Bandpass Function, Center Frequency, Half Power Frequency, Bandwidth, Quality Factor And Resonance

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Bandpass Function, Center Frequency, Half Power Frequency, Bandwidth, Quality Factor And Resonance

**(a)** Indicate the general representation of the *bandpass function* in the **s** domain
**(b)** Indicate the *frequency response* of the bandpass function
**(c)** Determine the *center frequency* of the frequency response
**(d)** Determine the *half-power frequencies* of the frequency response
**(e)** Determine the *bandwidth* of the frequency response
**(f)** Determine the *quality factor*
**(g)** Relate the *quality factor* to *resonance*
**(h)** If **H**(s) = **s**/(**s**^{2} + a**s** + b), determine a and b such that the magnitude of the frequency response |**H(ω)**| has a maximum at 100 Hz with a half-power bandwidth of 5 Hz. What is the quality factor?

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

Pj Problem of Interest is of type *change* (change - physical).

**(a)** General bandpass function in the **s** domain:

**H**(s) = k**s**/(**s**^{2} + a**s** + b)------(1) Where k, a, b > 0.

**(b)** Frequency response of bandpass function:
**H**(jω) = k**jω**/(**jω**^{2} + a**jω** + b) = k**jω**/(b - ω^{2} + ajω)------(2)

(c) |**H**(jω)|^{2} = k^{2}ω^{2}/[(b - ω^{2})^{2} + a^{2}ω^{2})]
= k^{2}/[a^{2} + (b - ω^{2})^{2}/ω^{2}]------(3)

The frequency (ω_{o}) at which |**H**| is maximum is the center frequency.

So from eq(3):|**H**| is maximum when (b - ω^{2}) = 0 that is, when ω = √b

So, center frequency, ω_{o} = √b.

At the center frequency, |**H**|_{max} = k/a.

(d) Consider the following equalities:

|**H**(ω_{l}|^{2} = |**H**(ω_{h}|^{2} = (1/2)|**H**(ω_{o})|^{2}-------(4)

ω_{l} and ω_{h} in eq(4) are called the *half-power lower frequency* and the *half-power upper frequency* respectively.

Substituting eq(3) into eq(4) and solving, we have:

ω_{l} and ω_{h} as the roots of:

(b - ω^{2})^{2}/ω^{2}] = a^{2}

So, ω_{l} = √(a^{2}/4 + b) - a/2

And ω_{h} = √(a^{2}/4 + b) + a/2
**(e)** Bandwidth, β = ω_{h} - ω_{l} =√(a^{2}/4 + b) + a/2 - √(a^{2}/4 + b) - a/2 = a.

ω_{h}ω_{l} = b = (ω_{o})^{2}
**(f)** Quality factor, Q = ω_{o}/β = √b/a

Quality factor measures sharpness of frequency response around the center frequency.
**(g)** Increasing quality factor tends towards resonance.
**(h)** Bandwidth = β = a = 2π(5) = 31.416

ω_{o} = 2π(100) = 628.3185 = √b

So, b = (628.3185)^{2} = 394784

Quality factor, Q = ω_{o}/β = 628.3185/31.416 = 20

Math

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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