RLC Circuits and Resonance Chapter 13 Thomas L. Floyd David M. Buchla DC/AC Fundamentals: A Systems...

RLC RLC Circuits and ResonanceCircuits and Resonance

Chapter 13

Thomas L. Floyd

David M. Buchla

DC/AC Fundamentals: A Systems DC/AC Fundamentals: A Systems ApproachApproach

DC/AC Fundamentals: A Systems ApproachThomas L. Floyd

© 2013 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

When a circuit contains an inductor and capacitor in series, the reactance of each opposes (i.e., cancels) the other. Total series LC reactance is found using:

The total impedance is found using:

The phase angle is found using:

Ch.13 Summary

Series RLC Circuits

R L C

VS

XX X CLtot

22tottot XRZ

R

X tot1tan



Rea

ctan

ce

f

XC XL

A series RLC circuit can be capacitive, inductive, or resistive, depending on the frequency.

The frequency where XC=XL is called the resonant frequency.

Series resonance

XC = XL

Below the resonant frequency, the circuit is predominantly capacitive.

Above the resonant frequency, the circuit is predominantly inductive.

XC > XL XL > XC

Ch.13 Summary

XL and XC Vs. Frequency



What is the total impedance and phase angle of the series RLC circuit below?

The total reactance is

The total impedance is 3.16 k

The circuit is capacitive, and I leads V by 71.6o.

The phase angle is 71.6o

Ch.13 Summary

Series RLC Circuit Impedance

1 k 2 k 5 k

R

VS

XCXL

kΩ 3 kΩ 5 kΩ 2 X L Ctot XX

2222 kΩ 3kΩ 1tottot XRZ

kΩ 1

kΩ 3tantan 11

R

X totθ



What is the magnitude of the impedance for the circuit below?

753

Ch.13 Summary


f = 100 kHz

470 330 mH 2000 pF

R

VS

CL

Ω 207H) kHz)(330 (10022 m fLXL

Ω 796pF) kHz)(2000 (1002

1

2

1

fCXC

Ω 589 Ω 796Ω 207 XX X CLtot

22

22

Ω) (589Ω) (470

totXRZ



XL

f

XC

X

Depending on the frequency, the circuit can appear to be capacitive or inductive. The circuit in the previous slide was capacitive because XC > XL.

XL

XC

Ch.13 Summary


100 kHz



What is the total impedance for the circuit when the frequency is increased to 400 kHz?

The circuit is now inductive.

786

Ch.13 Summary


Ω 298H) kHz)(330 (40022 m fLXL

Ω 991pF) kHz)(2000 (4002

1

2

1

fCXC

Ω 630 Ω 991Ω 829 XX X CLtot

22

22

Ω) (630Ω) (470

totXRZ

f = 400 kHz

470 330 mH 2000 pF

R

VS

CL



XL

f

XC

X

By changing the frequency, the circuit in the previous slide inductive (because XL > XC).

XL

XC

Ch.13 Summary

Impedance of Series RLC Circuits

400 kHz



The voltages across the RLC components must add to the source voltage in accordance with KVL. Because of the opposite phase shift due to L and C, VL and VC effectively subtract.

0

Notice that VC is out of phase with VL. When they are algebraically added, the result is…. VC

VL

This example is inductive.

Ch.13 Summary

Series RLC Circuit Voltages



At series resonance, XC and XL cancel. VC and VL also cancel because they are equal in magnitude and opposite in polarity. The circuit is purely resistive at resonance.

0

Algebraic sum is zero.

Ch.13 Summary

Series Resonance



A formula for resonance can be found by assuming XC = XL and solving. The result is:

What is the resonant frequency for the circuit?

196 kHz

Ch.13 Summary

Series Resonance

LCfr

2

1

m

pF) H)(2000 (3302

12

1

LCfr

470 330 mH 2000 pF

R

VS

CL



What is VR at resonance?

Ideally, at resonance the sum of VL and VC is zero.

5.0 Vrms

0 VVS

By KVL, VR = VS

Ch.13 Summary

Series Resonance

470 330 mH 2000 pF

CLR

VS

5 Vrms

5.0 Vrms



XL

f

XC

XThe general shape of the impedance versus frequency for a series RLC circuit is superimposed on the curves for XL and XC. Notice that at the resonant frequency, the circuit is resistive, and Z = R.

Z

Series resonance

Z = R

Ch.13 Summary




Summary of important concepts for series resonance:

• Capacitive and inductive reactances are equal.

• Total impedance is a minimum and is resistive.

• The current is maximum.

• The phase angle between VS and IS is zero.

• fr is calculated using:

Ch.13 Summary

Series Resonance

LCfr

2

1



Series resonant circuits can be used as filters. A band-pass filter allows signals within a range of frequencies to pass.

R

VoutVin

Resonant circuit

f

Vout

Series resonance

Circuit response

Ch.13 Summary

Series Resonant Filters

L C



The response curve has a peak; meaning the current is maximum at resonance and falls off at frequencies below and above resonance. The maximum current (at resonance) develops maximum voltage across the series resistor(s).

0.707

f1 fr f2

BW

The bandwidth (BW) of the filter is the range of frequencies over which the output is equal to or greater than 70.7% of its maximum value. f1 and f2 are commonly referred to as the critical frequencies, cutoff frequencies or half-power frequencies.

f

I or VoutPassband

1.0

Ch.13 Summary




Filter responses are often described in terms of decibels (dB). The decibel is defined as:

When circuit input and output voltages are known, the filter response can be calculated using:

Example: When output power is half the input power, the ratio of Pout/Pin = ½, and

Ch.13 Summary

Decibels

dB 32

110logdB

in

out

P

Plog10dB

in

out

V

VdB log20



Selectivity describes the basic frequency response of a resonant circuit. (The -3 dB frequencies are marked by the dots.)

f0

BW3

Least Selectivity

BW2

Medium Selectivity

BW1

Greatest Selectivity

The greater the Q of a filter at a given resonant frequency, the higher it’s selectivity.

Which curve represents the highest Q?

The one with the greatest selectivity.

Ch.13 Summary

Selectivity

Q

fBW r



By taking the output across the resonant circuit, a band-stop (or notch) filter is produced.

RVoutVin

f

Vout

Circuit response

Resonant circuit

f1 fr

BW

Stop band

0.707

1

f2

Ch.13 Summary


L

C



Conductance, susceptance, and admittance were defined in Chapter 10 as the reciprocals of resistance, reactance and impedance. As a review:

Ch.13 Summary

Conductance, Susceptance, and Admittance

Conductance is the reciprocal of resistance. R

G1

Susceptance is the reciprocal of reactance. X

B1

Admittance is the reciprocal of impedance. Z

Y1



Ch.13 Summary

Parallel RLC Circuit ImpedanceThe admittance can be used to find the impedance. Start by calculating the total susceptance:

CLtot BBB

The admittance is given by: 22totBGY

The impedance is the reciprocal of the admittance: Y

Ztot

1

The phase angle is:

G

Btot1tan

VS R L C



What is the total impedance of the parallel RLC circuit below?

First, determine the conductance and total susceptance as follows:

The total admittance is:

881

Ch.13 Summary

Parallel RLC Circuit Impedance

mS 0.3 CLtot BBB

mS 1.13 mS 0.3mS 1 22

22

tottot BGY

mS

YZtot

13.1

1

1

mS1kΩ 1

11

RG

ms 0.5kΩ 2

11

LL X

B

VS R XL XC

1 k 2 k 5 k



A typical current phasor diagram for a parallel RLC circuit is shown.

+90o

-90o

IC

IR

IL

The total current is given by:

The phase angle is given by:

Ch.13 Summary

AC Response of Parallel RLC Circuits

22 )( LCRtot IIII

R

CL

I

I1tanθ



The currents in the RLC components must add to the source current in accordance with KCL. Because of the opposite phase shifts of IL and IC (relative to VS) they effectively subtract.

Notice that IC is out of phase with IL. When they are algebraically added, the result is….

IC

IL

0

Ch.13 Summary

Parallel RLC Circuit Currents



Draw a diagram of the phasors having values of IR = 12 mA, IC = 22 mA and IL = 15 mA.

•Set up a grid.

0 mA

10 mA

20 mA

10 mA

20 mA

IR

IC

IL

•Plot the currents on the appropriate axes.

•Combine the reactive currents.

•Use the total reactive current and IR to find

the total current.

In this case, Itot = 16.6 mA

Ch.13 Summary

Parallel RLC Circuit Currents



0

Ideally, IC and IL cancel at parallel resonance because they are equal and opposite. Thus, the circuit is purely resistive at resonance.

The algebraic sum is zero.

Notice that IC is out of phase with IL. When they are algebraically added, the result is….

IC

IL

Ch.13 Summary

Parallel Resonance



The formula for the resonant frequency in both parallel and series circuits is the same:

What is the resonant frequency for the circuit?

49.8 kHz

Ch.13 Summary

Parallel Resonance

(ideal case)LC

fr

2

1

m

nF) H)(15 (6802π

12

1

LCfr

R L CVS680 mH1.0 k 15 nF

R L CVS

1 k 15 nF680 mH



In practical circuits, there is a small current through the coil at resonance and the resonant frequency is not exactly given by the ideal equation. The Q of the coil affects the equation for resonance:

For Q >10, the difference between the ideal and the non-ideal formula is less than 1%, and generally can be ignored.

Ch.13 Summary

Parallel Resonance in Non-ideal Circuits

(non-ideal)12

12

2

Q

Q

LCfr



In a parallel resonant circuit, impedance is maximum and current is minimum. The bandwidth (BW) can be defined in terms of the impedance curve.

f

Zmax

0.707Zmax

f1 fr f2

BW

Ztot

A parallel resonant circuit is commonly referred to as a tank circuit because of its ability to store energy like a storage tank.

Ch.13 Summary

Bandwidth of Resonant Circuits



Summary of important concepts for parallel resonance:

• Capacitive and inductive susceptance are equal.

• Total impedance is a maximum (ideally infinite).

• The current is minimum.

• The phase angle between VS and IS is zero.

• The resonant frequency (fr) is given by

Ch.13 Summary

Parallel Resonance

LCfr

2

1



Parallel resonant circuits can also be used for band-pass or band-stop filters. A basic band-pass filter is shown below.

R

VinVout

Parallel resonant band-pass filter

CLResonant circuit

0.707

f1 fr f2

BW

f

VoutPassband

1.0

Ch.13 Summary

Parallel Resonant Filters



For the band-stop filter, the positions of the resonant circuit and resistance are reversed as shown here.

f

Vout

f1 fr

BW

Stop band

0.707

1

R

Vin Vout

Parallel resonant band-stop filter

C

L

Resonant circuit

f2

Ch.13 Summary

Parallel Resonant Filters



•A band-stop filter rejects frequencies between two critical frequencies and passes all others.

•Band-pass and band-stop filters can be made from both series and parallel resonant circuits.

•The bandwidth of a resonant filter is determined by the Q and the resonant frequency.

•The output voltage at a critical frequency is 70.7% of the maximum.

A band-pass filter allows frequencies between two critical frequencies and rejects all others.

Ch.13 Summary

Key Ideas for Resonant Filters



A condition in a series RLC circuit in which the reactances ideally cancel and the impedance is a minimum.

The frequency at which resonance occurs; also known as the center frequency.

A condition in a parallel RLC circuit in which the reactances ideally are equal and the impedance is a maximum.

A parallel resonant circuit.

Ch.13 Summary

Key Terms

Series resonance

Resonant frequency (fr)

Parallel resonance

Tank circuit



The frequency at which the output power of a resonant circuit is 50% of the maximum value (the output voltage is 70.7% of maximum); another name for critical or cutoff frequency.

Ten times the logarithmic ratio of two powers.

A measure of how effectively a resonant circuit passes desired frequencies and rejects all others. Generally, the narrower the bandwidth, the greater the selectivity.

Ch.13 Summary

Key Terms

Half-power frequency

Decibel

Selectivity



1. In practical series and parallel resonant circuits, the total impedance of the circuit at resonance will be

a. capacitive

b. inductive

c. resistive

d. none of the above

Ch.13 Summary

Quiz



2. In a series resonant circuit, the current at the half-power frequency is

a. maximum

b. minimum

c. 70.7% of the maximum value

d. 70.7% of the minimum value

Ch.13 Summary

Quiz



3. The frequency represented by the red dashed line is the

a. resonant frequency

b. half-power frequency

c. critical frequency

d. all of the above

XL

f

XC

X

f

Ch.13 Summary

Quiz



4. In a series RLC circuit, if the frequency is below the resonant frequency, the circuit will appear to be

a. capacitive

b. inductive

c. resistive

d. answer depends on the particular

components

Ch.13 Summary

Quiz



5. In a series resonant circuit, the resonant frequency can be found from the equation

a.

b.

c.

d.

max0.707rf I

1

2rf

LC

1

2rf LC

r

BWf

Q

Ch.13 Summary

Quiz



6. In an ideal parallel resonant circuit, the total impedance at resonance is

a. zero

b. equal to the resistance

c. equal to the reactance

d. infinite

Ch.13 Summary

Quiz



7. In a parallel RLC circuit, the magnitude of the total current is always the

a. same as the current in the resistor.

b. phasor sum of all of the branch currents.

c. same as the source current.

d. difference between resistive and reactive

currents.

Ch.13 Summary

Quiz



8. If you increase the frequency in a parallel RLC circuit, the total current

a. will not change

b. will increase

c. will decrease

d. can increase or decrease depending on if it

is above or below resonance.

Ch.13 Summary

Quiz



9. The phase angle between the source voltage and

current in a parallel RLC circuit will be positive if

a. IL is larger than IC

b. IL is larger than IR

c. both a and b


Ch.13 Summary

Quiz



10. A highly selectivity circuit will have a

a. small BW and high Q.

b. large BW and low Q.

c. large BW and high Q.


Ch.13 Summary

Quiz



1. c

2. c

3. a

4. a

5. b

6. d

7. b

8. d

9. d

10. a

Ch.13 Summary

Answers

RLC Circuits and Resonance Chapter 13 Thomas L. Floyd David M. Buchla DC/AC Fundamentals: A Systems...

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