Chapter 16

R,L, and C Elements and the Impedance Concept

Introduction

To analyze ac circuits in the time domain is not very practical

It is more practical to:

Express voltages and currents as phasors

Circuit elements as impedances

Represent them using complex numbers

Introduction

AC circuits

Handled much like dc circuits using the same relationships and laws

Complex Number Review

A complex number has the form:

a + jb, where j = (mathematics uses i to represent imaginary numbers)

a is the real part

jb is the imaginary part

Called rectangular form

Complex Number Review

Complex number

May be represented graphically with a being the horizontal component

b being the vertical component in the complex plane

Conversion between Rectangular and Polar Forms

If C = a + jb in rectangular form, then C = CÐq, where

Complex Number Review

j ⁰ = 1

j ¹ = j

j ² = -1

j ³ = -j

j ⁴ = 1 (Pattern repeats for higher powers of j)

1/j = -j

Complex Number Review

To add complex numbers

Add real parts and imaginary parts separately

Subtraction is done similarly

Review of Complex Numbers

To multiply or divide complex numbers

Best to convert to polar form first

(AÐq)•(BÐf) = (AB)Ð(q + f)

(AÐq)/(BÐf) = (A/B)Ð(q - f)

(1/CÐq) = (1/C)Ð-q

Review of Complex Numbers

Complex conjugate of a + jb is a - jb

If C = a + jb

Complex conjugate is usually represented as C*

Voltages and Currents as Complex Numbers

AC voltages and currents can be represented as phasors

Phasors have magnitude and angle

Viewed as complex numbers

Voltages and Currents as Complex Numbers

A voltage given as 100 sin (314t + 30°)

Written as 100Ð30°

RMS value is used in phasor form so that power calculations are correct

Above voltage would be written as 70.7Ð30°

Voltages and Currents as Complex Numbers

We can represent a source by its phasor equivalent from the start

Phasor representation contains information we need except for angular velocity

Voltages and Currents as Complex Numbers

By doing this, we have transformed from the time domain to the phasor domain

KVL and KCL

Apply in both time domain and phasor domain

Summing AC Voltages and Currents

To add or subtract waveforms in time domain is very tedious

Convert to phasors and add as complex numbers

Once waveforms are added

Corresponding time equation of resultant waveform can be determined

Important Notes

Until now, we have used peak values when writing voltages and current in phasor form

It is more common to write them as RMS values

Important Notes

To add or subtract sinusoidal voltages or currents

Convert to phasor form, add or subtract, then convert back to sinusoidal form

Quantities expressed as phasors

Are in phasor domain or frequency domain

R,L, and C Circuits with Sinusoidal Excitation

R, L, and C circuit elements

Have different electrical properties

Differences result in different voltage-current relationships

When a circuit is connected to a sinusoidal source

All currents and voltages will be sinusoidal

R,L, and C Circuits with Sinusoidal Excitation

These sine waves will have the same frequency as the source

Only difference is their magnitudes and angles

Resistance and Sinusoidal AC

In a purely resistive circuit

Ohm’s Law applies

Current is proportional to the voltage

Resistance and Sinusoidal AC

Current variations follow voltage variations

Each reaching their peak values at the same time

Voltage and current of a resistor are in phase

Inductive Circuit

Voltage of an inductor

Proportional to rate of change of current

Voltage is greatest when the rate of change (or the slope) of the current is greatest

Voltage and current are not in phase

Inductive Circuit

Voltage leads the current by 90°across an inductor

Inductive Reactance

X_L, represents the opposition that inductance presents to current in an ac circuit

X_L is frequency-dependent

X_L = V/I and has units of ohms

X_L = wL = 2pfL

Capacitive Circuits

Current is proportional to rate of change of voltage

Current is greatest when rate of change of voltage is greatest

So voltage and current are out of phase

Capacitive Circuits

For a capacitor

Current leads the voltage by 90°

Capacitive Reactance

X_C, represents opposition that capacitance presents to current in an ac circuit

X_C is frequency-dependent

As frequency increases, X_C decreases

Capacitive Reactance

X_C= V/I and has units of ohms

Impedance

The opposition that a circuit element presents to current is impedance, Z

Z = V/I, is in units of ohms

Z in phasor form is ZÐq

q is the phase difference between voltage and current

Resistance

For a resistor, the voltage and current are in phase

If the voltage has a phase angle, the current has the same angle

The impedance of a resistor is equal to RÐ0°

Inductance

For an inductor

Voltage leads current by 90°

If voltage has an angle of 0°

Current has an angle of -90°

The impedance of an inductor

X_LÐ90°

Capacitance

For a capacitor

Current leads the voltage by 90°

If the voltage has an angle of 0°

Current has an angle of 90°

Impedance of a capacitor

X_CÐ-90°

Capacitance

Mnemonic for remembering phase

Remember ELI the ICE man

Inductive circuit (L)

Voltage (E) leads current (I)

A capacitive circuit (C)

Current (I) leads voltage (E)


	To analyze ac circuits in the time domain is not very practical
	It is more practical to:
		Express voltages and currents as phasors
		Circuit elements as impedances
		Represent them using complex numbers


	AC circuits
		Handled much like dc circuits using the same relationships and laws


	A complex number has the form:
		a + jb, where j = (mathematics uses i to represent imaginary numbers)
		a is the real part
		jb is the imaginary part
		Called rectangular form


	Complex number
		May be represented graphically with a being the horizontal component
		b being the vertical component in the complex plane


	j ⁰ = 1
	j ¹ = j
	j ² = -1
	j ³ = -j
	j ⁴ = 1 (Pattern repeats for higher powers of j)
	1/j = -j


	To add complex numbers
		Add real parts and imaginary parts separately
	Subtraction is done similarly


	To multiply or divide complex numbers
		Best to convert to polar form first
	(AÐq)•(BÐf) = (AB)Ð(q + f)
	(AÐq)/(BÐf) = (A/B)Ð(q - f)
	(1/CÐq) = (1/C)Ð-q


	Complex conjugate of a + jb is a - jb
	If C = a + jb
		Complex conjugate is usually represented as C*


	AC voltages and currents can be represented as phasors
	Phasors have magnitude and angle
		Viewed as complex numbers


	A voltage given as 100 sin (314t + 30°)
		Written as 100Ð30°
	RMS value is used in phasor form so that power calculations are correct
	Above voltage would be written as 70.7Ð30°


	We can represent a source by its phasor equivalent from the start
	Phasor representation contains information we need except for angular velocity


	By doing this, we have transformed from the time domain to the phasor domain
	KVL and KCL
		Apply in both time domain and phasor domain


	To add or subtract waveforms in time domain is very tedious
	Convert to phasors and add as complex numbers
	Once waveforms are added
		Corresponding time equation of resultant waveform can be determined


	Until now, we have used peak values when writing voltages and current in phasor form
	It is more common to write them as RMS values


	To add or subtract sinusoidal voltages or currents
		Convert to phasor form, add or subtract, then convert back to sinusoidal form
	Quantities expressed as phasors
		Are in phasor domain or frequency domain


	R, L, and C circuit elements
		Have different electrical properties
		Differences result in different voltage-current relationships
	When a circuit is connected to a sinusoidal source
		All currents and voltages will be sinusoidal


	These sine waves will have the same frequency as the source
		Only difference is their magnitudes and angles


	In a purely resistive circuit
		Ohm’s Law applies
		Current is proportional to the voltage


	Current variations follow voltage variations
		Each reaching their peak values at the same time
	Voltage and current of a resistor are in phase


	Voltage of an inductor
		Proportional to rate of change of current
	Voltage is greatest when the rate of change (or the slope) of the current is greatest
		Voltage and current are not in phase


	X_L, represents the opposition that inductance presents to current in an ac circuit
	X_L is frequency-dependent
	X_L = V/I and has units of ohms
	X_L = wL = 2pfL


	Current is proportional to rate of change of voltage
	Current is greatest when rate of change of voltage is greatest
		So voltage and current are out of phase


	X_C, represents opposition that capacitance presents to current in an ac circuit
	X_C is frequency-dependent
		As frequency increases, X_C decreases


	The opposition that a circuit element presents to current is impedance, Z
		Z = V/I, is in units of ohms
		Z in phasor form is ZÐq
		q is the phase difference between voltage and current


	For a resistor, the voltage and current are in phase
	If the voltage has a phase angle, the current has the same angle
	The impedance of a resistor is equal to RÐ0°


	For an inductor
		Voltage leads current by 90°
	If voltage has an angle of 0°
		Current has an angle of -90°
	The impedance of an inductor
		X_LÐ90°


	For a capacitor
		Current leads the voltage by 90°
	If the voltage has an angle of 0°
		Current has an angle of 90°
	Impedance of a capacitor
		X_CÐ-90°


	Mnemonic for remembering phase
		Remember ELI the ICE man
	Inductive circuit (L)
		Voltage (E) leads current (I)
	A capacitive circuit (C)
		Current (I) leads voltage (E)