Download SERIES RESONANT CIRCUITS RESONANCE

RESONANCE
SERIES RESONANT
CIRCUITS
5/2007
Enzo Paterno
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RESONANT CIRCUITS
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).
5/2007
Enzo Paterno
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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.
5/2007
Enzo Paterno
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SERIES RESONANT CIRCUITS
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
and 
We can calculate fr . Since XL = XC 
5/2007
Enzo Paterno
r
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SERIES RESONANT CIRCUITS
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:
Equal
Magnitude
The power factor of the circuit at resonance is:
FP = cos θ = cos 0 = 1
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Enzo Paterno
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SERIES RESONANT CIRCUITS
X
XC =
1
[Ω ]
2πfC
X L = 2πfL [Ω]
Xc [ Ω ] = XL [ Ω ]
fr =
0
5/2007
XL < XC
Capacitive network
1
2π LC
fr
Enzo Paterno
Resistive network
XL > XC
Inductive network
f [ Hz ]
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SERIES RESONANT CIRCUITS
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.
5/2007
Enzo Paterno
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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 range of frequencies
between f1 and f2 is referred to
the bandwidth (BW) of the
the resonant circuit.
@ f1 and f2
P = ½ Pmax = PHPF
BW = f2 – f1
BW ↓ S↑
BW ↑ S↓
The frequencies f1 and f2 (i.e. @ 0.707 Imax) are called the band frequencies,
cutoff frequencies, half-power frequencies or -3db frequencies.
5/2007
Enzo Paterno
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SERIES RESONANT CIRCUITS
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:
Qr =
Reactive power
Average power
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:
I X L X L ωr L 1
Qr = 2 =
=
=
I R
R
R
R
2
L
C
Q
Q = f(f), as f↑ XL↑
If R is only that of the coil, we speak of the Q
of the coil (given by the manufacturer ).
5/2007
Enzo Paterno
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SERIES RESONANT CIRCUITS
5/2007
Enzo Paterno
SERIES RESONANT CIRCUITS
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.
BW
BW = f 2 − f1  a = b =
2
f1 + f 2
fr =
2
BW
f1 = f r −
2
BW
f2 = fr +
2
The geometric mean of ωr:
ωr = ω1 ω2
5/2007
Enzo Paterno
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SERIES RESONANT CIRCUITS
BW = f2 – f1 ca be expressed in terms of R & L:
R ↓ BW ↓ S↑
R ↑ BW ↑ S↓
Qr can be expressed in terms of BW:
Qr ↑ BW ↓ S↑
Qr ↓ BW ↑ S↓
5/2007
fr
 Qr =
BW
Enzo Paterno
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SERIES RESONANT CIRCUITS
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.
a. fr = 2800 Hz, BW = 200 Hz
Qr =
141
b.
fr
2800
=
= 14
BW
200
1
L=
fr =
4π 2 fr 2C
2π LC
L = 31.832 mH
Qr =
1
1
R
1
L
R=
C
Qr
L
C
R = 40 Ω
 E=8v
5/2007
Enzo Paterno
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SERIES RESONANT CIRCUITS
Determine the frequency response for the voltage Vo for the circuit
below.
f1 = 50.3 – (5.6 / 2) = 47.5 kHz
f2 = 50.3 + (5.6 / 2) = 53.1 kHz
Vo = 0.707 Vomax = = 13.34 mV
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Enzo Paterno
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SERIES RESONANT CIRCUITS
PSPICE SIMULATION
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Enzo Paterno
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SERIES RESONANT CIRCUITS
18.85 mV
50.27 kHz
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Enzo Paterno
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SERIES RESONANT CIRCUITS - Formulas
BW
2
BW
f2 = fr +
2
f1 = f r −
r
ωr L
1
=
Qr =
R
R
BW =
fr
Qr
L
fr
=
C BW
ωr = ω1ω2
Enzo Paterno
GdB = 20 log
Vout
P
= 10 log out
Vin
Pin
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