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

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Figure 7.50 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)Frequency response, *H(jω)* = V_{o}/V_{i}(jω)

By the voltage divider rule:

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

So, H(jω) = V_{o}/V_{i}(jω) = [jωRC/(1 + jωRC)]----(1)

Phasor form:

jωRC = (ωRC)e^{jπ/2}

Phasor form:

1 + jωRC = (1 + (jωRC)^{2})^{1/2}(e^{jarctanωCR}

So, phasor form:

H(jω) = [ωRC/(1 + (jωRC)^{2}))^{1/2}](e^{j[π/2-arctan(ωCR)]}----(2)

where |H(jω)| = ωRC/(1 + (jωRC)^{2})^{1/2}

And
the phase angle:

<H(jω) = π/2-arctan(ωCR)

Equations (1) and (2) reveal 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 approaches infinity, the magnitude of the frequency response asymptotically approach 1.

(b) Cutoff frequency, ω_{0} = 1/RC.

When ω >> 1/RC, filter is called a *high pass filter*. A *high-pass filter* passes signals at high frequencies and filters out signals at low frequencies (ω << 1/RC).

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

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

In essence the *filter effect* results in the *scaling* and *phase angle shifting* of the input signal. In mathematical terms, for the following input signal in phasor form:

V_{i} = |V_{i}|e^{jφi}

The scaling of the output signal becomes:

V_{o} = |H|(V_{i}).

And the phase angle of the output signal becomes:

φ_{o} = <H + φ_{i}.

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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