Two-Way Crossover and Second Order Butterworth - Essay Example

Two-Way Crossover, Second Order Butterworth A two way crossover, second order Butterworth filter circuit is as shown in the figure below. High pass and low pass 2nd order Butterworth 2-way crossover Filter (ApICSLLc 2011) This is a passive filter circuit that combines inductance and capacitance to allow a specific range of signal frequency to pass and stops those out of its range. The simple LC network forms a series tuned network, where if we consider the low pass filter the series component (inductor) reactance increases while that of the shunt component (capacitor) decreases. This phenomenon causes a doubled increase in attenuation in comparison to a simple resistor-capacitor or resistor-inductor filter. (Winder 1997) At frequencies way above the pass band the rate of increase in attenuation is 12dB/octave at a 90 degrees phase shift which goes to 180 degrees at very high frequencies. The rate of the attenuation here depends on the filter order, which in turn is determined by the number of reactive components in the ladder. For instance in our case here there are two reactive components, hence the filter order is two and its rate of attenuation is given by nx6dB/octave = 12dB/octave since n=2. Its response at different values of n is as shown below. Attenuation versus frequency for a Butterworth filter with different values of n (Winder 1997, p.55) A two-way crossover has a combination of a high pass and a low pass filter which could be used to drive a tweeter and a loudspeaker at the same time. These two could be fed from the same model of an amplifier if and only if it can accommodate the frequency ranges of both of them. However this is a wide range hence it’s impossible to come up with such an amplifier, hence the two-way crossover is used. With the low pass filter designed as explained earlier its high pass counterpart can be derived from it following the simple fact that their frequency response are reciprocal of one another. This means that attenuation of a low pass filter at say a frequency of w=2 is the same as the equivalent high pass at w=0.5. Deriving from this the high pass filter components are the reciprocal of the normalized low pass filter, such that where there are capacitors in the low pass model they are replaced by inductors in the high pass model. Similarly, where there are inductors in the low pass model, they are replaced by capacitors in the high pass model. The use of a combination of the two models helps achieve low and high frequency paths without having to bear the losses associated with convectional power-splitter circuits. To understand how the circuit operates we can evaluate the behavior of the constituent components. One of these components is the capacitor which is a passive basic circuit element that stores electric energy. It has a current-voltage relationship given by,  . Voltage across it doesn’t change instantaneously and when there is no change of it at all (steady state) it acts as an open circuit. Further on the lower frequency range the capacitor is open-circuited and on the higher range it is short-circuited. On the other hand an inductor stores magnetic energy. It has a current-voltage relationship given by,  . Current across it doesn’t change instantaneously and when there is no change of it at all (steady state) it acts as short circuit. Further on the lower frequency range the inductor is short-circuited and on the higher range it is open-circuited. . (MIT’s online courses) In the low pass filter, at a source voltage of 0 Hz (direct current) the capacitor acts as an open circuit and the inductor as a short circuit therefore, a high current would flow if a load and the source voltage appears across it, to fulfill the Kirchhoff’s voltage law in a loop. With increase in frequency towards resonance, capacitive reactance decreases  and inductive reactance  increases. Reduced current due to the reactive impedances flows in the circuit, voltage drops vary with the impedance of each component. At resonance, the reactive impedances cancel each other, the current rises without limit and the voltage source appears short circuited. Since the inductor behaves as an open circuit at high frequencies, no current flows to give full attenuation. (Bigelow 2009). The circuit is affected differently by the components’ tolerances. For the inductors and the capacitors any deviation from the design value leads to a shift in the resonant frequency. For the resistive load (loudspeaker), a shift from the rated value can have an adverse effect on the filters response. Elliot (2004) says that “As an example, lets assume that a loudspeaker has a flat impedance curve, and is exactly 8 ohms. A Butterworth crossover may be designed that will have a -3dB frequency of 3 kHz. Should the impedance rise to 11 ohms, the -3dB frequency will increase to over 3.8 kHz, and the filter shape is changed. A Bessel filter changes to a Butterworth, or a Butterworth filter to a Chebyshev in terms of their responses.” In most cases the loudspeaker will not be manufactured at exactly the rated value. Similarly, the capacitive and inductive reactances are usually not a hundred percent reactive but will have a resistive component. This amplifies the complexity of the design process as the order increases since the tolerances becoming more exacting. This means that heat dissipation occurs across these components as it is caused by power losses due to resistive loads. However, for the capacitor the effect is minimal unlike the inductor whose coil may pose a considerable resistance to the circuit. To curb these losses, a wire of the highest gauge possible serves well in the inductor windings to prevent mechanical noise, it is worth impregnating the finished coil in varnish. For the capacitors the bipolar (non-polarized) electrolytic ones should be avoided as they are affected by high currents making them loose their capacitance over time. (Elliot 2004). In conclusion passive filters do not consume any power hence they are advantageous for low power systems but pose a challenge of being bulky due to the use of inductors especially if they are to be used in a high current environment. A suitable method of manufacturing this kind of a circuit is by load analysis where the specifications are obtained from a real environment scenario as opposed to theoretical design. Reference list: ApICSLLC 2011, Filter Design, Ohio, viewed 27 April 2011, < http://www.apicsllc.com/apics/Misc/filter2.htm> Bartelt, T 2011, ‘Voltage and Current Amplitudes in an LC Circuit’, in WISC-Online, viewed 27 April 2011, < http://www.wisc- online.com/Objects/ViewObject.aspx?ID=ACE12407>. Bigelow, K 2009, ‘Series LC Circuits’, in Play-Hookey, viewed 27 April 2011, < http://www.play-hookey.com/ac_theory/ac_lc_series.html>. Elliot, R 2004, ‘Design of Passive Crossovers’, in Elliot Sound Products, viewed 27 April 2011, < http://sound.westhost.com/lr-passive.htm >. Lacante, K 1991, ‘ A Basic Introduction to Filters-Active, Passive and Switched- Capacitor’ National Semiconductor, viewed 27 April 2011, < http://www.swarthmore.edu/NatSci/echeeve1/Ref/DataSheet/IntroToFilters.pdf> Massachusetts Institute of Technology 2011, Circuit Basics, Massachusetts, viewed 27 April 2011, . Mitchell, J n.d., Crossovers, Frazier Loudspeakers, viewed 27 April 2011, < http://www.frazierspeakers.com/download/cross.pdf>. Winder, S 1997, ‘Analog and digital filter design’, 2nd edn, Newness, New York. Read More