The electrical engineering handbook CH004

The electrical engineering handbook

Kerwin, W.J “Passive Signal Processing”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

Passive Signal Processing

4.1 Introduction Laplace Transform • Transfer Functions 4.2 Low-Pass Filter Functions Thomson Functions • Chebyshev Functions 4.3 Low-Pass Filters

Introduction • Butterworth Filters • Thomson Filters • Chebyshev Filters

4.4 Filter Design Scaling Laws and a Design Example • Transformation Rules, Passive Circuits

4.1 Introduction

This chapter will include detailed design information for passive RLC filters; including Butterworth, Thomson, and Chebyshev, both singly and doubly terminated As the filter slope is increased in order to obtain greater rejection of frequencies beyond cut-off, the complexity and cost are increased and the response to a step input

is worsened In particular, the overshoot and the settling time are increased The element values given are for normalized low pass configurations to 5th order All higher order doubly-terminated Butterworth filter element values can be obtained using Takahasi’s equation, and an example is included In order to use this information

in a practical filter these element values must be scaled Scaling rules to denormalize in frequency and impedance are given with examples Since all data is for low-pass filters the transformation rules to change from low-pass

to high-pass and to band-pass filters are included with examples

Laplace Transform

We will use the Laplace operator, s = s + jw Steady-state impedance is thus Ls and 1/Cs, respectively, for an inductor (L) and a capacitor (C), and admittance is 1/Ls and Cs In steady state s = 0 and therefore s = jw

Transfer Functions

We will consider only lumped, linear, constant, bilateral elements, and we will define the transfer function

T(s) as response over excitation

D s

( )

signal input

William J Kerwin

University of Arizona

Adapted from Instrumentation and Control: Fundamentals and Applications, edited by Chester L Nachtigal, pp 487–497, copyright 1990, John Wiley and Sons, Inc Reproduced by permission of John Wiley and Sons, Inc.

The roots of the numerator polynomial N(s) are the zeros of the system, and the roots of the denominator

D(s) are the poles of the system (the points of infinite response) If we substitute s = jw into T(s) and separate the result into real and imaginary parts (numerator and denominator) we obtain

(4.1)

Then the magnitude of the function, ÷T(jw)ï, is

(4.2)

and the phase is

(4.3)

Analysis

Although mesh or nodal analysis can always be used, since we will consider only ladder networks we will use

a method commonly called linearity, or working your way through. The method starts at the output and assumes either 1 volt or 1 ampere as appropriate and uses Ohm’s law and Kirchhoff ’s current law only

Example 4.1. Analysis of the circuit of Fig 4.1 for Vo = 1 Volt

FIGURE 4.1 Singly terminated 3rd order low pass filter ( W , H, F).

T j A jB

A jB

+

+

æ è ç

ö ø

÷

1 2 1 2

2 2 2 2

1 2

T j( w)

A

B A

1

1 2 2

T s V

i

o i

2

3

1

;

;

( )

V i

V1

I1

I3

V o

1/2

4/3

3/2 1

Example 4.2 Determine the magnitude and phase of T(s) in Example 4.1.

The values used for the circuit of Fig 4.1 were normalized; that is, they are all near unity in ohms, henrys, and farads These values simplify computation and, as we will see later, can easily be scaled to any desired set

of actual element values In addition, this circuit is low-pass because of the shunt capacitors and the series inductor By low-pass we mean a circuit that passes the lower frequencies and attenuates higher frequencies The cut-off frequency is the point at which the magnitude is 0.707 (–3 dB) of the dc level and is the dividing line between the passband and the stopband In the above example we see that the magnitude of V o/V i at w =

0 (dc) is 1.00 and that at w = 1 rad/s we have

(4.4)

and therefore this circuit has a cut-off frequency of 1 rad/s

Thus, we see that the normalized element values used here give us a cut-off frequency of 1 rad/s

The most common function in signal processing is the Butterworth It is a function that has only poles (i.e.,

no finite zeros) and has the flattest magnitude possible in the passband This function is also called maximally flat magnitude (MFM) The derivation of this function is illustrated by taking a general all-pole function of third-order with a dc gain of 1 as follows:

(4.5)

The squared magnitude is

(4.6)

1 Adapted from Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited Reproduced by permission of John Wiley and Sons Limited.

T s

T s

T s

-=

-1

1

1 1

2

2

2

w

w

w tan tan tan

rad s

w

w

w

=

+

=

= 1 1 1

0 707

6

T s

as bs cs

1

1

w

2

2 2 3 2

1 1

=

+

(4.7)

MFM requires that the coefficients of the numerator and the denominator match term by term (or be in the same ratio) except for the highest power

Therefore

(4.8)

We will also impose a normalized cut-off (–3 dB) at w = 1 rad/s; that is,

(4.9)

Thus, we find a = 1, then b = 2, c = 2 are solutions to the flat magnitude conditions of Eq 4.8 and our third-order Butterworth function is

(4.10)

Table 4.1 gives the Butterworth denominator polynomials up to n = 5

In general, for all Butterworth functions the normalized magnitude is

(4.11)

Note that this is down 3 dB at w = 1 rad/s for all n

This may, of course, be multiplied by any constant less than one for circuits whose dc gain is deliberately set to be less than one

Example 4.3. A low-pass Butterworth filter is required whose

cut-off frequency (–3 dB) is 3 kHz and in which the response must

be down 40 dB at 12 kHz Normalizing to a cut-off frequency of

1 rad/s, the –40-dB frequency is

thus

therefore n = 3.32 Since n must be an integer, a fourth-order filter is required for this specification

w

2

1

=

c2 – 2 b = 0 ; b2 – 2 ac = 0

a

.

+

=

1

2

1 1

0 707

T s

1

n

w

w

=

+

1 1

2

TABLE 4.1 Butterworth Polynomials

Source: Handbook of Measurement Science,

edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited Reproduced by permis-sion of John Wiley and Sons Limited.

s

s s

s s s

s s s s

s s s s s a

+

1

2 1

2 6131 3 4142 2 6131 1

3 2361 5 2361 5 2361 3 2361

2

12

kHz

=

+

n

There is an extremely important difference between the singly terminated (dc gain = 1) and the doubly

terminated filters (dc gain = 0.5) As was shown by John Orchard, the sensitivity in the passband (ideally at

maximum output) to all L, C components in an L, C filter with equal terminations is zero This is true regardless

of the circuit

This, of course, means component tolerances and temperature coefficients are of much less importance in

the equally terminated case For this type of Butterworth low-pass filter (normalized to equal 1-W terminations),

Takahasi has shown that the normalized element values are exactly given by

(4.12)

for any order n, where k is the L or C element from 1 to n.

Example 4.4. Design a normalized (w–3dB =1 rad/s) doubly terminated (i.e., source and load = 1 W)

Butter-worth low-pass filter of order 6; that is, n = 6.

The element values from Eq (4.12) are

The values repeat for C4, L5, C6 so that

C4 = L3, L5 = C2, C6 = L1

Thomson Functions

The Thomson function is one in which the time delay of

the network is made maximally flat This implies a linear

phase characteristic since the steady-state time delay is

the negative of the derivative of the phase This function

has excellent time domain characteristics and is used

wherever excellent step response is required These

func-tions have very little overshoot to a step input and have

far superior settling times compared to the Butterworth

functions The slope near cut-off is more gradual than

the Butterworth Table 4.2 gives the Thomson

denomi-nator polynomials The numerator is a constant equal to

the dc gain of the circuit multiplied by the denominator

constant The cut-off frequencies are not all 1 rad/s They

are given in Table 4.2

Chebyshev Functions

A second function defined in terms of magnitude, the Chebyshev, has an equal ripple character within the

passband The ripple is determined by e.

n

è

ö ø

2 p

L C L

1

2

3

p p p

H F H

TABLE 4.2 Thomson Polynomials

w –3dB (rad/s)

s + 1 1.0000

s3 +6s2 + 15s +15 1.7557

s4 + 10s3 + 45s2 + 105s + 105 2.1139

s5 + 15s4 +105s3 + 420s2 + 945s + 945 2.4274

Source: Handbook of Measurement Science, edited by Peter

Sydenham, copyright 1982, John Wiley and Sons Limited.

Reproduced by permission of John Wiley and Sons Limited.

where A = decibels of ripple; then for a given order n, we define v.

(4.14)

Table 4.3 gives denominator polynomials for the Chebyshev functions In all cases, the cut-off frequency (defined as the end of the ripple) is 1 rad/s The –3-dB frequency for the Chebyshev function is

(4.15)

The magnitude in the stopband (w > 1 rad/s) for the normalized filter is

(4.16)

for the singly terminated filter For equal terminations the above magnitude is multiplied by one-half [1/4 in

Eq (4.16)]

Example 4.5. What order of singly terminated Chebyshev filter having 0.25-dB ripple (A) is required if the

magnitude must be –60 dB at 15 kHz and the cut-off frequency (–0.25 dB) is to be 3 kHz? The normalized frequency for a magnitude of –60 dB is

Thus, for a ripple of A = 0.25 dB, we have from Eq (4.13)

and solving Eq (4.16) for n with w = 5 rad/s and *T(jw)* = –60 dB, we obtain n = 3.93 Therefore we must use

n = 4 to meet these specifications.

TABLE 4.3 Chebyshev Polynomials

Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982,

John Wiley and Sons Limited Reproduced by permission of John Wiley and Sons Limited.

s

+

sinh sinh sinh sinh sinh sinh sinh sinh sinh sinh sinh sinh sinh

n

3 4

0 75637 0 85355 1 84776 0 14645

0 61803 0 90451

v n

è

ö ø

e

1

1

ë

ê ê

ù û

ú ú

n

n

w

w

2

1 1

=

15

kHz

4.3 Low-Pass Filters1

Introduction

Normalized element values are given here for both singly and doubly terminated filters The source and load resistors are normalized to 1 W Scaling rules will be given in Section 4.4 that will allow these values to be modified to any specified impedance value and to any cut-off frequency desired In addition, we will cover the

transformation of these low-pass filters to high-pass or bandpass filters

Butterworth Filters

For n = 2, 3, 4, or 5, Fig 4.2 gives the element values for the singly terminated filters and Fig 4.3 gives the element values for the doubly terminated filters All cut-off frequencies (–3 dB) are 1 rad/s

1Adapted from Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons

Limited Reproduced by permission of John Wiley and Sons Limited.

FIGURE 4.2 Singly terminated Butterworth filter element values (in W, H, F) (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited Reproduced by permission of John Wiley and Sons Limited.)

FIGURE 4.3 Doubly terminated Butterworth filter element values (in W, H, F) (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited Reproduced by permission of John Wiley and Sons Limited.)

Thomson Filters

Singly and doubly terminated Thomson filters of order n = 2, 3, 4, 5 are shown in Figs 4.4 and 4.5 All time delays are 1 s The cut-off frequencies are given in Table 4.2

Chebyshev Filters

The amount of ripple can be specified as desired, so that only a selective sample can be given here We will use 0.1 dB, 0.25 dB, and 0.5 dB All cut-off frequencies (end of ripple for the Chebyshev function) are at 1 rad/s Since the maximum power transfer condition precludes the existence of an equally terminated even-order filter, only odd orders are given for the doubly terminated case Figure 4.6 gives the singly terminated Chebyshev

filters for n = 2, 3, 4, and 5 and Fig 4.7 gives the doubly terminated Chebyshev filters for n = 3 and n = 5.

4.4 Filter Design

We now consider the steps necessary to convert normalized filters into actual filters by scaling both in frequency and in impedance In addition, we will cover the transformation laws that convert low-pass filters to high-pass filters and low-pass to bandpass filters

Scaling Laws and a Design Example

Since all data previously given are for normalized filters, it is necessary to use the scaling rules to design a low-pass filter for a specific signal processing application

FIGURE 4.4 Singly terminated Thomson filter element values (in W, H, F) (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited Reproduced by permission of John Wiley and Sons Limited.)

FIGURE 4.5 Doubly terminated Thomson filter element values (in W, H, F) (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited Reproduced by permission of John Wiley and Sons Limited.)

Rule 1 All impedances may be multiplied by any constant without affecting the transfer voltage ratio Rule 2 To modify the cut-off frequency, divide all inductors and capacitors by the ratio of the desired frequency

to the normalized frequency

Example 4.6. Design a low-pass filter of MFMtype (Butterworth) to operate from a 600-W source into a 600-W load, with a cut-off frequency of 500 Hz The filter must be at least 36 dB below the dc level at 2 kHz, that is, –42 dB (dc level is –6 dB)

Since 2 kHz is four times 500 Hz, it corresponds to w = 4 rad/s in the normalized filter Thus at w = 4 rad/s

we have

FIGURE 4.6 Singly terminated Chebyshev filter element values (in W, H, F): (a) 0.1-dB ripple; (b) 0.25-dB ripple;

(c) 0.50-dB ripple (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and

Sons Limited Reproduced by permission of John Wiley and Sons Limited.)

-+

é ë

ê ê

ù û

ú ú

therefore, n = 2.99, so n = 3 must be chosen The 1/2 is present because this is a doubly terminated (equal

values) filter so that the dc gain is 1/2

Thus a third-order, doubly terminated Butterworth filter is required From Fig 4.3 we obtain the normalized network shown in Fig 4.8(a)

The impedance scaling factor is 600/1 = 600 and the frequency scaling factor is 2p500/1 = 2p500: that is, the ratio of the desired radian cut-off frequency to the normalized cut-off frequency (1 rad/s) Note that the impedance scaling factor increases the size of the resistors and inductors, but reduces the size of the capacitors The result is shown in Fig 4.8(b)

Transformation Rules, Passive Circuits

All information given so far applies only to low-pass filters, yet we frequently need high-pass or bandpass filters

in signal processing

FIGURE 4.7 Doubly terminated Chebyshev filter element values (in W, H, F).

FIGURE 4.8 Third-order Butterworth low-pass filter: (a) normalized (in W, H, F); (b) scaled (in W, H, mF).

0.10 1.0316 1.1474 1.0316 0.25 1.3034 1.1463 1.3034 0.50 1.5963 1.0967 1.5963

0.10 1.1468 1.3712 1.9750 1.3712 1.1468 0.25 1.3824 1.3264 2.2091 1.3264 1.3824 0.50 1.7058 1.2296 2.5408 1.2296 1.7058