EE133 Winter 2002 Cookbook Filter Guide Welcome to the Cookbook Filter Guide Don t have enough time to spice out that perfect filter before Aunt Thelma comes down for dinner Well this handout is for you The following pages detail a fast set of steps towards the design and creation of passive filters for practical use in communications and signal processing Amaze your friends and neighbors as you do twice the amount of work in half the time Here s a brief overview of the steps Specify your filter type Implement a low pass version of your filter Transform it to what you really want high pass bandpass bandstop Simulate and Iterate Step 1 What Filter Do I Want This is where you have to do most of your thinking It s no good to cook up a nice steak dinner if Auntie Thelma turned vegetarian last year Here are the specifications that are most often quoted with filters Filter Type Low Pass High Pass Band Pass Band Stop Center frequency rad s or Hz Bandwidth rad s or Hz Cut off Roll off rate dB Minimum Attenuation Required in Stopband Input Impedance ohms Output Impedance ohms Overshoot in Step Response Ringing in Step Response We will be creating filters through a method known as the insertion loss method Insertion Loss or the Power Loss Ratio PLR is defined as PowerSource 1 PLR 2 PowerDelivered 1 is our famliar reflection coefficient It turns out that this is expressible in the following form of M 2 PLR 1 N 2 Where M and N are two real polynomials Which simply means that we can define an arbitrary filter response and use this formula to match it to real components thus allowing us to make it physically realizable There are several standard filter responses each with their own advanatages and drawbacks Standard Filter Responses Butterworth AKA Maximally Flat or Binomial Filters Butterworth filters are general purpose filters Another common name for them is a maximally flat filter which refers to the relatively flat magnitude response in the passband Attenuation is 3dB at the design cutoff frequency with a 20dB decade roll off per pole above the cutoff frequency 1 EE133 Winter 2002 Cookbook Filter Guide Chebyshev AKA equal ripple magnitude Filter Chebyshev filters have a steeper attenuation above the cutoff frequency but at the expense of amplitude ripples in the pass band For a given number of poles a steeper cutoff can be achieved by allowing more pass band ripple The cutoff frequency is defined at the point at which the response falls below the ripple band of the pass band Bessel AKA Linear Phase or Maximally Flat Time Delay Filters Sometimes a design requires a filter to have a linear phase in order to avoid signal distortion In general a good phase output i e linear always comes at the expense of a good magnitude response i e fast attenuation Elliptic Filters For the previous filters as the frequency gets progressively further from the center frequency the attenuation increases Sometimes a design only requires a minimum attenuation in the stop band This relaxes some constraints on the response which allow a better cutoff rate However this filter has ripples in both the passband and stop band Step 2 Prototyping a Low Pass Design Having finally specified your filter it s time to prototype a low pass version of your filter Although it seems counterintuitive to spend time on a filter that doesn t even necessarily have the passband stop band characteristics of your desired filter it will become apparent that there is relation between the values derived for a low pass situation and the other filter types As an example we will design a low pass filter for a source impedance of 50 ohm a cut off frequency of 1MHz and which requires a minimum attenuation of 40dB at 10MHz Determine the type of filter and N the order of the filter First we have to determine which of the filter types we want to use Do we care about having linear phase Or is maxmium cut off attenuation the critical factor Once that s been done we can then determine the order of the filter necessary to fit the required attenuation spec Usually we refer to a graph like the one below showing the attenuation characteristics for various N versus normalized frequency for a particular filter type In this case we decide that a flat ma gnitude response in the most appropriate So then we look at the graph The definition of normalized frequency is w f norm 1 wc we find that wc 1MHz wo 10MHz which means the normalized frequency is 9 Looking on the graph we see that N 2 will easily satisfy our requirements for 40dB of attenuation 2 EE133 Winter 2002 Cookbook Filter Guide Design a Normalized Low Pass Filter using a Table Once that is done we can now design a second order prototype filter for a source impedance of 1 ohm a cut off frequency of 1 rad sec As shown in the figure below we use one of two equivalent ladder circuits Note the way the element values are numbered with g0 at the generator to gN 1 at the load How to read this chart g o generator resistance or a generator conductance gk inductance for series inductors or a capacitance for shunt capacitors g N 1 load resistance if g N is a shunt C or a load conductance if g N is a series L A key point is that the components alternate between shunt and series Note that during out prototyping inductors are always series capacitors are always shunt The only difference is whether or not the first element is series or shunt 3 EE133 Winter 2002 Cookbook Filter Guide To design a filter of a particular response i e Butterworth or Bessel there is a unique ratio of components to be used These ratios are usually kept is handy tables like the one below Element Values for Butterworth Maximally Flat Low Pass Filter Prototypes go 1 wc 1 N 1 to 10 N 1 2 3 4 5 6 7 8 9 10 g1 2 0000 1 4142 1 0000 0 7654 0 6180 0 5176 0 4450 0 3902 0 3473 0 3129 g2 1 0000 1 4142 2 0000 1 8478 1 6180 1 4142 1 2470 1 1111 1 0000 0 9080 g3 g4 g5 g6 g7 g8 g9 g10 g11 1 0000 1 0000 1 8478 2 0000 1 9318 1 8019 1 6629 1 5321 1 4142 1 0000 0 7654 1 6180 1 9318 2 0000 1 9615 1 8794 1 7820 1 0000 0 6180 1 4142 1 8019 1 9615 2 0000 1 9754 1 0000 0 5176 1 2470 1 6629 1 8794 1 9754 1 0000 0 4450 1 1111 1 5321 1 7820 1 0000 0 3902 1 0000 1 4142 1 0000 0 3473 0 9080 1 0000 0 3129 1 0000 For a second order we see that g1 and g2 must both equal 1 4142 and g3 the load must equal the go …