Frequency Response: Band Pass Filter And Resonant (Natural) Frequency

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{5}A_{51} Base Sequence = 12735 String Sequence = 12735 - 5 - 51 **

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Frequency Response: Band Pass Filter And Resonant (Natural) Frequency

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Figure 7.56 shows a circuit of a simple RLC filter. Determine:

(a) The frequency response of the RLC filter in terms of the *natural* or *resonant* frequency.

(b) The *bandwidth* in terms of the *natural* or *resonant* frequency and the *quality factor*.

**The string**:

S_{7}P_{5}A_{51} (Physical change).
**The math**:

Pj Problem of interest is of type *change*. Frequency problems are *change problems*. They are similar to velocity, acceleration and duration problems which are also *change problems*.

In general, *frequency response* is a measure of the variation in a load-related parameter as a function of the frequency of the excitation element. In electric circuits, the *load-related* parameter is usually the voltage across a load or the current through it and the excitation element is usually a sinusoidal signal. Consequently, any of the following is an acceptable definition of the frequency response of a circuit:

H_{V}(jω) = V_{L}(jω)/V_{s}(jω)

Where H_{V}(jω) is frequency response of load; V_{L}(jω) is voltage across load; V_{s}(jω) is frequency dependent voltage source.

H_{I}(jω) = I_{L}(jω)/I_{s}(jω)

Where H_{I}(jω) is frequency response of load; I_{L}(jω) is current through load; I_{s}(jω) is frequency dependent current source.

(a)H(jω) = V_{o}(jω)/Vi(jω)

By the voltage divider rule:

V_{o}(jω) = V_{i}(jω)[R/(R + (1/jωC) + jωL)]
= V_{i}(jω)[jωCR/(1 + jωCR + (jω)^{2}LC)]

So, H(jω) = [jωCR/(1 + jωCR + (jω)^{2}LC)]-------(1)

In phasor form:

H(jω) = ωCR/√[(1 -ω^{2}LC)^{2} + (ωCR)^{2}]<[π/2 -arctan(ωCR/(1 - ω^{2}LC))] -------(2)

(b)The natural or resonant frequency:

ω_{n} = √1/LC.

The quality factor:

Q = 1/(ω_{n}CR) = (1/R)(√(L/C)).

The damping ratio:

1/(2Q) = (R/2)(√(C/L))

Bandwidth, B = ω_{n}/Q.

So, H(jω) in equation (1) is expressed in terms of resonant frequency and quality factor as follows:

H(jω) = [(1/(Qω_{n}))jω]/[(jω/ω_{n})^{2} + (1/(Qω_{n}))jω + 1]

The *bandwith* (passband) is a frequency range. The *band pass filter* allows input signal frequencies within the range to pass through it.

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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