resonance...the resonant electrical circuit must have both inductance and capacitance. when...

5/2007 Enzo Paterno 1

SERIES RESONANT CIRCUITS

RESONANCE


A very important circuit, used in a wide variety of electrical and electronic systems today (i.e. radio & television tuners), is called the resonant / tuned circuit whose frequency response characteristic is shown below:

The response is a maximum @ fr.. fr. is called the resonant frequency. A tuning circuit will be tuned for maximum response so to receive the signal at its maximum energy ( @ fr ). In mechanical systems, this frequency is called the natural frequency (i.e. The Tacoma Narrows Bridge).

RESONANT CIRCUITS


RESONANT CIRCUITS

The resonant electrical circuit must have both inductance and capacitance. When resonance occurs due to the application of the proper frequency (fr), the energy absorbed by one reactive element is the same as that released by another reactive element within the system. Energy pulsates from one reactive element to the other. Once an ideal (pure L, C) system has reached a state of resonance, it requires no further reactive power since it is self-sustaining (i.e. mechanical system perpetual motion). In a practical circuit, however, there is some resistance associated with the reactive elements that will result in the eventual “damping” of the oscillations between reactive elements.



A resistive element will always be present due to the internal resistance of the source (RS), the internal resistance of the inductor (RL), and any added resistance to control the shape of the response curve (Rd).

Resonance will occur when XL = XC

Thus, @ f = fr ZT = R We can calculate fr . Since XL = XC and r


The current through the circuit at resonance is the maximum current with the input voltage and the current in phase:

The voltage across the inductor and the voltage across the capacitor at resonance are equal magnitude and are 180º out of phase:

The power factor of the circuit at resonance is: FP = cos θ = cos 0 = 1


Equal Magnitude


f [ Hz ] 0

X

Xc [ Ω ] = XL [ Ω ]

fr

LCfr

π21

=

XL < XC XL > XC


Capacitive network Inductive network Resistive network

][2

1Ω=

fCX C π

][2 Ω= fLX L π



Let us plot ZT :

The minimum impedance occurs at the resonant frequency and is equal to the resistance R. Note that the curve is not symmetrical about the resonant frequency.


SELECTIVITY - S

The plot of I = E/ZT , (E = K,) versus frequency, (called the selectivity curve), is shown below and is the inverse of the impedance-versus-frequency curve.

ZT is a minimum

The frequencies f1 and f2 (i.e. @ 0.707 Imax) are called the band frequencies, cutoff frequencies, half-power frequencies or -3db frequencies.

The range of frequencies between f1 and f2 is referred to the bandwidth (BW) of the the resonant circuit.

BW = f2 – f1

@ f1 and f2 P = ½ Pmax = PHPF

BW ↓ S↑ BW ↑ S↓



The quality factor Q of a series resonant circuit is defined as the ratio of the reactive power of either the inductor or the capacitor to the average power of the resistor at resonance:

power Averagepower Reactive

=rQ

The quality factor is an indication of how much energy is stored (continual transfer from one reactive element to the other) compared to that dissipated. The lower the level of dissipation, the larger the Qr factor and the more intense the region of resonance. A higher Q is desirable. Using inductive reactance the quality factor becomes:

CL

RRL

RX

RIXIQ rLL

r1

2

2

====ω

If R is only that of the coil, we speak of the Q of the coil (given by the manufacturer ).

Q = f(f), as f↑ XL↑ Q

5/2007 Enzo Paterno




For circuits where Qr > 10, a widely accepted approximation is that the resonant frequency bisects the bandwidth and that the resonant curve is symmetrical about the resonant frequency.

When designing a BPF, a Design rule of thumb is to Design for a Qr > 10.

221 fffr

+=

212BWbaffBW ==−=

2

2

2

1

BWff

BWff

r

r

+=

−=21 ωωω =r

The geometric mean of ωr:



BW = f2 – f1 ca be expressed in terms of R & L:

Qr can be expressed in terms of BW:

Qr ↑ BW ↓ S↑ Qr ↓ BW ↑ S↓

R ↓ BW ↓ S↑ R ↑ BW ↑ S↓

BWfQ r

r =

5/2007 13


a. Determine the Qr and bandwidth for the response curve below b. For C = 101.5 nF, find L and R for the series resonant circuit. c. Determine the applied voltage.

141

a. fr = 2800 Hz, BW = 200 Hz

14200

2800===

BWfQ r

r

mHLCf

LLC

fr

r

832.314

12

122

=

==ππ

Ω=

==

40

11

RCL

QR

CL

RQ

rr

b.

E = 8 v Enzo Paterno



Determine the frequency response for the voltage Vo for the circuit below.

Vo = 0.707 Vomax = = 13.34 mV

f1 = 50.3 – (5.6 / 2) = 47.5 kHz f2 = 50.3 + (5.6 / 2) = 53.1 kHz



PSPICE SIMULATION



18.85 mV

50.27 kHz

Enzo Paterno 17

r

BWf

CL

RRLQ rr

r ===1ω

r

r

QfBW =

2

2

2

1

BWff

BWff

r

r

+=

−=

SERIES RESONANT CIRCUITS - Formulas

21ωωω =r

in

out

in

outdB P

PVVG log10log20 ==

resonance...the resonant electrical circuit must have both inductance and capacitance. when...

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