) The voltage is on the vertical axis.  (^) Time is on the horizontal axis. The phase space in which the phasor is drawn is similar to polar coordinate graph paper.  (^) The radial coordinate represents the amplitude of the voltage.  (^) The angular coordinate is the phase angle.  (^) The vertical axis coordinate of the tip of the phasor represents the instantaneous value of the voltage.  (^) The horizontal coordinate does not represent anything. Alternating currents can also be represented by phasors.

Power

The rate at which electrical energy is delivered to a resistor in the circuit is given by  (^) P = i (^2) R  (^) i is the instantaneous current.  (^) The heating effect produced by an AC current with a maximum value of I max is not the same as that of a DC current of the same value.  (^) The maximum current occurs for a small amount of time.  (^) The average power delivered to a resistor that carries an alternating current is 2

P av  I rmsR

Notes About rms Values

rms values are used when discussing alternating currents and voltages because  (^) AC ammeters and voltmeters are designed to read rms values.  (^) Many of the equations that will be used have the same form as their DC counterparts.

Current in an Inductor

The equation obtained from Kirchhoff's loop rule can be solved for the current This shows that the instantaneous current iL in the inductor and the instantaneous voltage Δ vL across the inductor are out of phase by (/2) rad = 90o. max

sin

max max max max

cos

sin I

L L

V V

i ωt dt ωt

L ωL

V π V

i ωt

ωL ωL

Phase Relationship of Inductors in an AC Circuit

The current is a maximum when the voltage across the inductor is zero.  (^) The current is momentarily not changing For a sinusoidal applied voltage, the current in an inductor always lags behind the voltage across the inductor by 90° ( π /2).

Inductive Reactance

The factor ωL has the same units as resistance and is related to current and voltage in the same way as resistance. Because ωL depends on the frequency, it reacts differently, in terms of offering resistance to current, for different frequencies. The factor is the inductive reactance and is given by:  (^) XL = ωL

Inductive Reactance, cont.

Current can be expressed in terms of the inductive reactance: As the frequency increases, the inductive reactance increases  (^) This is consistent with Faraday’s Law:  (^) The larger the rate of change of the current in the inductor, the larger the back emf, giving an increase in the reactance and a decrease in the current. max rms max rms L L

V V

I or I

X X