Passive Filters, Physics tutorial

Introduction:

Passive filters include combinations of resistors, inductors and capacitors. They don't have active components nor do they rely on external power supplies. Series Inductors block high-frequency signals and conduct low-frequency signals, whereas series capacitors conduct high frequency signals and block low frequency signals.

The filter in which signal passes through the inductor, or in which capacitor gives the path to ground, presents less attenuation to low frequency signals than high-frequency signals and is low-pass filter. If signal passes through the capacitor, or has path to ground through inductor, then filter presents less attenuation to high-frequency signals than low-frequency signals and is high-pass filter. Resistors on their own contain no frequency-selective properties, but are added to inductors and capacitors to find out time-constants of circuit, and thus frequencies to which it responds.

Order of the filter is determined by number of reactive elements of filter while inductors and capacitors are reactive elements of filter, an LC tuned circuit utilized in the band-pass or band-stop filter is considered the single element although it comprises of two components.

Band-pass filter:

There are five basic kinds of filter and these are bandpass, notch, lowpass, high-pass, and all-pass filters. If you take Bandpass filter as a typical illustration, number of possible bandpass response characteristics is infinite, but all share same basic form.

2057_Bandpass Filters curve.jpg

The sketches explain numerous bandpass amplitude response curves and only first curve (a) can be referred to as ideal bandpass response. Curve has completely constant gain within pass band, zero gain outside pass band, and abrupt boundary between the two.

The Curves (b) through (f) represent examples of the few bandpass amplitude response curves which approximate ideal curves with varying degrees of accuracy while some bandpass responses are extremely smooth, other have wavy ripple that is actually gain variations in their passband and others have ripple in their stopband as well.

Bandpass filters have two stopband; one above and one below the passband. Stopband is range of frequencies over which unwanted signals are attenuated.

It is difficult to find out through observation alone exactly where boundaries of passband and stopband lie as they are seldom obvious and consequently, frequency at which the stopband starts is generally determined by system specific requirements of the given attenuation at given frequency that states beginning of stopband. Rate of change of attenuation between passband and stopband also varies from one filter to next and slope of the curve in this region depends strongly on order of filter. Higher order filters present steeper cut-off slopes.

Bandpass filters are utilized in electronic systems to divide the signal at one frequency or within band of frequencies from signals at other frequencies. As example, the filter whose purpose was to pass the desired signal at frequency f1, while attenuating as much as possible unwanted signal at frequency f2. This function could be done by the suitable bandpass filter with centre frequency f1. Such a filter could also reject unwanted signals at other frequencies outside of passband, so it could be helpful in situations where signal of interest has been contaminated by signals at several different frequencies.

Band-reject filter:

Band-reject filter has effectively opposite function of bandpass filter and typical illustration of filter elements is diagrammed below.

256_Band Reject Filter.png

This filter also called as Notch Filter has transfer function:

HN(s) = VOUT/VIN = s2 +1/s2 + s + 1

Notch filters are utilized to remove unwanted frequency from signal with as little effect as possible on all other frequencies.

The quantities utilized to explain behaviour of band-pass filter are also appropriate for the notch filter. The curve (a) shows the ideal notch response, while other curves show different approximations to ideal characteristic.

1289_Band Reject Filter curve.jpg

Low-pass filter:

Third filter type is low-pass filter. Low-pass filter passes low frequency signals, and rejects signals at frequencies above filter's cutoff frequency.

The circuit diagram below shows the circuit elements connected to realize a low pass filter.

330_Low Pass Filter.jpg

The transfer function is:

HLP(s) = VOUT/VIN = 1/(s2 + s + 1)

Transfer function has more gain at low frequencies than at high frequencies as ω approaches 0, HLP approaches 1; as ω approaches infinity, HLP approaches 0. Amplitude and phase response curves are given with possible amplitude response curves and various approximations to unrealizable ideal low-pass amplitude characteristics take different forms, few are monotonic by always having negative slope whereas others have ripple in passband and/or in stopband.

Low-pass filters are utilized whenever high frequency components should be removed from the signal. This can be shown with light-sensing photodiode. If light levels are low, output of photodiode could be very small, permitting it to be partially obscured by noise of sensor and its amplifier, whose spectrum can extend to very high frequencies. If the low-pass filter is placed at output of amplifier, and if its cutoff frequency is high enough to permit desired signal frequencies to pass, overall noise level can be decreased.

1306_Low Pass Filter curves.jpg

High-pass filter

Opposite of low-pass is high-pass filter. High pas filter rejects signals below its cut-off frequency and a high-pass filter can be realized with circuit diagram below.

2479_highpass filter.png

The transfer function for this filter is:

HHP(S) = VOUT/VIN = s2/(S2 + S + 1)

1471_High Pass Filter curves.jpg

Amplitude response of high-pass is mirror image of lowpass filter response and examples of high pass filter responses are shown with ideal response in (a) and different approximations to ideal shown in (b) through (f).

High-pass filters significant and finds application when rejection of low frequency signals is desirable. The application of high pass filters is in high fidelity loudspeaker systems where high-frequency audio drivers called as tweeters can be damaged by low frequency audio signals of sufficient energy.

The high-pass filter between broadband audio signal and tweeter input terminals will prevent low frequency program material from reaching tweeter. In conjunction with the low-pass filter for low frequency driver and maybe other filters for other drivers, high-pass filter is part of what is called as a crossover network.

Phase-shift filter:

The response of final filter type has no effect at all on amplitude of the input signal over broad band of frequencies but instead changes phase of signal without affecting amplitude. This kind of filter is known as an all-pass or phase shift filter.

Effect of the shift in phase is explained by two sinusoidal waveforms where curves are identical except that they are out of phase. Output of bandpass filter has undergone the time delay relative to input signal.

1102_Phase Shift Filter curve.jpg

Relation between time delay and phase shift is TD = θ/2πω, so if phase shift is constant with frequency, time delay will decrease as frequency increases. All-pass filters are usually utilized to introduce phase shifts in signals to cancel or partially cancel any unwanted phase shifts previously imposed on signals by other circuitry or transmission media. The illustration shows the curve of phase vs frequency for all-pass filter with the transfer function

HAP(s) = (S2 - S + 1)/(s2 + s + 1)

Absolute value of gain is equal to unity at all frequencies, but phase changes as the function of frequency.

All of the transfer functions share same denominator. All of the numerators are composed of terms found in denominator: high-pass numerator is first term (s2) in denominator, bandpass numerator is second term, low-pass numerator is third term, and notch numerator is sum of denominator's first and third terms (s2 + 1). Numerator for all-pass transfer function is little different in that it comprises all the denominator terms, but one of the terms has negative sign.

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