Fundamentals of Measurement Technology (13) Prof Wang Boxiong
Fundamentals of Measurement Technology (13) Prof. Wang Boxiong
5.4.3 Types of filters 1. Low-pass filters e C (a) Electrical K (b) Mechanical /se k psI/in n psI ARA (c)Hydraulic Fig. 5.40 Low-pass filters
5.4.3 Types of filters 1. Low-pass filters Fig. 5.40 Low-pass filters
5.4.3 Types offilters · RC lower- pass filter de Rc-o+e =e dt (5.63) H +1 (5.64) where the frequency f=_1 2TRO corresponds to the point of amplitude attenuation of -3dB, and is the upper cut-off frequency
5.4.3 Types of filters • RC lower-pass filter o i o e e dt de RC + = ( ) 1 1 ( ) + = = s s e e H s i o (5.63) (5.64) where the frequency RC f 2 1 = corresponds to the point of amplitude attenuation of –3dB, and is the upper cut-off frequency
5.4.3 Types offilters H(f) For simple first-order systems, the attenuation √2 is quite gradual with frequency 6dB/octave 1/2πr By adding more"stages P(f) 1/2xr 0 (see Fig. 5.42(a)the 45 sharpness of cutoff may be increased. But the 0)° disadvantage: loading effects should be take Fig. 5.41 Amplitude and phase characteristics of the first-order Into account low-pass filters
5.4.3 Types of filters For simple first-order systems, the attenuation is quite gradual with frequency,6dB/octave. By adding more “stages” (see Fig.5.42(a))the sharpness of cutoff may be increased. But the disadvantage: loading effects should be take into account. Fig. 5.41 Amplitude and phase characteristics of the first-order low-pass filters
5.4.3 Types offilters Many electrical filters are active devices based on op-amp technology. Passive filters have very low noise, require no power supplies, and have a wide dynamic range: Active filters are much more adjustable and versatile, can cover very wide frequency ranges, have very high input and very low output impedances, and can be configured for simple switching from low-pass to high-pass and combination for band-pass or band-reject behavior
5.4.3 Types of filters Many electrical filters are active devices based on op-amp technology. Passive filters have very low noise, require no power supplies, and have a wide dynamic range; Active filters are much more adjustable and versatile, can cover very wide frequency ranges, have very high input and very low output impedances, and can be configured for simple switching from low-pass to high-pass and combination for band-pass or band-reject behavior
5.4.3 Types offilters Fig. 5.44(b)illustrates a state-variable filter with electrically adjustable parameters. The state-variable filter provides three simultaneous outputs: a low-pass, a high-pass, and a band-pass s2/o2+2as/o,+1 (566) OOR-C/V (567) /On+2/on+1 bp (202RC /V )s (5.68) s/n+2/n+1
5.4.3 Types of filters Fig.5.44(b) illustrates a “state-variable filter” with electrically adjustable parameters. The state-variable filter provides three simultaneous outputs: a low-pass, a high-pass, and a band-pass: ( ) / 2 / 1 1 2 2 + + = − i n n ep s s s e e ( ) ( ) / 2 / 1 100 / 2 2 2 2 2 2 + + = − n n c i hp s s R C V s s e e ( ) ( ) / 2 / 1 20 / 2 2 + + = − n n C i b p s s RC V s s e e (5.66) (5.67) (5.68)
5.4.3 Types offilters C b Bessel Butterworth Eight pole filters Fig. 5.42 Sharper-cutoff low-pass filters(1)
5.4.3 Types of filters Fig. 5.42 Sharper-cutoff low-pass filters(1)
5.4.3 Types offilters 5 LHH ARY t 0.05 dB passband ripple 15 Eight pole In Butterworth fifi 30 tow pass f -35 40 50 -55 Eight- pole 0.5 d8 Chebyshev low Fass Attenuation 70 Seven pole -78 dB- 580 Cauer elliptic f=165′ f。=1.06 Typical seventh-order 10511523456811523456789 Cauer filte ( Not to scale Fig 5.42 Sharper-cutoff low-pass filters(2)
5.4.3 Types of filters Fig. 5.42 Sharper-cutoff low-pass filters(2)
5.4.3 Types offilters Many methods can be adopted in filter design to raise the filter order. four basic designing methods Butterworth, Chebyshev, Bessel, Cauer or elliptical filters With the increase in filter order the transition band become steeper, the attenuation increases thus the filtering effect strengthened Theoretically, it is possible to cascade several RC networks to enhance filter order and to accelerate the attenuation In practice, loading effect between different cascaded stages must be considered. To solve the loading effect, a better way is to use operational amplifiers to construct active filters
5.4.3 Types of filters • Many methods can be adopted in filter design to raise the filter order. four basic designing methods: Butterworth, Chebyshev, Bessel, Cauer or elliptical filters. With the increase in filter order, the transition band become steeper, the attenuation increases, thus the filtering effect strengthened. • Theoretically, it is possible to cascade several RC networks to enhance filter order and to accelerate the attenuation. In practice, loading effect between different cascaded stages must be considered. To solve the loading effect, a better way is to use operational amplifiers to construct active filters
5.4.3 Types offilters RC-network is connected to the input terminal of an op-amp The filter's cut-off frequency is then R and its gain K=1+ 2ITRC Connecting a high-pass network to the feedback loop at the amplifier yields a low-pass filter. As shown in Fig. 5.43(b), the filter has a cut-off frequency c2 2,c and its gain x Re f∫ R
5.4.3 Types of filters RC-network is connected to the input terminal of an op-amp. The filter’s cut-off frequency is then : RC f c 2 1 2 = and its gain 1 1 R R K F = + Connecting a high-pass network to the feedbackloop at the amplifier yields a low-pass filter. As shown in Fig. 5.43 (b), the filter has a cut-off frequency : R C f F c 2 1 2 = and its gain R1 R K F =
