Chapter 12Chapter 12RL Circuits
Sinusoidal Response of RL
Circuits• The inductor voltage leads the source voltage
• Inductance causes a phase shift between voltage and current that depends on the relative values of the resistance and the inductive reactancethe resistance and the inductive reactance
Illustration of sinusoidal response with general phase relationships of VR, V
L, and I relative to the
source voltage.
VR
and I are in phase; VR
lags VS; and V
Lleads V
S. V
Rand V
Lare 90º out of phase with each other.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Relationships of the Current and
Voltages in a Series RL Circuit
• Resistor voltage is in
phase with the current
• Inductor voltage leads
the current by 90°the current by 90°
• There is a phase
difference of 90°
between the resistor
voltage, VR, and the
inductor voltage, VL
Phase relation of current and voltages in a series RL circuit
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
I, R
VL, XL
Z, Vs
Phase Angle
Reference
Impedance and Phase Angle of
Series RL Circuits
• Impedance of any RL circuit is the total opposition
to sinusoidal current and its unit is the ohm
• The phase angle is the phase difference between
the total current and the source voltagethe total current and the source voltage
• The impedance of a series RL circuit is
determined by the resistance (R) and the inductive
reactance (XL)
Impedance of a series RL circuit
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
The Impedance Triangle
• The impedance magnitude of the series RL
circuit in terms of resistance and reactance:
Z = √R2 + X2LZ = √R2 + X2L
– The magnitude of the impedance (Z) is
expressed in ohms
• The phase angle is:
θ = tan-1(XL/R)
The Impedance Triangle
• In ac analysis, both R and XL are treated a phasor quantities, with XL appearing at a +90° angle with respect to R
• θ is the phase angle between applied voltage and current• θ is the phase angle between applied voltage and current
Ohm’s Law
• Application of Ohm’s Law to series RL
circuits involves the use of the phasor
quantities Z, Vs, and Itotquantities Z, Vs, and Itot
Vs = ItotZ
Itot = Vs/Z
Z = Vs/Itot
Illustration of how the variation of impedance affects the voltages and current as the
source frequency is varied. The source voltage is held at a constant amplitude
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Observing changes in Z and XL
with frequency by watching the meters and recalling
Ohm’s law
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Variation of Impedance and
Phase Angle with Frequency
• Inductive reactance varies directly with frequency
• Z is directly dependent on frequency
• Phase angle θ also varies directly with frequency• Phase angle θ also varies directly with frequency
0° = Purely Resistive
As the frequency increases, the phase angle θθθθ increases.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Parallel RL Circuits - Skip
Power in RL Circuits
• When there is both resistance and inductance,
some of the energy is alternately stored and
returned by the inductance and some is dissipated
by the resistanceby the resistance
• The amount of energy converted to heat is
determined by the relative values of the resistance
and the inductive reactance
• The Power in the inductor is reactive power:
Pr = I2XL
Power Triangle for RL Circuits
• The apparent power (Pa) is the resultant of the true
power (Ptrue) and the reactive power (PR)
• Recall Power Factor: PF = cos θ
Significance of the Power Factor
• Many practical loads have inductance as a result of their particular function, and it is essential for their proper operation
• Examples are: transformers, electric motors and speakers
• A higher power factor is an advantage in delivering power more efficiently to a load
• Recall Power Factor:
– PF = PTRUE/PA
– PF = RT/Z
– PF = cos θ
Illustration of the effect of the power factor on system requirements such as source
rating (VA) and conductor size.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
RL Lag Circuit (Low Pass Filter)
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Illustration of how the frequency affects the phase lag and the output voltage in an RL
lag network with the amplitude of Vin
held constant.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
RL Lead Circuit (High Pass Filter)
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Illustration of how the frequency affects the phase lead and the output voltage in an RL
lead network with the amplitude of Vin
held constant.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
RL Circuit as a Low-Pass Filter
• An inductor acts as a short to dc
• As the frequency is increased, so does the
inductive reactance
– As inductive reactance increases, the output voltage
across the resistor decreases
– A series RL circuit, where output is taken across the
resistor, finds application as a low-pass filter
– Cutoff Frequency (fc) is where the output voltage is
at 70.7% of its maximum value => L
Rfc
π2=
Example of low-pass filtering action.
As the input frequency increases, the output voltage decreases.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
RL Circuit as a High-Pass Filter
• For the case when output voltage is measured
across the inductor
– At dc, the inductor acts a short, so the output voltage
is zerois zero
– As frequency increases, so does inductive reactance,
resulting in more voltage being dropped across the
inductor
– The result is a high-pass filter
– Cutoff Frequency (fc) is where the output voltage is
at 70.7% of its maximum value => L
Rfc
π2=
Example of high-pass filtering action.
As the input frequency increases, the output voltage increases.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Troubleshooting: Effect of an open coil.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Troubleshooting: Effect of an open resistor.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Troubleshooting: Effect of an open component in a parallel circuit with Vs
constant.
Thomas L. Floyd
Electronics Fundamentals, 6e
Electric Circuit Fundamentals, 6e
Copyright ©2004 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.