4.5 Design of Analogue Filters

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137

Explanatory Example

For ECG filtering, design a passive Chebyshev- 2nd order filter using an RLC element

which has a passband cut-off frequency of 200 Hz and at this an attenuation of 3 dB.

To do this, the corresponding normalised low-pass is first determined again. Since

at an attenuation of 3 dB the magnitude square of the frequency response has the

value of 0.5, it follows from Equation 4.26 that here ϵ = 1 must be. From Equation 4.45

it follows for the poles:

P±0 = ± sin(π/4) ⋅sinh(tI±0) + j cos(π/4) ⋅cosh(tI±0)

P±1 = ± sin(3π/4) ⋅sinh(tI±1) + j cos(3π/4) ⋅cosh(tI±1)

tI±0 = tI±1 = −0.5 arcsinh(1) = −0.449687

(4.58)

resp.

P±0 = ∓0.321787 + j0.776887

P±1 = ∓0.321787 −j0.776887

(4.59)

and for |A0|, when all poles are in the left P- half plane, from Equation 4.57:

A0 =

1

ϵ ⋅2^{n−1 =}

1

1 ⋅2^{2−1 = 1}

2 ^.

(4.60)

Thus the poles are conjugate complex. Those lying in the left P half plane are se-

lected, i.e. P+0 = −0.3217871 + j0.776887 and P+1 = −0.3217871 −j0.776887. The

transfer function of the normalised low-pass filter can now be given:

AnTP(P) = A0

1

(P −P+0)(P −P+1) ^.

(4.61)

In order to realise the desired low pass with a cut-off frequency fD of 200 Hz from

the normalised low pass, the normalised angular frequency Ω= 2πF must be replaced

by a suitable frequency transformation according to Table 4.5. In this case we get F =

f/fD resp.

Ω= ^ω

ωD

,

mit ωD = 2π ⋅200 Hz .

(4.62)

For the desired low-pass we obtain the complex transfer function

AnTP (P = jΩ= j ^ω

ωD

) =

1/^√2

1 + 0,9101795 ^jω

ωD ^{+ 1.4142137( jω}

ωD ^{)2 .}

(4.63)

A 2nd order RLC low pass can be realised by a simple voltage divider. The transfer

functions of the calculated Chebyshev-lowpass and the RLC element must be identical

for this. This is possible if, for example, the damping factor 1/^√2 is not taken into

account, i.e.

A^󸀠

TP^{(jω) := ATP(jω) ∗√2 =}

1

1 + 0.9101795 ^jω

ωD ^{+ 1.4142137( jω}

ωD ⁾²

= ARLC(jω) =

1

1 + jωRC + (jω)²LC ^.

(4.64)