In applications that use filters to shape the frequency spectrum of a signal such as in communications or control systems, the shape or width of the rolloff also called the “transition band”, for a simple firstorder filter may be too long or wide and so active filters designed with more than one “order” are required. These types of filters are commonly known as “Highorder” or “n^{th}order” filters.
The complexity or filter type is defined by the filters “order”, and which is dependant upon the number of reactive components such as capacitors or inductors within its design. We also know that the rate of rolloff and therefore the width of the transition band, depends upon the order number of the filter and that for a simple firstorder filter it has a standard rolloff rate of 20dB/decade or 6dB/octave.
Then, for a filter that has an n^{th} number order, it will have a subsequent rolloff rate of 20n dB/decade or 6n dB/octave. So a firstorder filter has a rolloff rate of 20dB/decade (6dB/octave), a secondorder filter has a rolloff rate of 40dB/decade (12dB/octave), and a fourthorder filter has a rolloff rate of 80dB/decade (24dB/octave), etc, etc.
Highorder filters, such as third, fourth, and fifthorder are usually formed by cascading together single firstorder and secondorder filters.
For example, two secondorder low pass filters can be cascaded together to produce a fourthorder low pass filter, and so on. Although there is no limit to the order of the filter that can be formed, as the order increases so does its size and cost, also its accuracy declines.
Decades and Octaves
One final comment about Decades and Octaves. On the frequency scale, a Decade is a tenfold increase (multiply by 10) or tenfold decrease (divide by 10). For example, 2 to 20Hz represents one decade, whereas 50 to 5000Hz represents two decades (50 to 500Hz and then 500 to 5000Hz).
An Octave is a doubling (multiply by 2) or halving (divide by 2) of the frequency scale. For example, 10 to 20Hz represents one octave, while 2 to 16Hz is three octaves (2 to 4, 4 to 8 and finally 8 to 16Hz) doubling the frequency each time. Either way, Logarithmic scales are used extensively in the frequency domain to denote a frequency value when working with amplifiers and filters so it is important to understand them.
Logarithmic Frequency Scale
Since the frequency determining resistors are all equal, and as are the frequency determining capacitors, the cutoff or corner frequency ( ƒ_{C} ) for either a first, second, third or even a fourthorder filter must also be equal and is found by using our now old familiar equation:
As with the first and secondorder filters, the third and fourthorder high pass filters are formed by simply interchanging the positions of the frequency determining components (resistors and capacitors) in the equivalent low pass filter. Highorder filters can be designed by following the procedures we saw previously in the Low Pass filter and High Pass filter tutorials. However, the overall gain of highorder filters is fixed because all the frequency determining components are equal.
Filter Approximations
So far we have looked at a low and high pass firstorder filter circuits, their resultant frequency and phase responses. An ideal filter would give us specifications of maximum pass band gain and flatness, minimum stop band attenuation and also a very steep pass band to stop band rolloff (the transition band) and it is therefore apparent that a large number of network responses would satisfy these requirements.
Not surprisingly then that there are a number of “approximation functions” in linear analogue filter design that use a mathematical approach to best approximate the transfer function we require for the filters design.
Such designs are known as Elliptical, Butterworth, Chebyshev, Bessel, Cauer as well as many others. Of these five “classic” linear analogue filter approximation functions only the Butterworth Filter and especially the low pass Butterworth filter design will be considered here as its the most commonly used function.
Low Pass Butterworth Filter Design
The frequency response of the Butterworth Filter approximation function is also often referred to as “maximally flat” (no ripples) response because the pass band is designed to have a frequency response which is as flat as mathematically possible from 0Hz (DC) until the cutoff frequency at 3dB with no ripples. Higher frequencies beyond the cutoff point rollsoff down to zero in the stop band at 20dB/decade or 6dB/octave. This is because it has a “quality factor”, “Q” of just 0.707.
However, one main disadvantage of the Butterworth filter is that it achieves this pass band flatness at the expense of a wide transition band as the filter changes from the pass band to the stop band. It also has poor phase characteristics as well. The ideal frequency response, referred to as a “brick wall” filter, and the standard Butterworth approximations, for different filter orders are given below.
Ideal Frequency Response for a Butterworth Filter
Note that the higher the Butterworth filter order, the higher the number of cascaded stages there are within the filter design, and the closer the filter becomes to the ideal “brick wall” response.
In practice however, Butterworth’s ideal frequency response is unattainable as it produces excessive passband ripple.
Where the generalised equation representing a “nth” Order Butterworth filter, the frequency response is given as:
Where: n represents the filter order, Omega ω is equal to 2πƒ and Epsilon ε is the maximum pass band gain, (A_{max}). If A_{max} is defined at a frequency equal to the cutoff 3dB corner point (ƒc), ε will then be equal to one and therefore ε^{2} will also be one. However, if you now wish to define A_{max} at a different voltage gain value, for example 1dB, or 1.1220 (1dB = 20*logA_{max}) then the new value of epsilon, ε is found by:

Transpose the equation to give:
The Frequency Response of a filter can be defined mathematically by its Transfer Function with the standard Voltage Transfer Function H(jω) written as:

Note: ( jω ) can also be written as ( s ) to denote the Sdomain. and the resultant transfer function for a secondorder low pass filter is given as:
Normalised Low Pass Butterworth Filter Polynomials
To help in the design of his low pass filters, Butterworth produced standard tables of normalised secondorder low pass polynomials given the values of coefficient that correspond to a cutoff corner frequency of 1 radian/sec.
n  Normalised Denominator Polynomials in Factored Form 
1  (1+s) 
2  (1+1.414s+s^{2}) 
3  (1+s)(1+s+s^{2}) 
4  (1+0.765s+s^{2})(1+1.848s+s^{2}) 
5  (1+s)(1+0.618s+s^{2})(1+1.618s+s^{2}) 
6  (1+0.518s+s^{2})(1+1.414s+s^{2})(1+1.932s+s^{2}) 
7  (1+s)(1+0.445s+s^{2})(1+1.247s+s^{2})(1+1.802s+s^{2}) 
8  (1+0.390s+s^{2})(1+1.111s+s^{2})(1+1.663s+s^{2})(1+1.962s+s^{2}) 
9  (1+s)(1+0.347s+s^{2})(1+s+s^{2})(1+1.532s+s^{2})(1+1.879s+s^{2}) 
10  (1+0.313s+s^{2})(1+0.908s+s^{2})(1+1.414s+s^{2})(1+1.782s+s^{2})(1+1.975s+s^{2}) 
Filter Design – Butterworth Low Pass
Find the order of an active low pass Butterworth filter whose specifications are given as: A_{max} = 0.5dB at a pass band frequency (ωp) of 200 radian/sec (31.8Hz), and A_{min} = 20dB at a stop band frequency (ωs) of 800 radian/sec. Also design a suitable Butterworth filter circuit to match these requirements.
Firstly, the maximum pass band gain A_{max} = 0.5dB which is equal to a gain of 1.0593, remember that: 0.5dB = 20*log(A) at a frequency (ωp) of 200 rads/s, so the value of epsilon ε is found by:
Secondly, the minimum stop band gain A_{min} = 20dB which is equal to a gain of 10 (20dB = 20*log(A)) at a stop band frequency (ωs) of 800 rads/s or 127.3Hz.
Substituting the values into the general equation for a Butterworth filters frequency response gives us the following:
Since n must always be an integer ( whole number ) then the next highest value to 2.42 is n = 3, therefore a “a thirdorder filter is required” and to produce a thirdorder Butterworth filter, a secondorder filter stage cascaded together with a firstorder filter stage is required.
From the normalised low pass Butterworth Polynomials table above, the coefficient for a thirdorder filter is given as (1+s)(1+s+s^{2}) and this gives us a gain of 3A = 1, or A = 2. As A = 1 + (Rf/R1), choosing a value for both the feedback resistor Rf and resistor R1 gives us values of 1kΩ and 1kΩ respectively as: ( 1kΩ/1kΩ ) + 1 = 2.
We know that the cutoff corner frequency, the 3dB point (ω_{o}) can be found using the formula 1/CR, but we need to find ω_{o} from the pass band frequency ω_{p} then,
So, the cutoff corner frequency is given as 284 rads/s or 45.2Hz, (284/2π) and using the familiar formula 1/CR we can find the values of the resistors and capacitors for our thirdorder circuit.
Note that the nearest preferred value to 0.352uF would be 0.36uF, or 360nF.
Thirdorder Butterworth Low Pass Filter
and finally our circuit of the thirdorder low pass Butterworth Filter with a cutoff corner frequency of 284 rads/s or 45.2Hz, a maximum pass band gain of 0.5dB and a minimum stop band gain of 20dB is constructed as follows.
So for our 3rdorder Butterworth Low Pass Filter with a corner frequency of 45.2Hz, C = 360nF and R = 10kΩ