SIMPLE RC FILTERS

A filter allows ac of some frequencies to pass through it with the amplitude (voltage or current) essentially unchanged, but the amplitudes of other frequencies are decreased. (An active filter, involving transistors, can increase certain frequencies, as well as decreasing some.) A simple example, using just one resistor and one capacitor, which is called a low-pass RC filter, is shown in Fig. 11.1. It is a type of first order Butterworth filter. 

As mentioned previously (pages 56 and 114), the circle symbol represents a source of ac voltage. The resistor and capacitor together make a voltage divider. 

At low frequency, the capacitor acts as a very high resistance, and with dc it is an open circuit (essentially infinite resistance). Therefore, more voltage will appear across the output terminals — hence the name low-pass.  

On page 99 of the Capacitors chapter, the concept of reactance was explained. This is a sort of effective resistance for a capacitor in an ac circuit, and its symbol is Xq. Since the RC filter is similar to a potentiometer with a pair of resistors 

Putting Z into eqn. 11.1 above, then the output voltage becomes  

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The slope of the fall-off characteristic of a simple RC filter is 6 db/octave or 20db/decade. Thus for a 10-to-l change in frequency there is a 10-to-l change in signal strength. If simple RC low-pass filters are used to eliminate noise, one must be careful that the cutoff frequency of the filter does not overlap principal frequency components of the desired signal. Because of the... 

As an example of the use of Eq. (3-321), we shall calculate the output power density spectrum of an RC filter whose input consists of white noise. The RC filter in question is shown in Fig. 3-14. It is a simple matter to verify that the impulse response of this filter is given by... 

When simple electrical RC filters are treated, the truncated exponential e, x,H(x) is indispensable. Its transform is given by (2n) 1/2(1 — jco)/( 1 + co2). If the truncated exponential is reflected about the origin, eliminating H(x) and leaving e x, the imaginary part of the transform disappears. We obtain the transform (2/7c)1/2/(l + co2). This is the resonance contour, Cauchy distribution, or Lorentzian shape encountered previously in Section III.B. 

The design of optimum electrical filters is a highly developed discipline that we shall not address here. It is worthwhile, however, to discuss briefly the simple single-stage RC filter that is widely used. When all factors are... 

As discussed in Chapter 1, any linear analog filter network may be used. Its performance may always be described by convolution with a filter function, here called sE. Many modern digital filters may be thus described. The instrument employed in the present work used a simple single-stage RC filter of the type analyzed in Section II.D of Chapter 2. 

The error/deviation from the actual curve is usually very small if we replace it with its asymptotes (for first-order filters). For example, the worst-case error for the gain of the simple RC network is only -3dB, and occurs at the break frequency. 

A low-pass filter is one that passes frequencies from dc (0 Hz) to f and significantly attenuates all other frequencies. The passband of the low-pass filter is shown in the shaded area of Eig. 1(b). Fig. 1 shows the most basic low-pass filter circuit consisting of just one resistor R and one capacitor C and its response curve. This basic RC filter has a single pole and its decrease rate reaches -20 dB/decade where the frequency of signals goes beyond the critical frequency. (A decade is a ten times change in frequency). The critical frequency of the simple low-pass RC filter occurs where f = U2nRC. The output at the critical frequency is 70.7% of the input (Thomas L. Floyd David M. Buchla 2007). Fig. 1(b) illustrates the basic one pole response (-20 dB/decade). 

FIGURE 1.5 Bode magnitude and phase responses fora simple RC lowpass filter. 

Filters with steeper slopes than the simple RC types can be made with combinations of inductors and capacitors, as in Fig. 11.4. These can be "second order Butterworth" types as shown here, or they can involve more inductors or capacitors, plus resistors, as needed. 

Hum (60 Hz) in a signal may be greatly attenuated by using a simple RC twin-tee circuit as a rejection filter, provided that the desired signal has no major components at or near 60 Hz. The design conditions for the circuit (Valley and Wallman, 1948) are... 

Noise elimination -> Noise can be eliminated using either electronic circuits or software procedures. The hardware methods based on simple RC circuits can work well in potentiostats. More complex circuits may have names such as noise filter, noise jammer, or noise killer. Very helpful in protecting the electrochemical cell from external noise is the application of a Faraday box (cage). When smoothing of data is done with the soft-... 

A very simple linear system is the RC filter consisting of a resistor R and a capacitor C. An RC low-pass filter is seen in Fig. 2. 

Figure 3 shows a simple single-section RC low-pass filter network. KirchhofFs law tells us that... 

Passive filter A kind of usually simple filters composed of elements such as resistors (R), capacitors (C), and inductors (L) that do not depend upon an external power supply. There are different passive filters such as the so-called RC, RL, LC, and RLC varieties. Inductors block high-frequency signals and conduct low-frequency signals, while capacitors do the reverse. Resistors have no frequency-selective properties, but are added to inductors and capacitors to determine the time-constant of the circuit. 

An appropriate filter would consist of a high-pass filter and a series of bandpass filters or band-reject filters. The high-pass filter would be designed to pass from just below 5 kHz and the band-pass filters would be tuned to pass 5 kHz, 15 kHz, 25 kHz, etc. A simple system as shown in Figure 9.10b could be tried It consists of an RC high-pass filter and a series-resonant set... 

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