CALIFORNIA INSTITUTE OF TECHNOLOGY PHYSICS MATHEMATICS AND ASTRONOMY DIVISION Sophomore Physics Laboratory (PH5/15) Analog Electronics Active Filters Copyright c Virgínio de Oliveira Sannibale, 23 (Revision December 212)
Chapter 6 Active Filters Introduction An electronic circuit that modifies the frequency spectrum of an arbitrary signal is called filter A filter that modifies the spectrum producing amplification is said to be an active filter Vis à vis its definition, it is convenient to study the filter characteristics in terms of the frequency response of its associated two port network H(ω) = (ω) (ω), where and are respectively the input voltage and the output voltage of the network, and ω the angular frequency Depending on the design, active filters have some important advantages: they can provide gain, they can provide isolation because of the typical characteristic impedances of amplifiers, they can be cascaded because of the typical characteristic impedances of amplifiers, they can avoid the use of inductors greatly simplifying the design of the filters Here some disadvantages: 127
128 CHAPTER 6 ACTIVE FILTERS they are limited by the amplifiers band-with, and noise, they need power supplies, they dissipate more heat than a passive circuit Let s make some simple definitions useful to classify different types of filters 61 Classification of Ideal Filters Based on their magnitude response H(ω), Some basic ideal filters can be classified as follows: H(ω) H(ω) 1 1 ω Low-pass ω ω High-pass ω H(ω) 1 H(ω) 1 H(ω) 1 ω ω 1 ω ω ω 1 ω Band-pass Stop-band/band-reject ω ω Notch Practical filters approximate more or less the ideal definitions
62 FILTERS AS RATIONAL FUNCTIONS 129 A 1 DA 1 DA 3 A 3 DA 2 A 2 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7ω 8 ω Figure 61: Graphical definition of the filter performance specifications, and hypothetical filter response (red curve) that satisfy the specification Usually, the filter requirements are specified defining the band frequencies with their gains (attenuation or amplification) gain ripples, and slope transitions in terms of power of the frequency Figure 61 shows a quite general graphical definition of the design parameters of a filter with an hypothetical design For a complete specification one should also define the requirement for phase response 62 Filters as Rational Functions Let s consider filters whose transfer function can be expressed as rational function or standard form H(ω) = α α 1 jωα 2 (jω) 2 α N (jω) N β β 1 jωβ 2 (jω) 2 β M (jω) M For the filter not to diverge M N H(ω) < for any value of ω Writing the transfer function as a polynomial factorization we obtain H(ω) = k (ω z 1) n 1 (ωz 2 ) n 2 (ω z N ) n N (ω p 1 ) m 1 (ω p 2 ) m 2 (ω p M ) m M
13 CHAPTER 6 ACTIVE FILTERS Denominator roots p 1, p 2,, p n are called poles, and numerator roots z 1, z 2,, z m are called zeros The integers n 1, n 2,, n N, and m 1, m 2,, m N are therefore the multiplicity of poles and zeros Poles and zeros values determine the shape of the filter, and apart from zero frequency, one could say that poles provide attenuation and zeros amplification The transition from transmission to attenuation, and vice versa, in the filter magnitude H(ω) is characterized by an asymptote slope which determine the so called filter order For example, considering the RC low pass filter with ω = 1/RC, we have one pole p 1 = jω H(ω) = ω ω jω first oder low pass filter with cut-off freq ω For the RC high pass filter with ω = 1/RC, we have one pole p 1 = jω and one zero z 1 = H(ω) = ω ω jω first oder high pass filter with cut-off freq ω In the next sub-sections, we will analyze into more details filters with the following transfer function H(ω) = H ω 2 jω ω 1 Q 1 ω 2 1 ω 2 jω ω Q ω2
62 FILTERS AS RATIONAL FUNCTIONS 131 621 Second Order Low-Pass Filter Figure below shows the second order low pass filter bode plot, with resonant frequency ω res, and characteristic frequency ω Magnitude [db] H max 1 1 2 Filter ω res ω 3 4 1 2 ω res ω 1 3 1 4 Phase [Deg] 5 φ res 9 1 15 1 2 ω 1 3 1 4 res ω Frequency [rad/s] The second order low-pass filter written in standard form Transfer Function Resonance Maximum DC High Freq Gain Gain H ω 2 ω 2 jω ω Q ω2 ω 1 1 Q 2Q 2 H 1 1 4Q 2 H
132 CHAPTER 6 ACTIVE FILTERS 622 Second Order High-Pass Filter Magnitude [db] H max 1 1 2 Filter ω res ω 3 1 2 ω 1 3 ω res 1 4 15 Phase [Deg] 1 9 φ res 5 1 2 ω 1 3 ω res 1 4 Frequency [rad/s] The second order high-pass filter written in standard form Transfer Function Resonance Maximum DC High Freq Gain Gain H ω 2 ω 2 jω ω Q ω2 ω 1 1 2Q 2 H Q 1 1 4Q H
62 FILTERS AS RATIONAL FUNCTIONS 133 623 Band-Pass Filter Magnitude [db] H 5 max 5 1 15 2 Filter ω res ω 25 3 1 2 1 3 ω ω res 1 4 5 Phase [Deg] φ res 5 1 2 ω 1 3 ω res 1 4 Frequency [rad/s] The band-pass filter written in standard form is Transfer Function Resonance Maximum DC Gain High Freq Gain H jω ω Q ω 2 jω ω Q ω2 ω H
134 CHAPTER 6 ACTIVE FILTERS For example, depending on the output we consider, the already studied LRC series circuit is a low-pass, a band-pass, or a high-pass filter with the transfer function described above When we will study difference filters topologies we will reduce their transfer function into one of the standard form above 63 Common Circuit Filters Topologies This is a brief and not exhaustive at all list of filter topologies that use resistors, capacitors, and operational amplifiers to implement the filters types described above: Infinite gain, multiple feedback (IGMF) Generalized Sallen-Key (GSK) State Variable (SV) Switched Capacitor Filters (SC) Cascading these implementation allows to increase the filter order
64 INFINITE GAIN MULTIPLE FEEDBACK CONFIGURATION (IGMF)135 64 Infinite Gain Multiple Feedback Configuration (IGMF) I 1 Y 1 I 4 Y 4 Y 5 I3 A B V A Y 3 V I 2 Y 2 V Figure 62: Infinite Gain Multiple Feedback Filter Let s consider the circuit in Figure 62 with generic admittances Y 1, Y 2, Y 3, Y 4, and Y 5 Applying the KCL to node A and considering the circuit virtual ground (V = ), we have V A Y 3 ( V A ) Y 4 ( V A ) Y 1 V A Y 2 = (61) Again, applying KCL to node B and for the virtual ground we have Y 5 V A Y 3 = V A = Y 5 Y 3 Replacing the last expression into eq 61 and after some algebra we obtain the generic transfer function for the circuit Y = 1 Y 3 Y 5 (Y 1 Y 2 Y 3 Y 4 ) Y 3 Y 4 Choosing the proper type of admittances we can construct different types of active filters, low-pass band-pass, and high-pass It is worthwhile noticing that IGMF configuration allows to implement low-pass, bandpass, and high-pass filter with capacitors, resistor and no inductors This simplifies considerably the design of the filters
136 CHAPTER 6 ACTIVE FILTERS 641 Low-pass Filter R 4 C 5 R R V 1 3 i C 2 G Figure 63: Low-pass filter configuration of the infinite gain multiple feedback filter A possible choice to implement a low-pass filter as shown in Figure 63 is Y 1 = 1 R 1, Y 2 = jωc 2, Y 3 = 1 R 3, Y 4 = 1 R 4, Y 5 = jωc 5, and the transfer function of the circuit becomes 1/R 1 R 3 = jωc 5 (1/R 1 jωc 2 1/R 3 1/R 4 )1/R 3 R 4 Rearranging the expression to obtain a rational fraction in ω we finally obtain = 1 R 1 R 3 C 2 C 5 ω 2 jω 1 1 (1/R 1 1/R 3 1/R 4 ) C 2 R 3 R 4 C 2 C 5 Comparing the denominator of the previous equation with the denominator of the transfer function in section 621 we find that the frequency ω, the quality factor Q, and the DC gain H are respectively ω = 1 R 3 R 4 C 2 C 5, Q = ω C 2 (1/R 1 1/R 3 1/R 4 ), H = R 4 R 1
64 INFINITE GAIN MULTIPLE FEEDBACK CONFIGURATION (IGMF)137 642 High-pass Filter C 4 R 5 C 1 C 3 R 2 G Figure 64: High-pass filter configuration of the infinite gain multiple feedback filter A possible choice to implement a high-pass filter as shown in Figure 64is Y 1 = jωc 1, Y 2 = 1 R 2, Y 3 = jωc 3, Y 4 = jωc 4, Y 5 = 1, R 5 and the transfer function of the circuit becomes jωc = 1 jωc 3 1/R 5 (jωc 1 1/R 2 jωc 3 jωc 4 ) jωc 3 jωc 4 Rearranging the expression to obtain a rational fraction in ω we obtain = ω 2 (C 1/C 4 ) ω 2 1 1 jω(c 1 C 3 C 4 ) R 5 C 3 C 4 R 2 R 5 C 3 C 4 Comparing the denominator of the previous equation with the denominator of the transfer function in section 622 we find that the frequency ω, the quality factor Q, High frequency gain H are respectively ω = 1 R 2 R 5 C 3 C 4, Q = ω R 5 C 3 C 4 (C 1 C 3 C 4 ), H = C 1 C 4
138 CHAPTER 6 ACTIVE FILTERS 643 Band-pass Filter C 4 R 5 R 1 C 3 R 2 Figure 65: Band-pass filter configuration of the infinite gain multiple feedback filter A possible choice to implement a Band-pass filter is shown in Figure 65 The admittances are Y 1 = 1 R 1, Y 2 = 1 R 2, Y 3 = jωc 3, Y 4 = jωc 4, Y 5 = 1 R 5, and the transfer function of the circuit becomes jωc = 3 /R 1 1/R 5 (1/R 1 1/R 2 jωc 3 jωc 4 ) jωc 3 jωc 4 Rearranging the expression to get a rational fraction in ω we finally obtain ( ) C3 C = R jω 4 5 R 5 C 3 C 4 R 1 ω 2 jω C 3 C 4 R 1R 2 C 3 C 4 R 5 R 1 R 2 R 5 C 3 C 4 Comparing the denominator of the previous equation with the denominator of the transfer function in section 623 we find that the resonance frequency, and the quality factor are respectively R ω = 1 R 2 R, Q = 5 C 3 C ω 4, H R 1 R 2 R 5 C 3 C 4 C 3 = R 5 C 4 R 1
65 GENERALIZED SALLEN-KEY FILTER TOPOLOGY (GSK) 139 65 Generalized Sallen-Key Filter Topology (GSK) I 4 Y 4 I 1 I 3 Y 1 A V A Y 2 B V I 3 Y 3 V C Y 6 I 5 Y 5 I 6 Figure 66: Generalized Sallen-Key Topology Let s consider the circuit in Figure 66 with generic admittances Y 1, Y 2, Y 3, Y 4, Y 5, and Y 6 Applying the KCL to node A, we have ( V A ) Y 1 ( V A ) Y 4 (V V A ) Y 2 = (62) Applying KCL to node B (V V A ) Y 2 V Y 3 = V A = Y 2 Y 3 Y 2 V Applying KCL to node C (V V ) Y 6 V Y 5 = V = V = Y 6 Y 6 Y 5 V Replacing the expression found for V A, and V into eq (62) and after quite some boring algebra, we obtain ( = 1 Y 5 Y 6 ) Y 1 Y 2 Y 6 Y 1 Y 6 (Y 2 Y 3 )Y 3 Y 6 (Y 2 Y 4 )Y 2 Y 4 Y 5 Let s analyze some admittances configuration of the this filter topology
14 CHAPTER 6 ACTIVE FILTERS 651 GSK Second Order Low-pass Filter C 4 R 1 R 2 C 3 R 6 R 5 Figure 67: Low-pass filter configuration of the Generalized Sallen-Key filter A possible choice to implement a low-pass filter as shown in Figure 67 is Y 1 = 1 R 1, Y 2 = 1 R 2, Y 3 = jωc 3, Y 4 = jωc 4, Y 5 = 1 R 5, Y 6 = 1 R 6, and the transfer function of the circuit becomes 1 ( = 1 R ) 6 R 1 R 2 C 3 C ( 4 R 5 1 ω 2 jω 1 1 ) R 6 1 R 1 C 4 R 2 C 4 R 2 C 3 R 5 R 1 R 2 C 3 C 4 Comparing the denominator of the previous equation with the denominator of the transfer function in section 621we find that the frequency square ω 2, the quality factor Q, and the DC gain H are respectively ( ω 2 = 1 R, Q = 1 R 2 R 5 C 3 C 4 ω, H = R 1 C 4 R 2 C 3 R 5 (R 1 R 2 ) C 3 R 1 R 6 C 4 1 R 6 R 5 )
65 GENERALIZED SALLEN-KEY FILTER TOPOLOGY (GSK) 141 652 Simple Case If R 1 = R 2 = R, C 3 = C 4 = C, and R 5 = R 6 =, then = ω 2 ω 2 jωω ω 2, ω 2 = 1 R 2 C 2,, Q = 1, which is the transfer function of a second order low-pass filter with low quality factor 653 GSK Second Order High-pass Filter R 4 C 2 C 1 R 3 R 6 R 5 Figure 68: High-pass filter configuration of the Generalized Sallen-Key filter To implement a low-pass filter as shown in Figure 68 one needs to choose the admittances as follows Y 1 = jωc 1, Y 2 = jωc 2, Y 3 = 1 R 3, Y 4 = 1 R 4, Y 5 = 1 R 5, Y 6 = 1 R 6, and the transfer function of the circuit becomes
142 CHAPTER 6 ACTIVE FILTERS ( = 1 R ) 6 R 5 ω ( 2 1 ω² jω 1 1 ) R 6 R 3 C 2 R 3 C 1 R 4 C 1 R 5 1 R 3 R 4 C 1 C 2 Comparing the denominator of the previous equation with the denominator of the transfer function in section 622 we find that the frequency square ω 2, the quality factor Q, and the DC gain H are respectively ω 2 = 1 R 3 C 1 R 4 C 2, Q = ω R 3 R 4 R 5 C 1 C 2 R 5 (C 1 C 2 ) R 3 C 1 R 6 R 4, H = 654 Simple Case If R 1 = R 2 = R, C 3 = C 4 = C, and R 5 = R 6 =, then = ω 2 ω 2 jωω ω 2, ω 2 = 1 R 2 C 2,, Q = 1, which is the transfer function of a second order high-pass filter with low quality factor 655 GSK Band-pass Filter ( 1 R ) 6 R 5 R 4 C 2 V R 1 i R 3 C 3 R6 R 5 Figure 69: Band-pass filter configuration of the Generalized Sallen-Key filter
65 GENERALIZED SALLEN-KEY FILTER TOPOLOGY (GSK) 143 To implement a band-pass filter as shown in Figure 69 one needs to choose the admittances as follows Y 1 = 1 R 1, Y 2 = jωc 2, Y 3 = 1 R 3 jωc 3, Y 4 = 1 R 4, Y 5 = 1 R 5, Y 6 = 1 R 6, and the transfer function of the circuit becomes ( = 1 R ) 6 R 5 R 1 C ( 3 C2 ω 2 C 3 jω 1 1 1 ) R 6 C 2 C 3 R 1 C 3 R 3 C 2 R 4 C 3 R 4 R 5 Comparing the denominator of the previous equation with the denominator of the transfer function in section 623 we find that the frequency square ω 2, the quality factor Q, and the DC gain H are respectively jω R 1R 4 C 2 C 3 R 1 R 3 R 4 Q = ω C 2 C 3 R 1 R 3 R 4 R 5 (C 2 C 3 ) R 3 R 4 R 5 C 2 R 1 R 4 R 5 C 3 R 1 R 3 R 5 C 2 R 1 R 3 R 6, 656 Simple Case ω 2 = R 1R 4,H = 1 ( C 2 C 3 R 1 R 3 R 4 R 1 C 3 1 R 6 R 5 ) Q ω If R 1 = R 3 = R 4 = R, C 2 = C 3 = C, and R 5 = R 6 =, then = jω ω Q ω 2 jω ω Q ω2 2 2, ω = RC,, Q = 3, which is the transfer function of a second order high-pass filter with low quality factor
144 CHAPTER 6 ACTIVE FILTERS 66 State Variable Filter Topology (SV) The state variable filter provides a low pass, a band pass, and a high pass filter outputs At the same time, it allows to change the gain, the cut-off frequencies, and the quality factor independently, but it requires 4 Op- Amps R 1 R 2 R3 R 4 R 5 C 1 C 2 G V HP G V BP G V LP R 7 R 6 Figure 61: State variable filter circuit TBF 67 Practical Considerations 671 Component Values How do we select the values of capacitance and resistance? Here are some considerations that should help the filter design: reducing the resistance values reduces the thermal noise and therefore the filter noise, reducing resistance values minimizes the op-amp voltage offsets,
67 PRACTICAL CONSIDERATIONS 145 increasing the resistance reduce the current load on the op-amps, increasing the resistances usually allows to decrease the capacitance and therefore it make easier to find capacitors because of the small capacitance values needed, reducing the capacitance minimizes the capacitance fluctuations due to temperature, increasing the capacitance allows to reduce resistance values and therefore the thermal noise As we can clearly see, some of the consideration cannot be used at the same time Based on the design requirements one can decide which of the consideration above are more important to finally meet the design requirements Rules of Thumb Particularly critical design often overrule these following rules: Capacitor with capacitance less of ~1 pf should be avoided, Try to use resistor with resistance between few kilo-ohms to few hundreds of kilo-ohms 672 Components technology Capacitors The use of low loss dielectric is very important to obtain good results If possible one should use plastic film capacitors or CG/NPO ceramic capacitors, 1% tolerance for temperature stability Resistor Low thermal noise resistors such as metal film resistors 1% tolerance for temperature stability should be used
146 CHAPTER 6 ACTIVE FILTERS
Bibliography [1] Hank Zumbahlen, State Variable Filters, Mini Tutorial MT-223, Analog Devices 147
148 BIBLIOGRAPHY