111 Filters (Passive) 111.1 Fundamentals 111.2 Applications. Simple RL and RC Filters • Simple RLC Filters • Compound Filters • Constant- k Filters • m -Derived Filters A filter is a frequency-sensitive two-port circuit that transmits with or without amplification signals in a band of frequencies and rejects (or attenuates) signals in other bands. The electric filter was invented during the First World War by two engineers working independently of each other — the American engineer G. A. Campbell and the German engineer K. W. Wagner. O. Zobel followed in the 1920s. These devices were developed to serve the growing areas of telephone and radio communication. Today, filters are found in all types of electrical and electronic applications from power to communications. Filters can be both active and passive. In this section we will confine our discussion to those filters that employ no active devices for their operation. The main advantage of passive filters over active ones is that they require no power (other than the signal) to operate. The disadvantage is that they often employ inductors that are bulky and expensive. 111.1 Fundamentals The basis for filter analysis involves the determination of a filter circuit’s sinusoidal steady state response from its transfer function T ( j w ). Some references use H ( j w ) for the transfer function. The filter’s transfer function T ( j w ) is a complex function and can be represented through its gain Ω T ( j w ) Ω and phase – T ( j w ) characteristics. The gain and phase responses show how the filter alters the amplitude and phase of the input signal to produce the output response. These two characteristics describe the frequency response of the circuit since they depend on the frequency of the input sinusoid. The signal-processing performance of devices, circuits, and systems is often specified in terms of their frequency response. The gain and phase functions can be expressed mathematically or graphically as frequency-response plots. Figure 111.1 shows examples of gain and phase responses versus frequency, w . The terminology used to describe the frequency response of circuits and systems is often based on the form of the gain plot. For example, at high frequencies the gain in Figure 111.1 falls off so that output signals in this frequency range are reduced in amplitude. The range of frequencies over which the output is significantly attenuated is called the stopband. At low frequencies the gain is essentially constant and there is relatively little attenuation. The frequency range over which there is little attenuation is called a passband. The frequency associated with the boundary between a passband and an adjacent stopband is called the cutoff frequency ( w C = 2 p f C ) . In general, the transition from the passband to the stopband, called the transition band , is relatively gradual, so the precise location of the cutoff frequency is a matter of definition. The most widely used approach defines the cutoff frequency as the frequency at which the gain has decreased by a factor of from its maximum value in the passband. 120707/.= Albert J. Rosa University of Denver 1586_book.fm Page 1 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC 111 -2 The Engineering Handbook, Second Edition This particular definition is based on the fact that the power delivered to a resistor by a sinusoidal current or voltage waveform is proportional to the square of its amplitude. At a cutoff frequency the gain is reduced by a factor of and the square of the output amplitude, and thusly also its power, is reduced by a factor of one half. For this reason the cutoff frequency is also called the half-power frequency. There are four prototypical filters. These are low pass (LP), high pass (HP), band pass (BP), and bandstop (BS). Figure 111.2 shows how the amplitude of an input signal consisting of three separate equal- amplitude frequencies is altered by each of the four-prototypical filter responses. The low-pass filter passes frequencies below its cutoff frequency w C , called its passband, and attenuates the frequencies above the cutoff, called its stopband. The high-pass filter passes frequencies above the cutoff frequency w C and attenuates those below. The band-pass filter passes those frequencies that lie between two cutoff frequen- cies, w C 1 and w C 2 , its passband, and attenuates those frequencies that lie outside the passband. Finally, the bandstop filter attenuates those frequencies that lie in its reject or stopband, between w C 1 and w C 2 , and passes all others. The bandwidth of a gain characteristic is defined as the frequency range spanned by its passband. For the band-pass case in Figure 111.2, the bandwidth is the difference in the two cutoff frequencies. BW = w C 2 - w C 1 (111.1) This equation applies to the low-pass response with the lower cutoff frequency w C 1 set to zero. In other words, the bandwidth of a low-pass circuit is equal to its cutoff frequency (BW = w C ). The bandwidth of a high-pass characteristic is infinite since the upper cutoff frequency w C 1 is infinity. For the bandstop case, Equation (111.1) defines the bandwidth of the stopband rather than the passbands. Frequency-response plots are usually made using logarithmic scales for the frequency variable because the frequency ranges of interest often span several orders of magnitude. A logarithmic frequency scale compresses the data range and highlights important features in the gain and phase responses. The use of a logarithmic frequency scale involves some special terminology. A frequency range whose end points have a 2:1 ratio is called an octave and one with 10:1 ratio is called a decade. Straight-line approximations FIGURE 111.1 Low-pass filter characteristics showing passband, stopband, and the cutoff frequency, w C . Gain Phase −45° 0° −90° A A 2 0 Passband Stopband ω c ω ω ω c √ 12/ 1586_book.fm Page 2 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC Filters (Passive) 111 -3 of these plots, called Bode (Bow-dee) plots, are often used to describe the general behavior of the devices, circuits, or systems. In frequency-response plots the gain Ω T ( j w ) Ω is often expressed in decibels (dB), defined as Ω T ( j w ) Ω dB = 20 log 10 Ω T ( j w ) Ω (111.2) Gains expressed in decibels can be either positive, negative, or zero. A gain of zero dB means that Ω T ( j w ) Ω = 1 — that is, the input and output amplitudes are equal. A positive dB gain means the output amplitude exceeds the input since Ω T ( j w ) Ω > 1, whereas a negative dB gain means the output amplitude is smaller than the input since Ω T ( j w ) Ω < 1. A cutoff frequency usually occurs when the gain is reduced from its maximum passband value by a factor or 3 dB. Figure 111.3 shows the asymptotic gain characteristics of ideal and real low-pass filters. The gain of the ideal filter is unity (0 dB) throughout the passband and zero ( -• dB) in the stopband. It also has an infinitely narrow transition band. The asymptotic gain responses of real low-pass filters show that we can only approximate the ideal response. As the order of the filter or number of poles n increases, the approximation improves since the asymptotic slope or “rolloff” in the stopband is - 20 ¥ n dB/decade. On the other hand, adding poles requires additional stages in a cascade realization, so there is a trade-off between (1) filter complexity and cost and (2) how closely the filter gain approximates the ideal response. Figure 111.4 shows how low-pass filter requirements are often specified. To meet the specification, the gain response must lie within the unshaded region in the figure, as illustrated by the two responses shown in Figure 111.4. The parameter T max is the passband gain. In the passband the gain must be within 3 dB of T max and must equal at the cutoff frequency w C . In the stopband the gain must decrease and remain below a gain of T min for all w ≥ w min . A low-pass filter design requirement is usually defined by specifying values for these four parameters. The parameters T max and w C define the passband response, whereas T min and w min specify how rapidly the stopband response must decrease. FIGURE 111.2 Four prototype filters and their effects on an input signal consisting of three frequencies. Passband Stopband LOW PASS GAIN Amplitude Amplitude Amplitude Amplitude Amplitude Input Transfer function Output GAIN GAIN GAIN ω ω 1 ω 1 ω c ω 2 ω 2 ω 3 ω 3 ω 1 ω 2 ω 3 ω 1 ω 2 ω 3 ω 1 ω 2 ω 3 ω 1 ω 2 ω 3 ω 1 ω 2 ω 3 ω 1 ω 2 ω 3 ω 1 ω 2 ω 3 ω ω ω ω ω ω ω ω Amplitude Amplitude Amplitude Passband Stopband Passband Passband Stopband ω 1 ω c ω 2 ω 3 ω Stopband Stopband ω 1 ω c1 ω 2 ω c2 ω 3 ω Passband HIGH PASS BAND PASS BANDSTOP 12/ T max /2 1586_book.fm Page 3 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC 111 -4 The Engineering Handbook, Second Edition 111.2 Applications Simple RL and RC Filters A first-order LP filter has the following transfer function: (111.3) FIGURE 111.3 The effect of increasing the order n of a filter relative to an ideal filter. FIGURE 111.4 Parameters for specifying low-pass filter requirements. |T(jω)| dB Passband Stopband −40 0.1 1.0 10 0 −20 −60 Ideal n = 1 n = 2 n = 3 Real ω C ω |T(jω)| dB T MAX T MIN 3dB Passband Stopban d ω MIN ω c ω Ts K s ()= +a 1586_book.fm Page 4 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC Filters (Passive) 111-5 where for a passive filter K £ a and a = w C . This transfer function can be realized in several ways including using either of the two circuits shown in Figure 111.5. For sinusoidal response the respective transfer functions are (111.4) For these filters the passband gain is equal to one and the cutoff frequency is determined by R/L for the RL filter and 1/RC for the RC filter. The gain ΩT( jw)Ω and phase –T( jw) plots of these circuits are shown back in Figure 111.1. A first-order HP filter is given by the following transfer function: (111.5) where, for a passive filter, K £ 1 and a is the cutoff frequency. This transfer function can also be realized in several ways including using either of the two circuits shown in Figure 111.6. For sinusoidal response the respective transfer functions are (111.6) For the LP filters the passband gain is one and the cutoff frequency is determined by R/L for the RL filter and 1/RC for the RC filter. The gain ΩT( jw)Ω and phase –T( jw) plots of these circuits are shown in Figure 111.7. FIGURE 111.5 Single-pole LP filter realizations: (a) RL, (b) RC. FIGURE 111.6 Single-pole HP filter realizations: (a) RL, (b) RC. (a) (b) R C L R R C L (a) (b) R Tj RL jRL Tj /RC j/RC RL RC () () ; ( ) () w w w w = + = + / / 1 1 Ts Ks s ()= +a Tj j jRL Tj j jRC RL RC () () ; ( ) () w w w w w w = + = +//1 1586_book.fm Page 5 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC 111-6 The Engineering Handbook, Second Edition Simple RLC Filters Simple second-order LP, HP, or BP filters can be made using series or parallel RLC circuits. Series or parallel RLC circuits can be connected to produce the following transfer functions: (111.7) where for a series RLC circuit and . w 0 is called the undamped natural frequency and is related to the cutoff frequency in the HP and LP case and is the center frequency in the BP case. z is called the damping ratio and determines the nature of the roots of the equation that translates to how quickly a transition is made from the passband to the stopband. z in the BP case helps define the bandwidth of the circuit, that is, . Figure 111.8 shows how a series RLC circuit can be connected to achieve the transfer functions given in Equation 111.7. The gain ΩT( jw)Ω and phase –T( jw) plots of these circuits are shown in Figure 111.9 through Figure 111.11. FIGURE 111.7 High-pass filter characteristics showing passband, stopband, and the cutoff frequency, w c . Gain Phase 45° 0° 90° A A 2 0 Passband Stopband ω c ω ω ω c √ Tj K j LP ()w wzwww = -+ + 2 00 2 2 Tj K j HP ()w w wzwww = - -+ + 2 2 00 2 2 Tj Kj j BP ()w w wzwww = -+ + 2 00 2 2 w 0 = LC z= RC L2 Bw = 2 0 zw 1586_book.fm Page 6 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC Filters (Passive) 111-7 FIGURE 111.8 RLC circuit connections to achieve LP, HP, or BP responses. FIGURE 111.9 Second-order low-pass gain responses. FIGURE 111.10 Second-order band-pass gain responses. R CL HP BP LP + V OHP – + V OBP –+ V OLP – + V IN – 0.01 0.1 1.0 10 100 10000.001 |T(0)| |T(0)| |T(0)| |T(0)| dB + 20 dB |T(0)| dB + 0 dB |T(0)| dB − 20 dB |T(0)| dB − 40 dB 10 100 Gain Gain (dB) 10|T(0)| ζ = 0.5 ω ω 0 ζ = 5 ζ = 0.05 Asymptote s 0.01 0.1 1.0 10 100 10000.001 Gain Gain (dB) |K| 10 ω 0 K dB + 20dB ω 0 K dB − 20dB ω 0 K dB − 40dB ω 0 K dB + 0dB ω 0 ω 0 ω |K| 10 ω 0 |K| ω 0 |K| 100 ω 0 B ζ=5 ζ = 0.5 ζ = 5 B ζ=0.05 ζ = 0.05 B ζ=0.5 1586_book.fm Page 7 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC 111-8 The Engineering Handbook, Second Edition Compound Filters Compound filters are higher-order filters obtained by cascading lower-order designs. Ladder circuits are an important class of compound filters. Two of the more common passive ladder circuits are the constant- k and the m-derived filters (either of which can be configured using a T-section, p-section, or L-section, or combinations thereof), the bridge-T network and parallel-T network, and the Butterworth and Chebyshev realizations. Only the first two known as image-parameter filters will be discussed in this section. Figure 111.12(a) shows a standard ladder network consisting of two impedances, Z 1 and Z 2 , organized as an L-section filter. Figure 111.12(b) and Figure 111.12(c) show how the circuit can be redrawn to represent a T-section or ’-section filter, respectively. T- and ’-section filters (also referred to as “full sections”) are usually designed to be symmetrical so that either can have its input and output reversed without changing its behavior. The “L-section” (also known as a “half section”) is unsymmetrical, and orientation is important. Since cascaded sections “load” each other, the choice of termination impedance is important. The image impedance, Z i , of a symmetrical filter is the impedance with which the filter must be terminated in order to “see” the same impedance at its input terminals. In general the image impedance is the desired load or source impedance to which the filter matches. The image impedance of a filter can be found from (111.8) where Z 1O is the input impedance of the filter with the output terminals open circuited, and Z 1S is its input impedance with the output terminals short-circuited. For symmetrical filters the output and input can be reversed without any change in its image impedance — that is, (111.9) The concept of matching filter sections and terminations to a common image impedance permits the development of symmetrical filter designs. The image impedances of T- and ’-section filters are given as FIGURE 111.11 Second-order high-pass gain responses. 0.01 0.1 1.0 10 100 10000.001 |T(∞)| |T(∞)| dB + 20 dB |T(∞)| dB + 0 dB |T(∞)| dB − 20 dB |T(∞)| dB − 40 dB |T(∞)| 10 |T(∞)| 100 Gain Gain (dB) 10|T(∞)| ζ = 5 ω 0 ω Asymptotes ζ = 0.05 ζ = 0.5 ZZZ iOS = 11 ZZZ ZZZ ZZZ iOS i OS iii 111 2 22 12 == == and ZZZ ZZZ iT O S ==+ 11 1 2 12 1 4 1586_book.fm Page 8 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC Filters (Passive) 111-9 and (111.10) These expressions also describe the input (or output) impedance when the filter’s output (or input) is terminated with appropriate image impedance, i.e., Z ¸ or Z T . The image impedance of an L-section filter, being unsymmetrical, depends on the terminal pair being calculated. For the L-section shown in Figure 111.12(a), image impedances are (111.11) These equations show that the image impedance of an L-section at its input is equal to the image impedance of a T-section, whereas the image impedance of an L-section at its output is equal to the image impedance of a ’-section. This relationship is important in achieving an optimum termination when cascading L-sections with T- and/or ’-sections to form a composite filter. Since Z 1 and Z 2 vary significantly with frequency, the image impedances of T- and ’-sections will also change. This condition does not present any particular problem in combining any number of equivalent filter sections together, since their impedances va4ry equally at all frequencies. But this does make it difficult to terminate these filters exactly, causing a limitation of these types of filters. However, there is a frequency within the filter’s passband where the image impedance becomes purely resistive. It is useful FIGURE 111.12 Ladder networks: (a) standard L-section, (b) T-section, (c) ’-section. Z 1 2Z 2 2Z 2 Z 1 2Z 2 2Z 2 Z 1 2Z 2 2Z 2 2Z 2 2 2 1 1 Z 1 Z 2 Z 1 Z 2 Z 1 Z 2 2 2 1 1 Z 1 /2 Z 2 2 2 1 1 Z 1 /2 Z 1 /2 Z 2 Z 1 /2 Z 1 /2 Z 2 Z 1 /2 (a) (b) (c) ZZZ ZZ ZZZ iOS’ == + 11 12 1 2 12 14(/) ZZZZZ Z ZZ ZZZ Z iL iT iL i11 2 12 2 12 1 2 12 1 4 14 =+= = + = ’ (/) and 1586_book.fm Page 9 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC 111-10 The Engineering Handbook, Second Edition to terminate the filter with this value of resistance since it provides good matching over much of the filter’s passband. To develop the theory of constant-k and m-derived filters, consider the circuit of Figure 111.13. The current transfer function in the sinusoidal steady state is given by T( jw) = ΩT( jw)Ω–T( jw) = I 2 /I 1 : (111.12) where a is the attenuation in dB, b is the phase shift in radians, and g is the image transfer function. For the circuit shown in Figure 111.13, the following relationship can be derived: (111.13) This relationship and those in Equation (111.12) will be used to develop the constant-k and m-derived filters. Constant-k Filters During the 1920s O. Zobel developed an important class of symmetrical filters called constant-k filters with the conditions that Z 1 and Z 2 are purely reactive — that is, ±X( jw) and Z 1 Z 2 = k 2 = R 2 (111.14) In modern references an R replaces the k. Note that the units of k are ohms. The advantage of this type of filter is that the image impedance in the passband is a pure resistance, whereas in the stopband it is purely reactive. Hence if the termination is a pure resistance and equal to the image impedance, all the power will be transferred to the load since the filter itself is purely reactive. Unfortunately, the value of the image impedance varies significantly with frequency, and any termination used will result in a mismatch except at one frequency. In LC constant-k filters, Z 1 and Z 2 have opposite signs, so that . The image impedances become (111.15) Therefore, in the stopband and passband, we have the following relations for standard T- or ’-sections, where n represents the number of identical sections: FIGURE 111.13 Circuit for determining the transfer function of a T-section filter. 2 21 1 Z i Z 1 /2 Z 2 Z 1 /2 Z i V I 1 I 2 + − Tj Tj Tj Ij Ij ee e j () () () () () www w w ab g =– === - 2 1 tanh /g= ZZ SO11 ZZ jX jX X X R 12 1 2 12 2 =± =+ =m ZR ZZ Z R Z iT i = = ’ 14 1 12 1 (/)and /4Z 2 () 1586_book.fm Page 10 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC [...]... sections: (a) LP ’-section, (b) HP T-section, (c) BP T-section Stopband Passband a = 2n cosh-1 - Z 1 / 4Z 2 a=0 b = ±n p, ± 3n p, b = 2n sin -1 - Z 1 / 4Z 2 (111.16) The ultimate roll-off of constant-k filters is 20 dB per decade per reactive element or 60 dB per decade for the T- or ’-section, 40 dB per decade per L-section Figure 111.14 shows normalized plots of a and b versus -Z 1 / 4Z 2 These curves... constant-k but had a higher attenuation near the cutoff frequency The impedances in the m-derived filter were related to those in the constant-k as © 2005 by CRC Press LLC 1586_book.fm Page 12 Monday, May 10, 2004 3:53 PM 11 1-1 2 The Engineering Handbook, Second Edition 1/2mZ 1 1/2mZ 1 mZ1 (1 − m)Z1 4m 2Z2 m Z2 m (a) 4m (1 − m)Z1 2Z2 m (b) FIGURE 111.16 m-derived filters: (a) T-section, (b) ’-section... low-pass, high-pass, band-pass, or bandreject filters Figure 111.15 shows examples of a typical LP ’-section, an HP T-section, and a BP T-section m-Derived Filters The need to develop a filter section that could provide high attenuation in the stopband near the cutoff frequency prompted the development of the m-derived filter O Zobel developed a class of filters that had the same image impedance as the. .. ground-level pads, or in underground vaults A specification of 7200/ 12,470Y V for the high-voltage winding of a single-phase transformer means the transformer may be connected in a line-to-neutral “wye” connection for a system with a line-to-line voltage of 12,470 V or in a line-to-line “delta” connection for a system with a line-to-line voltage of 7200 V A specification of 240/120 V for the low-voltage... represents the phase Thevenin matrix looking into the node Once Zth and pre-fault voltages at the node are available, Vf can be written in terms of I depending upon conditions imposed by the fault Then Equation (112.1) can be solved for I The pre-fault system model represents the system behavior before the fault occurs On the pre-fault model, a power flow calculation may be used to obtain the voltages V0 The. .. evaluated For an ungrounded node, the phase-to-phase voltages instead of phase-to-neutral voltages are employed © 2005 by CRC Press LLC 1586_book.fm Page 10 Monday, May 10, 2004 3:53 PM 11 2-1 0 The Engineering Handbook, Second Edition Protection and Coordination Over-current protection is the most common protection applied to the distribution system With overcurrent protection, the protective device trips... in the stopband than can be obtained using constant-k filters Equation 111.18 relates the cutoff frequency to the infinite attenuation frequency m = 1- 2 wC 2 w• (111.18) Figure 111.17 shows the attenuation curve for a single m-derived LP stage for two values of m The smaller m becomes, the steeper the attenuation near the cutoff, but also the lesser the attenuation at higher frequencies Constant-k filters... respectively) to the common point p Zf is the impedance between p and n Consider solving for a three-phase-to-ground fault Let Vikn and Vfkn denote pre-fault and post-fault phase-toground voltages of phase k, respectively Then the boundary conditions are: I f = Ia + Ib + Ic , (112.3) ( ) (112.4) ( ) (112.5) ( ) (112.6) i f i DVan = Van - Van = Van - I a Z a + I f Z f , i f i DVbn = Vbn - Vbn = Vbn - I b Z b... means the transformer provides a three-wire connection with 120 V mid-tap voltage and 240 V full-winding voltage A specification of 480Y/277 V for the low voltage winding means the winding is permanently wye-connected with a fully insulated neutral avilable for a three-phase, four-wire service to deliver 480 V line-to-line and 277 V line-to-neutral Distribution substations consist of one or more step-down... increases, the risk of common-mode noise increases and the need for EMI solutions rises As shown in Figure 113.2, the common-mode ground current Iao = Cl-g dv/dt This characteristic of common-mode current makes the adjustable-speed drive a prime source of common-mode noise because of its abrupt voltage transitions on the drive output terminal The conducted noise will be created as the individual pulses on the . of the more common passive ladder circuits are the constant- k and the m-derived filters (either of which can be configured using a T-section, p-section, or L-section, or combinations thereof), the. = aa bpp b =- = =± ± - - - 24 0 324 1 12 1 12 nZZ nn n ZZ -ZZ 12 4/ 1586_book.fm Page 11 Monday, May 10, 2004 3:53 PM © 2005 by CRC Press LLC 11 1-1 2 The Engineering Handbook, Second Edition (111.17) where. LLC 11 1-1 0 The Engineering Handbook, Second Edition to terminate the filter with this value of resistance since it provides good matching over much of the filter’s passband. To develop the theory