Frequency response: Resonance, Bandwidth, Q factor

[Pages:11]Frequency response: Resonance, Bandwidth, Q factor

Resonance.

Let's continue the exploration of the frequency response of RLC circuits by investigating the series RLC circuit shown on Figure 1.

I

R

C

+

Vs

R VR

-

Figure 1

The magnitude of the transfer function when the output is taken across the resistor is

H () VR =

RC

(1.1)

( ) Vs

1- 2LC 2 + (RC )2

At the frequency for which the term 1- 2LC = 0 the magnitude becomes

H () = 1

(1.2)

The dependence of H () on frequency is shown on Figure 2 for which L=47mH and C=47?F and for various values of R.

6.071/22.071 Spring 2006, Chaniotakis and Cory

1

Figure 2.

The frequency 0 =

1 is called the resonance frequency of the RLC network. LC

The impedance seen by the source Vs is

Z = R + jL + 1

jC

(1.3)

=

R+

j

L

-1 C

Which at = 0 =

1 becomes equal to R . LC

? Therefore at the resonant frequency the impedance seen by the source is purely resistive.

? This implies that at resonance the inductor/capacitor combination acts as a short circuit.

? The current flowing in the system is in phase with the source voltage.

The power dissipated in the RLC circuit is equal to the power dissipated by the resistor.

Since the voltage across a resistor (VR cos(t)) and the current through it ( IR cos(t)) are

in phase, the power is

p(t) = VR cos(t)IR cos(t) = VR IR cos2 (t)

(1.4)

6.071/22.071 Spring 2006, Chaniotakis and Cory

2

And the average power becomes

P()

=

1 2

VR

I

R

=

1 2

I

2 R

R

(1.5)

Notice that this power is a function of frequency since the amplitudes VR and IR are frequency dependent quantities.

The maximum power is dissipated at the resonance frequency

Pmax

=

P( =0 )

=

1 VS2 2R

(1.6)

6.071/22.071 Spring 2006, Chaniotakis and Cory

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Bandwidth.

At a certain frequency the power dissipated by the resistor is half of the maximum power

which as mentioned occurs at 0 =

1 . The half power occurs at the frequencies for LC

which the amplitude of the voltage across the resistor becomes equal to 1 of the 2

maximum.

P1/ 2

=

1 4

V2 max R

(1.7)

Figure 3 shows in graphical form the various frequencies of interest.

1/ 2

Figure 3

Therefore, the ? power occurs at the frequencies for which

1=

RC

( ) 2

1- 2LC 2 + (RC )2

Equation (1.8) has two roots

1

=

-

R 2L

+

R 2L

2

+

1 02

2

=

R 2L

+

R 2L

2

+

1 02

6.071/22.071 Spring 2006, Chaniotakis and Cory

(1.8)

(1.9) (1.10)

4

The bandwidth is the difference between the half power frequencies

Bandwidth = B = 2 - 1

(1.11)

By multiplying Equation (1.9) with Equation (1.10) we can show that 0 is the geometric mean of 1 and 2 .

0 = 12

(1.12)

As we see from the plot on Figure 2 the bandwidth increases with increasing R. Equivalently the sharpness of the resonance increases with decreasing R.

For a fixed L and C, a decrease in R corresponds to a narrower resonance and thus a higher selectivity regarding the frequency range that can be passed by the circuit.

As we increase R, the frequency range over which the dissipative characteristics dominate the behavior of the circuit increases. In order to quantify this behavior we define a parameter called the Quality Factor Q which is related to the sharpness of the peak and it is given by

Q = 2

maximum energy stored

= 2 ES

total energy lost per cycle at resonance

ED

(1.13)

which represents the ratio of the energy stored to the energy dissipated in a circuit.

The energy stored in the circuit is

ES

=

1 2

LI 2

+

1 CVc2 2

(1.14)

For Vc = Asin(t) the current flowing in the circuit is I = C dVc = CAcos(t) . The dt

total energy stored in the reactive elements is

ES

=

1 2

L 2C 2 A2

cos2 (t) +

1 CA2 sin2 (t) 2

(1.15)

At the resonance frequency where = 0 =

1 the energy stored in the circuit LC

becomes

ES

=

1 CA2 2

(1.16)

6.071/22.071 Spring 2006, Chaniotakis and Cory

5

The energy dissipated per period is equal to the average resistive power dissipated times the oscillation period.

ED = R

I2

2 0

=

R

02C 2 2

A2

2

0

=

2

1 2

RC 0 L

A2

(1.17)

And so the ratio Q becomes

Q = 0L = 1 R 0RC

(1.18)

? The quality factor increases with decreasing R ? The bandwidth decreases with decreasing R

By combining Equations (1.9), (1.10), (1.11) and (1.18) we obtain the relationship between the bandwidth and the Q factor.

Therefore:

B = L = 0 RQ

(1.19)

A band pass filter becomes more selective (small B) as Q increases.

6.071/22.071 Spring 2006, Chaniotakis and Cory

6

Similarly we may calculate the resonance characteristics of the parallel RLC circuit.

IR(t)

I s(t)

R

L

C

Figure 4

Here the impedance seen by the current source is

Z //

=

j L (1- 2LC) +

j L

R

(1.20)

At the resonance frequency 1- 2LC = 0 and the impedance seen by the source is purely resistive. The parallel combination of the capacitor and the inductor act as an open circuit. Therefore at the resonance the total current flows through the resistor.

If we look at the current flowing through the resistor as a function of frequency we obtain according to the current divider rule

1

IR = IS

1

ZR +1+

1

ZR ZC ZL

=

IS

j L (R - 2LCR) +

j L

(1.21)

And the transfer function becomes

H () = IR =

L

( ) IS

R - 2LCR 2 + (L)2

(1.22)

Again for L=47mH and C=47?F and for various values of R the transfer function is plotted on Figure 5.

For the parallel circuit the half power frequencies are found by letting H () = 1 2

6.071/22.071 Spring 2006, Chaniotakis and Cory

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1=

L

( ) 2

R - 2LCR 2 + (L)2

Solving Equation (1.23) for we obtain the two ? power frequencies.

1

=

-

1 2RC

+

1 2RC

2

+

1 02

2

=

1 2RC

+

1 2RC

2

+

1 02

(1.23)

(1.24) (1.25)

Figure 5

And the bandwidth for the parallel RLC circuit is

The Q factor is

BP

=

2

- 1

=

1 RC

Q

=

0 BP

= 0RC

= R 0 L

6.071/22.071 Spring 2006, Chaniotakis and Cory

(1.26) (1.27)

8

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