Frequecy Response: Low Pass Filter And Cut-Off 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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Frequecy Response: Low Pass Filter And Cut-Off Frequency

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Figure 7.49 shows a circuit of a simple RC filter. Determine:

(a) The phasor form of the frequency response, *H(jω)* in terms of ω, R and C.

(b) The cutoff frequency of the RC filter.

**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)/V_{i}(jω)

By the voltage divider rule,:

V_{o}(jω) = V_{i}(jω)[(1/jωC)/(R + 1/jωC)] = V_{i}(jω)[1/(1 + jωRC)].

So, H(jω) = [1/(1 + jωRC)]----(1)

In phasor form:

1 = 1e^{j0}

And

1 + jωRC = [1 + (jωRC)^{2})]^{1/2}(e^{jarctan(ωCR})

So, in Phasor form:

H(jω) = [1/(1 + (jωRC)^{2})]^{1/2}(e^{-jarctan(ωCR})----(2)

Where |H(jω)| = [1/(1 + (jωRC)^{2})]^{1/2}

And the phase angle is:

<H(jω) = -arctan(ωCR).

Equations (1) and (2) reveals that at ω = 0,

V_{o}(jω = 0) = V_{i}(jω = 0).

That is, no filtering at ω = 0 (DC signal).

As the signal frequency increases, the magnitude of the frequency response decreases.

(b) *Cutoff frequency, ω _{0}* = 1/RC.

At low frequencies (When ω << 1/RC), filter is called a

The value of |H(jω)| at the cufoff frequency, is 1/√2 = .707.

The cufoff frequency depends only on the values of R and C. As a result filtering characteristics can be selected for various values of R and C.

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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