RLC Circuit and Resonance

Electric Circuit 2Endah S. Ningrum

Mechatronics-EEPIS-ITS

Impedance of Series RLC Circuits A series RLC circuit contains both inductance and capacitance Since XL and XC have opposite effects on the circuit phase angle,

the total reactance (Xtot)is less than either individual reactance

CLtot XXX

Impedance of Series RLC Circuits When XL>XC, the circuit is predominantly inductive, causes the total

current to lag the source voltage. When XC> XL, the circuit is predominantly capacitive, causes the

total current to lag the source voltage. Total reactance |XL – XC| Total impedance for a series RLC circuit is:

Example Problem

Determine the total impedance and the phase angle in the next figure, with f=1kHz, R=560kW, L=100mH, and C=0,56mF

Solution

628100122

28456.012

1

2

1

mHkHzLfX

FkHzCfX

L

C

Example Solution

344284628CLtot XXX

6.31560

344tantan

657344560

11

2222

R

X

XRZ

tot

tottot

Analysis of Series RLC Circuits

A series RLC circuit is: Capacitive when XC>XL

Inductive when XL>XC

Resonant when XC=XL

At resonance Ztot = R XL is a straight line

y = mx + b XC is a hyperbola xy = k

Example Problem

For each of the following frequencies of the source voltage, find the impedance and the phase angle for the circuit in the next figure. Note the change in the impedance and the phase angle with frequency, with, R=3,3k, L=100mH, and C=0,022F for:

a. f=1khZ b. f=3,5kHz c. f=5kHz

Example Solution

Of a.

628100122

23.7022.012

1

2

1

mHkHzLfX

kFkHzCf

X

L

C

kkXXX CLtot 6.623.7628

4.633.3

6.6tantan

38.76.63.3

11

2222

k

k

R

X

kkkXRZ

tot

tottot

Example Solution

Of b.

kmHkHzLfX

kFkHzCf

X

L

C

20.21005.322

07.2022.05.32

1

2

1

13007.220.2 kkXXX CLtot

26.23.3

130tantan

30.31303.3

11

2222

kR

X

kkXRZ

tot

tottot

Example Solution

Of c.

kmHkHzLfX

kFkHzCf

X

L

C

14.3100522

45.1022.052

1

2

1

kkkXXX CLtot 69.145.114.3

1.273.3

69.1tantan

71.369.13.3

11

2222

k

k

R

X

kkkXRZ

tot

tottot

Voltage Across the Series Combination of L and C In a series RLC circuit, the

capacitor voltage and the inductor voltage are always 180° out of phase with each other

Because they are 180° out of phase, VC and VL subtract from each other

The voltage across L and C combined is always less that the larger individual voltage across either element The voltage across the series

combination of C and L is always less than the larger individual voltage across either C or L

Voltage Across the Series Combination of L and C In a series RLC circuit, the

capacitor voltage and the inductor voltage are always 180° out of phase with each other

Because they are 180° out of phase, VC and VL subtract from each other

The voltage across L and C combined is always less that the larger individual voltage across either element Inductor voltage and capacitor

voltage effectively subtract because they are out of phase

Problem

Find the voltage across each element in the next figure and draw a complete voltage phasor diagram. Also find the voltage across L and C combined

Solution

Example

kkkXXX CLtot 356025

kkkXRZ tottot 8.823575 2222

Example Solution

Ak

V

Z

VI

tot

S 12128.8

10

VkAXIV

VkAXIV

VkARIV

CC

LL

R

26.760121

03.325121

08.975121

VVVVVV LCCL 23.403.326.7

25

75

35tantan 11

k

k

R

X tot

Series Resonance Resonance is a condition in a series RLC circuit in which the

capacitive and inductive reactances are equal in magnitude The result is a purely resistive impedance The formula for series resonance is:

At the resonant frequency (f ), the reactances are equal in magnitude and effectively cancel, leaving Z = R

Series Resonance

CLff

CLfCfLf

CfLf

XX

r

r

rrr

rr

CL

22

2

22

4

1

1422

2

12

CLf r

2

1

Series Resonance At the resonant frequency fr, the voltages across C and L are equal

in magnitude. Since they are 180º out of phase with each other, they cancel,

leaving 0 V across the C,L combination (point A to point B). The section of the circuit from A to B effectively looks like a short at

resonance (neglecting winding resistance).

V&I Ampitudes in Series RLC Circuit

Below the Resonant Frequency

At the Resonant Frequency

Above the Resonant Frequency

Series Resonance

Generalized current and voltage magnitudes as a function of frequency in a series RLC circuit. VC and VL can be much larger than the source voltage. The shapes of the graphs depend on particular circuit values.

Series RLC Impedance as a Function of Frequency

Example Problem

Find I, VR, VL, and VC at the resonance in the next figure , with VS=50mV, R=22, XL=100, and XC=100

Solution

mAmV

R

VI S 27.2

22

50

mVmAXIV

mVmAXIV

mVmARIV

CC

LL

R

22710027.2

22710027.2

502227.2

Example Problem

Determine the impedance at the following frequencies, for the R=10, L=100mH, C=0,01F

a. fr b. 1kHz below fr c. 1kHz above fr

Example Solution for a.

Solution for b.

10RZ

kHz

FmHCLf r 03.5

01.01002

1

2

1

kHzkHzkHzkHzff r 03.4103.51

kmHkHzLfX

kFkHzCf

X

L

C

53.210003.422

95.301.003.42

1

2

1

kkkXXX CLtot 42.195.353.2

kkXRZ tottot 42.142.110 2222

Example

Solution for c.

kHzkHzkHzkHzff r 03.6103.51

kmHkHzLfX

kFkHzCf

X

L

C

79.310003.622

64.201.003.62

1

2

1

kkkXXX CLtot 15.164.279.3

kkXRZ tottot 15.115.110 2222

Phase Angle of a Series RLC Circuit

A Basic Series Resonant Bandpass Filter Current is maximum at

resonant frequency Bandwidth (BW) is the

range between two cutoff frequencies (f1 to f2)

Within the bandwidth frequencies, the current is greater than 70.7% of the highest resonant value

Bandwidth of Series Resonant Circuits

Frequency Response of Series Resonant Bandpass Filter

Example of the frequency response of a series resonant band-pass filter with the input voltage at a constant 10 V rms. The winding resistance of the coil is neglected.

Formula for Bandwidth

Bandwidth for either series or parallel resonant circuits is the range of frequencies between the upper and lower cutoff frequencies for which the response curve (I or Z) is 0.707 of the maximum value

Ideally the center frequency is:

Half Power Point The 70.7% cutoff point is also referred to as:

• The Half Power Point

• -3dB Point

Half Power Point

max

2max

2max

22/1

2max

22/12/1

2maxmax

5,05,0707,0

707,0

PRIRIP

RIRIP

RIP

ff

ffff

Half Power Point

in

out

in

out

P

PdB

V

VdB

log10

log20 dB

V

V3707,0log20

707,0log20

max

max

Selectivity

Selectivity defines how well a resonant circuit responds to a certain frequency and discriminates against all other frequencies

The narrower the bandwidth steeper the slope, the greater the selectivity

This is related to the Quality (Q) Factor (performance) of the inductor at resonance. A higher Q Factor produces a tighter bandwidth

Selectivity

W

LLL

RR

X

R

X

RI

XIQ

Q

2

2

power true

power reactive

DissipatedEnergy

StroredEnergy

How Q Affects Bandwitdh

Q

fBW r

Note: XL at the resonant frequency

A Basic Series Resonant Bandstop Filter

A basic series resonant band-stopFilter Circuit

Generalized response curve for a band-stop filter

Frequency Response of Series Resonant Bandstop Filter

Example of the frequency response of a series resonant band-stop filter with Vin at a constant 10 V rms. The winding resistance is neglected.

Parallel RLC Circuits A parallel resonant circuit

stores energy in the magnetic field of the coil and the electric field of the capacitor. The energy is transferred back and forth between the coil and capacitor

Parallel RLC Circuits

G

BY

Z

BGY

BBB

tot

tot

CLtot

1

22

tan

1

R

CL

CLRtot

I

I

III

1

22

tan

Where :1. B : Susceptance2. Y : Admitance3. G : Conductance

Current Across the Parallel Combination of L and C

R

CL

CLRtot

I

I

III

1

22

tanLCCL III Where :

Current Across the Parallel Combination of L and C

Conversion of Series-Parallel to Parallel

1

1

2

2

2

QRR

Q

QLL

Weqp

eq

mHmHmHL

LL

eq

eq

1.1001.11010

11010

2

2

For Q>=10

Parallel Resonant Circuits For parallel resonant circuits, the impedance is maximum (in theory,

infinite) at the resonant frequency Total current is minimum at the resonant frequency Bandwidth is the same as for the series resonant circuit; the critical

frequency impedances are at 0.707Zmax

A basic parallel resonant band-pass filter

Parallel Resonant Circuits For parallel resonant circuits, the impedance is maximum (in theory,

infinite) at the resonant frequency Total current is minimum at the resonant frequency Bandwidth is the same as for the series resonant circuit; the critical

frequency impedances are at 0.707Zmax

Generalized frequency response curves for a parallel resonant band-pass filter

Parallel Resonant Circuits

Example of the response of a parallel resonant band-pass filter with the input voltage at a constant 10 V rms

Parallel Resonant Circuits

A basic parallel resonant band-stop filter

Applications A simplified portion of a TV receiver showing filter usage

Applications A simplified diagram of a superheterodyne AM radio broadcast

receiver showing the application of tuned resonant circuits