chapter 36. ac circuits - uw-madison department …...author duncan created date 4/1/2009 2:49:13 pm

Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley

Chapter 36. AC Circuits Today, a “grid” of AC electrical distribution systems spans the United States and other countries. Any device that plugs into an electric outlet uses an AC circuit. In this chapter, you will learn some of the basic techniques for analyzing AC circuits. Chapter Goal: To understand and apply basic techniques of AC circuit analysis.


AC Circuits Alternating current refers to currents and voltages which are harmonic functions.

“AC power” refers to the 60 hz alternating voltage available from a common outlet.

AC circuit analysis has a more general motivation and application. Any voltage or current may be represented as a sum of harmonic voltages or currents and the response of a linear circuit understood in terms of the response for different frequencies.


Fourier analysis Any function may be represented by a sum of sin and cos functions.


AC Sources and Phasors

An AC signal is described by an amplitude, a frequency, and a phase.


AC Current in a resistor If an ideal resistor is subject to an alternating voltage, an alternating current flows. Ohm’s law V=IR applies to the instantaneous currents and voltages.


AC Current in a resistor

The resistor voltage vR is given by

where VR is the peak or maximum voltage. The current through the resistor is

where IR = VR/R is the peak current.


EXAMPLE 36.1 Finding resistor voltages


AC current in a capacitor

A capacitor differentiates a voltage signal.

An AC voltage applied to a capacitor corresponds to an AC current at the same frequency but with a different phase.


Capacitor Circuits Suppose the instantaneous voltage across a capacitor is:

The instantaneous current in the circuit is

The AC current to and from a capacitor leads the capacitor voltage by π/2 rad, or 90°.


Capacitive Reactance The capacitive reactance XC is defined as

The units of reactance, like those of resistance, are ohms. Reactance relates the peak voltage VC and current IC:

NOTE: Reactance differs from resistance in that it does not relate the instantaneous capacitor voltage and current because they are out of phase. That is, vC ≠ iCXC.


EXAMPLE 36.3 Capacitor current


Power filtering with a capacitor current

A capacitor has a high impedance at low frequency (behaves like a huge resistor) and low impedance at high frequency (behaves like a short).

The low pass filter, a DC input voltage charges the capacitor in time RC and there after the output voltage equals the input. An additional high frequency voltage component is shorted out.


AC inductor circuit

An ideal inductor is circuit active only under AC conditions.

An ideal inductor integrates a signal.


Inductor Circuits If the instantaneous voltage across a single inductor is

the instantaneous inductor current is

The AC current through an inductor lags the inductor voltage by π/2 rad, or 90°.


Inductive Reactance The inductive reactance XL is defined as

Reactance relates the peak voltage VL and current IL:

NOTE: Reactance differs from resistance in that it does not relate the instantaneous inductor voltage and current because they are out of phase. That is, vL ≠ iLXL.


Inductive Reactance

In contrast to a capacitor, an inductor offers a vanishing impedance at low frequency and a high impedance at high frequency.


Inductive filters

An inductor lets low frequencies slip through while choking off high frequencies.

In the low pass filter a DC input voltage transfers to the output – the inductor is just a conducting wire – while a high frequency is choked. In the high pass filter, the inductor shorts out low frequency voltages.

Low pass filter

High pass filter


EXAMPLE 36.5 Current and voltage of an inductor

QUESTION:


EXAMPLE 36.5 Current and voltage of an inductor


AC circuit combinations

The response of a combination of circuit elements can be understood by series and parallel addition of the response of the elements by accounting simultaneously for amplitude and phase effects.


Applications


The Series RLC Circuit The impedance Z of a series RLC circuit is defined as

Impedance, like resistance and reactance, is measured in ohms. The circuit’s peak current is related to the source emf and the circuit impedance by

Z is at a minimum, making I a maximum, when XL = XC, at the circuit’s resonance frequency:


The resonance of a RLC circuit


Applications


Power in AC Circuits

The root-mean-square current Irms is related to the peak current IR by

Similarly, the root-mean-square voltage and emf are

The average power supplied by the emf is


EXAMPLE 36.7 Lighting a bulb


Impedance

This chapter introduces analysis of time dependent linear circuits. A passive circuit element is described by an internal resistance and a frequency dependent reactance. The combination is the impedance Z= R+iX.

Not demonstrated here but for your amusement: series and parallel combinations of passive elements may be understood by reduction using familiar rules for the parallel (add inverses) and series (addition) addition of resistance but with complex impedance.

chapter 36. ac circuits - uw-madison department …...author duncan created date 4/1/2009 2:49:13 pm

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