
- •Contents
- •Acknowledgments
- •Preface
- •What a Crossover Does
- •Why a Crossover Is Necessary
- •Beaming and Lobing
- •Passive Crossovers
- •Active Crossover Applications
- •Bi-Amping and Bi-Wiring
- •Loudspeaker Cables
- •The Advantages and Disadvantages of Active Crossovers
- •The Advantages of Active Crossovers
- •Some Illusory Advantages of Active Crossovers
- •The Disadvantages of Active Crossovers
- •The Next Step in Hi-Fi
- •Active Crossover Systems
- •Matching Crossovers and Loudspeakers
- •A Modest Proposal: Popularising Active Crossovers
- •Multi-Way Connectors
- •Subjectivism
- •Sealed-Box Loudspeakers
- •Reflex (Ported) Loudspeakers
- •Auxiliary Bass Radiator (ABR) Loudspeakers
- •Transmission Line Loudspeakers
- •Horn Loudspeakers
- •Electrostatic Loudspeakers
- •Ribbon Loudspeakers
- •Electromagnetic Planar Loudspeakers
- •Air-Motion Transformers
- •Plasma Arc Loudspeakers
- •The Rotary Woofer
- •MTM Tweeter-Mid Configurations (d’Appolito)
- •Vertical Line Arrays
- •Line Array Amplitude Tapering
- •Line Array Frequency Tapering
- •CBT Line Arrays
- •Diffraction
- •Sound Absorption in Air
- •Modulation Distortion
- •Drive Unit Distortion
- •Doppler Distortion
- •Further Reading on Loudspeaker Design
- •General Crossover Requirements
- •1 Adequate Flatness of Summed Amplitude/Frequency Response On-Axis
- •2 Sufficiently Steep Roll-Off Slopes Between the Filter Outputs
- •3 Acceptable Polar Response
- •4 Acceptable Phase Response
- •5 Acceptable Group Delay Behaviour
- •Further Requirements for Active Crossovers
- •1 Negligible Extra Noise
- •2 Negligible Impairment of System Headroom
- •3 Negligible Extra Distortion
- •4 Negligible Impairment of Frequency Response
- •5 Negligible Impairment of Reliability
- •Linear Phase
- •Minimum Phase
- •Absolute Phase
- •Phase Perception
- •Target Functions
- •All-Pole and Non-All-Pole Crossovers
- •Symmetric and Asymmetric Crossovers
- •Allpass and Constant-Power Crossovers
- •Constant-Voltage Crossovers
- •First-Order Crossovers
- •First-Order Solen Split Crossover
- •First-Order Crossovers: 3-Way
- •Second-Order Crossovers
- •Second-Order Butterworth Crossover
- •Second-Order Linkwitz-Riley Crossover
- •Second-Order Bessel Crossover
- •Second-Order 1.0 dB-Chebyshev Crossover
- •Third-Order Crossovers
- •Third-Order Butterworth Crossover
- •Third-Order Linkwitz-Riley Crossover
- •Third-Order Bessel Crossover
- •Third-Order 1.0 dB-Chebyshev Crossover
- •Fourth-Order Crossovers
- •Fourth-Order Butterworth Crossover
- •Fourth-Order Linkwitz-Riley Crossover
- •Fourth-Order Bessel Crossover
- •Fourth-Order 1.0 dB-Chebyshev Crossover
- •Fourth-Order Linear-Phase Crossover
- •Fourth-Order Gaussian Crossover
- •Fourth-Order Legendre Crossover
- •Higher-Order Crossovers
- •Determining Frequency Offsets
- •Filler-Driver Crossovers
- •The Duelund Crossover
- •Crossover Topology
- •Crossover Conclusions
- •Elliptical Filter Crossovers
- •Neville Thiele MethodTM (NTM) Crossovers
- •Subtractive Crossovers
- •First-Order Subtractive Crossovers
- •Second-Order Butterworth Subtractive Crossovers
- •Third-Order Butterworth Subtractive Crossovers
- •Fourth-Order Butterworth Subtractive Crossovers
- •Subtractive Crossovers With Time Delays
- •Performing the Subtraction
- •Active Filters
- •Lowpass Filters
- •Highpass Filters
- •Bandpass Filters
- •Notch Filters
- •Allpass Filters
- •All-Stop Filters
- •Brickwall Filters
- •The Order of a Filter
- •Filter Cutoff Frequencies and Characteristic Frequencies
- •First-Order Filters
- •Second-Order and Higher-Order Filters
- •Filter Characteristics
- •Amplitude Peaking and Q
- •Butterworth Filters
- •Linkwitz-Riley Filters
- •Bessel Filters
- •Chebyshev Filters
- •1 dB-Chebyshev Lowpass Filter
- •3 dB-Chebyshev Lowpass Filter
- •Higher-Order Filters
- •Butterworth Filters up to 8th-Order
- •Linkwitz-Riley Filters up to 8th-Order
- •Bessel Filters up to 8th-Order
- •Chebyshev Filters up to 8th-Order
- •More Complex Filters—Adding Zeros
- •Inverse Chebyshev Filters (Chebyshev Type II)
- •Elliptical Filters (Cauer Filters)
- •Some Lesser-Known Filter Characteristics
- •Transitional Filters
- •Linear-Phase Filters
- •Gaussian Filters
- •Legendre-Papoulis Filters
- •Laguerre Filters
- •Synchronous Filters
- •Other Filter Characteristics
- •Designing Real Filters
- •Component Sensitivity
- •First-Order Lowpass Filters
- •Second-Order Filters
- •Sallen & Key 2nd-Order Lowpass Filters
- •Sallen & Key Lowpass Filter Components
- •Sallen & Key 2nd-Order Lowpass: Unity Gain
- •Sallen & Key 2nd-Order Lowpass Unity Gain: Component Sensitivity
- •Filter Frequency Scaling
- •Sallen & Key 2nd-Order Lowpass: Equal Capacitor
- •Sallen & Key 2nd-Order Lowpass Equal-C: Component Sensitivity
- •Sallen & Key 2nd-Order Butterworth Lowpass: Defined Gains
- •Sallen & Key 2nd-Order Lowpass: Non-Equal Resistors
- •Sallen & Key 2nd-Order Lowpass: Optimisation
- •Sallen & Key 3rd-Order Lowpass: Two Stages
- •Sallen & Key 3rd-Order Lowpass: Single Stage
- •Sallen & Key 4th-Order Lowpass: Two Stages
- •Sallen & Key 4th-Order Lowpass: Single-Stage Butterworth
- •Sallen & Key 4th-Order Lowpass: Single-Stage Linkwitz-Riley
- •Sallen & Key 5th-Order Lowpass: Three Stages
- •Sallen & Key 5th-Order Lowpass: Two Stages
- •Sallen & Key 5th-Order Lowpass: Single Stage
- •Sallen & Key 6th-Order Lowpass: Three Stages
- •Sallen & Key 6th-Order Lowpass: Single Stage
- •Sallen & Key Lowpass: Input Impedance
- •Linkwitz-Riley Lowpass With Sallen & Key Filters: Loading Effects
- •Lowpass Filters With Attenuation
- •Bandwidth Definition Filters
- •Bandwidth Definition: Butterworth Versus Bessel
- •Variable-Frequency Lowpass Filters: Sallen & Key
- •First-Order Highpass Filters
- •Sallen & Key 2nd-Order Filters
- •Sallen & Key 2nd-Order Highpass Filters
- •Sallen & Key Highpass Filter Components
- •Sallen & Key 2nd-Order Highpass: Unity Gain
- •Sallen & Key 2nd-Order Highpass: Equal Resistors
- •Sallen & Key 2nd-Order Butterworth Highpass: Defined Gains
- •Sallen & Key 2nd-Order Highpass: Non-Equal Capacitors
- •Sallen & Key 3rd-Order Highpass: Two Stages
- •Sallen & Key 3rd-Order Highpass in a Single Stage
- •Sallen & Key 4th-Order Highpass: Two Stages
- •Sallen & Key 4th-Order Highpass: Butterworth in a Single Stage
- •Sallen & Key 4th-Order Highpass: Linkwitz-Riley in a Single Stage
- •Sallen & Key 4th-Order Highpass: Single-Stage With Other Filter Characteristics
- •Sallen & Key 5th-Order Highpass: Three Stages
- •Sallen & Key 5th-Order Butterworth Filter: Two Stages
- •Sallen & Key 5th-Order Highpass: Single Stage
- •Sallen & Key 6th-Order Highpass: Three Stages
- •Sallen & Key 6th-Order Highpass: Single Stage
- •Sallen & Key Highpass: Input Impedance
- •Bandwidth Definition Filters
- •Bandwidth Definition: Subsonic Filters
- •Bandwidth Definition: Combined Ultrasonic and Subsonic Filters
- •Variable-Frequency Highpass Filters: Sallen & Key
- •Designing Filters
- •Multiple-Feedback Filters
- •Multiple-Feedback 2nd-Order Lowpass Filters
- •Multiple-Feedback 2nd-Order Highpass Filters
- •Multiple-Feedback 3rd-Order Filters
- •Multiple-Feedback 3rd-Order Lowpass Filters
- •Multiple-Feedback 3rd-Order Highpass Filters
- •Biquad Filters
- •Akerberg-Mossberg Lowpass Filter
- •Akerberg-Mossberg Highpass Filters
- •Tow-Thomas Biquad Lowpass and Bandpass Filter
- •Tow-Thomas Biquad Notch and Allpass Responses
- •Tow-Thomas Biquad Highpass Filter
- •State-Variable Filters
- •Variable-Frequency Filters: State-Variable 2nd Order
- •Variable-Frequency Filters: State-Variable 4th-Order
- •Variable-Frequency Filters: Other Orders of State-Variable
- •Other Filters
- •Aspects of Filter Performance: Noise and Distortion
- •Distortion in Active Filters
- •Distortion in Sallen & Key Filters: Looking for DAF
- •Distortion in Sallen & Key Filters: 2nd-Order Lowpass
- •Distortion in Sallen & Key Filters: 2nd-Order Highpass
- •Mixed Capacitors in Low-Distortion 2nd-Order Sallen & Key Filters
- •Distortion in Sallen & Key Filters: 3rd-Order Lowpass Single Stage
- •Distortion in Sallen & Key Filters: 3rd-Order Highpass Single Stage
- •Distortion in Sallen & Key Filters: 4th-Order Lowpass Single Stage
- •Distortion in Sallen & Key Filters: 4th-Order Highpass Single Stage
- •Distortion in Sallen & Key Filters: Simulations
- •Distortion in Sallen & Key Filters: Capacitor Conclusions
- •Distortion in Multiple-Feedback Filters: 2nd-Order Lowpass
- •Distortion in Multiple-Feedback Filters: 2nd-Order Highpass
- •Distortion in Tow-Thomas Filters: 2nd-Order Lowpass
- •Distortion in Tow-Thomas Filters: 2nd-Order Highpass
- •Noise in Active Filters
- •Noise and Bandwidth
- •Noise in Sallen & Key Filters: 2nd-Order Lowpass
- •Noise in Sallen & Key Filters: 2nd-Order Highpass
- •Noise in Sallen & Key Filters: 3rd-Order Lowpass Single Stage
- •Noise in Sallen & Key Filters: 3rd-Order Highpass Single Stage
- •Noise in Sallen & Key Filters: 4th-Order Lowpass Single Stage
- •Noise in Sallen & Key Filters: 4th-Order Highpass Single Stage
- •Noise in Multiple-Feedback Filters: 2nd-Order Lowpass
- •Noise in Multiple-Feedback Filters: 2nd-Order Highpass
- •Noise in Tow-Thomas Filters
- •Multiple-Feedback Bandpass Filters
- •High-Q Bandpass Filters
- •Notch Filters
- •The Twin-T Notch Filter
- •The 1-Bandpass Notch Filter
- •The Bainter Notch Filter
- •Bainter Notch Filter Design
- •Bainter Notch Filter Example
- •An Elliptical Filter Using a Bainter Highpass Notch
- •The Bridged-Differentiator Notch Filter
- •Boctor Notch Filters
- •Other Notch Filters
- •Simulating Notch Filters
- •The Requirement for Delay Compensation
- •Calculating the Required Delays
- •Signal Summation
- •Physical Methods of Delay Compensation
- •Delay Filter Technology
- •Sample Crossover and Delay Filter Specification
- •Allpass Filters in General
- •First-Order Allpass Filters
- •Distortion and Noise in 1st-Order Allpass Filters
- •Cascaded 1st-Order Allpass Filters
- •Second-Order Allpass Filters
- •Distortion and Noise in 2nd-Order Allpass Filters
- •Third-Order Allpass Filters
- •Distortion and Noise in 3rd-Order Allpass Filters
- •Higher-Order Allpass Filters
- •Delay Lines for Subtractive Crossovers
- •Variable Allpass Time Delays
- •Lowpass Filters for Time Delays
- •The Need for Equalisation
- •What Equalisation Can and Can’t Do
- •Loudspeaker Equalisation
- •1 Drive Unit Equalisation
- •3 Bass Response Extension
- •4 Diffraction Compensation Equalisation
- •5 Room Interaction Correction
- •Equalisation Circuits
- •HF-Cut and LF-Boost Equaliser
- •Combined HF-Boost and HF-Cut Equaliser
- •Adjustable Peak/Dip Equalisers: Fixed Frequency and Low Q
- •Adjustable Peak/Dip Equalisers With High Q
- •Parametric Equalisers
- •The Bridged-T Equaliser
- •The Biquad Equaliser
- •Capacitance Multiplication for the Biquad Equaliser
- •Equalisers With Non-Standard Slopes
- •Equalisers With −3 dB/Octave Slopes
- •Equalisers With −3 dB/Octave Slopes Over Limited Range
- •Equalisers With −4.5 dB/Octave Slopes
- •Equalisers With Other Slopes
- •Equalisation by Filter Frequency Offset
- •Equalisation by Adjusting All Filter Parameters
- •Component Values
- •Resistors
- •Through-Hole Resistors
- •Surface-Mount Resistors
- •Resistors: Values and Tolerances
- •Resistor Value Distributions
- •Obtaining Arbitrary Resistance Values
- •Other Resistor Combinations
- •Resistor Noise: Johnson and Excess Noise
- •Resistor Non-Linearity
- •Capacitors: Values and Tolerances
- •Obtaining Arbitrary Capacitance Values
- •Capacitor Shortcomings
- •Non-Electrolytic Capacitor Non-Linearity
- •Electrolytic Capacitor Non-Linearity
- •Active Devices for Active Crossovers
- •Opamp Types
- •Opamp Properties: Noise
- •Opamp Properties: Slew Rate
- •Opamp Properties: Common-Mode Range
- •Opamp Properties: Input Offset Voltage
- •Opamp Properties: Bias Current
- •Opamp Properties: Cost
- •Opamp Properties: Internal Distortion
- •Opamp Properties: Slew Rate Limiting Distortion
- •Opamp Properties: Distortion Due to Loading
- •Opamp Properties: Common-Mode Distortion
- •Opamps Surveyed
- •The TL072 Opamp
- •The NE5532 and 5534 Opamps
- •The 5532 With Shunt Feedback
- •5532 Output Loading in Shunt-Feedback Mode
- •The 5532 With Series Feedback
- •Common-Mode Distortion in the 5532
- •Reducing 5532 Distortion by Output Stage Biasing
- •Which 5532?
- •The 5534 Opamp
- •The LM4562 Opamp
- •Common-Mode Distortion in the LM4562
- •The LME49990 Opamp
- •Common-Mode Distortion in the LME49990
- •The AD797 Opamp
- •Common-Mode Distortion in the AD797
- •The OP27 Opamp
- •Opamp Selection
- •Crossover Features
- •Input Level Controls
- •Subsonic Filters
- •Ultrasonic Filters
- •Output Level Trims
- •Output Mute Switches, Output Phase-Reverse Switches
- •Control Protection
- •Features Usually Absent
- •Metering
- •Relay Output Muting
- •Switchable Crossover Modes
- •Noise, Headroom, and Internal Levels
- •Circuit Noise and Low-Impedance Design
- •Using Raised Internal Levels
- •Placing the Output Attenuator
- •Gain Structures
- •Noise Gain
- •Active Gain Controls
- •Filter Order in the Signal Path
- •Output Level Controls
- •Mute Switches
- •Phase-Invert Switches
- •Distributed Peak Detection
- •Power Amplifier Considerations
- •Subwoofer Applications
- •Subwoofer Technologies
- •Sealed-Box (Infinite Baffle) Subwoofers
- •Reflex (Ported) Subwoofers
- •Auxiliary Bass Radiator (ABR) Subwoofers
- •Transmission Line Subwoofers
- •Bandpass Subwoofers
- •Isobaric Subwoofers
- •Dipole Subwoofers
- •Horn-Loaded Subwoofers
- •Subwoofer Drive Units
- •Hi-Fi Subwoofers
- •Home Entertainment Subwoofers
- •Low-Level Inputs (Unbalanced)
- •Low-Level Inputs (Balanced)
- •High-Level Inputs
- •High-Level Outputs
- •Mono Summing
- •LFE Input
- •Level Control
- •Crossover In/Out Switch
- •Crossover Frequency Control (Lowpass Filter)
- •Highpass Subsonic Filter
- •Phase Switch (Normal/Inverted)
- •Variable Phase Control
- •Signal Activation Out of Standby
- •Home Entertainment Crossovers
- •Fixed Frequency
- •Variable Frequency
- •Multiple Variable
- •Power Amplifiers for Home Entertainment Subwoofers
- •Subwoofer Integration
- •Sound-Reinforcement Subwoofers
- •Line or Area Arrays
- •Cardioid Subwoofer Arrays
- •Aux-Fed Subwoofers
- •Automotive Audio Subwoofers
- •Motional Feedback Loudspeakers
- •History
- •Feedback of Position
- •Feedback of Velocity
- •Feedback of Acceleration
- •Other MFB Speakers
- •Published Projects
- •Conclusions
- •External Signal Levels
- •Internal Signal Levels
- •Input Amplifier Functions
- •Unbalanced Inputs
- •Balanced Interconnections
- •The Advantages of Balanced Interconnections
- •The Disadvantages of Balanced Interconnections
- •Balanced Cables and Interference
- •Balanced Connectors
- •Balanced Signal Levels
- •Electronic vs Transformer Balanced Inputs
- •Common-Mode Rejection Ratio (CMRR)
- •The Basic Electronic Balanced Input
- •Common-Mode Rejection Ratio: Opamp Gain
- •Common-Mode Rejection Ratio: Opamp Frequency Response
- •Common-Mode Rejection Ratio: Opamp CMRR
- •Common-Mode Rejection Ratio: Amplifier Component Mismatches
- •A Practical Balanced Input
- •Variations on the Balanced Input Stage
- •Combined Unbalanced and Balanced Inputs
- •The Superbal Input
- •Switched-Gain Balanced Inputs
- •Variable-Gain Balanced Inputs
- •The Self Variable-Gain Balanced Input
- •High Input Impedance Balanced Inputs
- •The Instrumentation Amplifier
- •Instrumentation Amplifier Applications
- •The Instrumentation Amplifier With 4x Gain
- •The Instrumentation Amplifier at Unity Gain
- •Transformer Balanced Inputs
- •Input Overvoltage Protection
- •Noise and Balanced Inputs
- •Low-Noise Balanced Inputs
- •Low-Noise Balanced Inputs in Real Life
- •Ultra-Low-Noise Balanced Inputs
- •Unbalanced Outputs
- •Zero-Impedance Outputs
- •Ground-Cancelling Outputs
- •Balanced Outputs
- •Transformer Balanced Outputs
- •Output Transformer Frequency Response
- •Transformer Distortion
- •Reducing Transformer Distortion
- •Opamp Supply Rail Voltages
- •Designing a ±15 V Supply
- •Designing a ±17 V Supply
- •Using Variable-Voltage Regulators
- •Improving Ripple Performance
- •Dual Supplies From a Single Winding
- •Mutual Shutdown Circuitry
- •Power Supplies for Discrete Circuitry
- •Design Principles
- •Example Crossover Specification
- •The Gain Structure
- •Resistor Selection
- •Capacitor Selection
- •The Balanced Line Input Stage
- •The Bandwidth Definition Filter
- •The HF Path: 3 kHz Linkwitz-Riley Highpass Filter
- •The HF Path: Time-Delay Compensation
- •The MID Path: Topology
- •The MID Path: 400 Hz Linkwitz-Riley Highpass Filter
- •The MID Path: 3 kHz Linkwitz-Riley Lowpass Filter
- •The MID Path: Time-Delay Compensation
- •The LF Path: 400 Hz Linkwitz-Riley Lowpass Filter
- •The LF Path: No Time-Delay Compensation
- •Output Attenuators and Level Trim Controls
- •Balanced Outputs
- •Crossover Programming
- •Noise Analysis: Input Circuitry
- •Noise Analysis: HF Path
- •Noise Analysis: MID Path
- •Noise Analysis: LF Path
- •Improving the Noise Performance: The MID Path
- •Improving the Noise Performance: The Input Circuitry
- •The Noise Performance: Comparisons With Power Amplifier Noise
- •Conclusion
- •Index

176 Lowpass and Highpass Filter Characteristics
Table 7.8: Frequencies and Q’s for 2 dB-Chebyshev lowpass filters up to 8th order. For highpass filters use the reciprocal of the frequency. Stages are arranged in order of increasing Q; odd-order filters have the 1st-order section at the end with no Q shown.
Order |
Freq 1 |
Q 1 |
Freq 2 |
Q 2 |
Freq 3 |
Q 3 |
Freq 4 |
Q 4 |
|
|
|
|
|
|
|
|
|
2 |
0.9072 |
1.1286 |
|
|
|
|
|
|
3 |
0.9413 |
2.5516 |
0.3689 |
n/a |
|
|
|
|
4 |
0.4707 |
0.9294 |
0.9637 |
4.5939 |
|
|
|
|
5 |
0.6270 |
1.7751 |
0.9758 |
7.2323 |
0.2183 |
n/a |
|
|
6 |
0.3161 |
0.9016 |
0.7300 |
2.8443 |
0.9828 |
10.4616 |
|
|
7 |
0.4609 |
1.6464 |
0.7971 |
4.1151 |
0.9872 |
14.2802 |
0.1553 |
n/a |
8 |
0.2377 |
0.8924 |
0.5719 |
2.5327 |
0.8425 |
5.5835 |
0.9901 |
18.6873 |
|
|
|
|
|
|
|
|
|
Table 7.9: Frequencies and Q’s for 3 dB-Chebyshev lowpass filters up to 8th-order. For highpass filters use the reciprocal of the frequency. Stages are arranged in order of increasing Q; odd-order filters have the 1st-order section at the end with no Q shown.
Order |
Freq 1 |
Q 1 |
Freq 2 |
Q 2 |
Freq 3 |
Q 3 |
Freq 4 |
Q 4 |
|
|
|
|
|
|
|
|
|
2 |
0.8414 |
1.3049 |
|
|
|
|
|
|
3 |
0.9160 |
3.0678 |
0.2986 |
n/a |
|
|
|
|
4 |
0.4426 |
1.0765 |
0.9503 |
5.5770 |
|
|
|
|
5 |
0.6140 |
2.1380 |
0.9675 |
8.8111 |
0.1775 |
n/a |
|
|
6 |
0.2980 |
1.0441 |
0.7224 |
3.4597 |
0.9771 |
12.7899 |
|
|
7 |
0.4519 |
1.9821 |
0.7920 |
5.0193 |
0.9831 |
17.4929 |
0.1265 |
n/a |
8 |
0.2243 |
1.0337 |
0.5665 |
3.0789 |
0.8388 |
6.8251 |
0.9870 |
22.8704 |
|
|
|
|
|
|
|
|
|
the question. If such sharp filtering is really required (and that situation is not very likely in crossover design), then it will probably be cheaper to convert to the digital domain and use DSPtechniques, which can provide stable filtering of pretty much any kind you can imagine.
Comparisons of 2nd-, 3rd-, and 4th-order order Chebyshev filters of various kinds can be seen in Figures 7.16 to 7.18.
More Complex Filters—Adding Zeros
All the filter types we have looked at so far can be made by plugging together 2nd-order and 1storder lowpass or highpass stages in cascade. This is true no matter how high the filter order. They are technically known as “all-pole filters”, which basically means that they combine different sorts

Lowpass and Highpass Filter Characteristics 177
of roll-off, but always the response stays rolled-off; it does not come back up again. There are no deep notches in the amplitude response; once the roll-off is established, it just keeps on going down. However, filters with notches in their stopband that plunge to the infinite depths (in theory, at least) have their uses, and their study makes up a large proportion of filter theory.
When a faster roll-off than Butterworth is required, without the passband amplitude ripples of the
Chebyshev, one possibility is the Inverse Chebyshev, which adds a notch in the response just outside the passband. Notches have slopes that get steeper and steeper as you approach the actual notch frequency, so the roll-off is much accelerated.
Elliptical (Cauer) filters permit ripple in the passband and have notches in the response just outside the passband, and offer even steeper roll-off slopes. They are also very economical on hardware. Suppose you need a serious lowpass filter that must roll-off from −0.5 dB to −66 dB in a single octave (not very likely in crossover design, but stay with me). This would need a 13th-order Butterworth filter
or an 8th-order Chebyshev, but a 5th-order Cauer filter can do the job with much greater economy in components and also in power, because fewer amplifiers are required.
In filter design the notch frequencies are known as “zeros” because they are the frequencies at which the complex equations describing the filter response give a value of zero—in other words, infinite attenuation. Real filters do not have infinitely deep notches, as the depth usually depends on component tolerances and amplifier gain-bandwidths.An exception is the Bainter filter, which gives beautifully deep notches without tweaking; more on the Bainter later.
The design procedures for these filters are not at all straightforward, and I am simply going to show some design examples. These can have their component values scaled in the usual away to obtain different cutoff frequencies.
It is of course always possible to add notches to a filter response by, for example, cascading a Butterworth filter with a notch filter placed suitably in the stopband. This is however not as efficient as Inverse Chebyshev or Cauer filters (though it is conceptually much simpler), because in the latter the notch is properly integrated with the filter response, and so better passband flatness and sharper rolloffs are obtained.
Inverse Chebyshev Filters (Chebyshev Type II)
The Inverse Chebyshev filter, also known as the Chebyshev Type II filter, does not have amplitude ripples in the passband; instead it has notches (zeros) in the stopband. Like the Chebyshev filter, it is directed towards getting a faster roll-off than a Butterworth filter while meeting other conditions; it offers a maximally flat passband, a moderate group delay, and an equi-ripple stopband. The cutoff
frequency is usually defined to be at the −3 dB level, though other definitions can be used. The Inverse Chebyshev filter uses zeros, so it is not an all-pole filter. These filters are complicated to design, and it would not be a good use of space to try and plod through the procedures here. Instead I am presenting a finished design which can be easily scaled for different frequencies.
Figure 7.34 shows a simple 5th-order Inverse Chebyshev filter with a cutoff frequency of 1 kHz. It is made up of two lowpass notch filters followed by a 1st-order lowpass filter. The most familiar

178 Lowpass and Highpass Filter Characteristics
Figure 7.34: A 5th-order Inverse Chebyshev lowpass filter made up of two lowpass notch filters followed by a 1st-order lowpass. Cutoff frequency 1 kHz.
notch filter is the symmetrical sort, where the gains on either side of the notch are the same; there are however also lowpass notch filters where, as the frequency increases, the response dives down into the notch but comes up again to level out at a lower gain, and highpass notch filters where the gain is lower on the low-frequency side of the notch. There is more on this in Chapter 12 on bandpass and notch filters.
Here the two lowpass notch filters are of the Deliyannis-Friend type, [12][13] which consists basically of a multiple-feedback (MFB) lowpass filter with an extra signal path via the non-inverting input that generates the notch by cancellation. It is not exactly obvious, but these filter stages are non-inverting in the passband. The notch depth is critically dependent on the accuracy of the ratio set by R4, R5.
The complete filter has an overall gain in the passband of +1.3 dB. The filter structure is based on an example given by Van Valkenburg. [14]
The amplitude response in Figure 7.35 shows how the first notch in the stopband, at 1.22 kHz, is very narrow and makes the roll-off very steep indeed. However, the response naturally also starts to come back up rapidly, but this is suppressed by the second notch (at 1.88 kHz), which has a lower Q and is therefore broader and buys time, so to speak, for the final 1st-order filter to start bringing in a useful amount of attenuation. Each time the response comes back up it reaches −17 dB; this is what is meant by an equi-ripple stopband. Variations on this type of filter with non-equal stopband ripples may not be officially Inverse Chebyshev filters, but they can be useful in specific cases.
To scale this circuit for different frequencies you can alter the capacitors, keeping C1 = C2 and
C3 = C4 and the ratios of C1, C3 and C5 the same, or you can change resistors R1, R2, R3 and R6, R7,
R8 and R11, keeping all their ratios the same. You can of course change both capacitors and resistors to get appropriate circuit impedances. R4 and R5 can be altered, but their ratio must remain the same, and likewise with R9 and R10; remember that the accuracy of this ratio controls notch depth. R11 and
C5 can be altered as you wish, quite separately from any other alterations to the circuit, so long as its time-constant is unchanged.
More details on the design of Inverse Chebyshev filters can be found in Van Valkenburg. [15]

Lowpass and Highpass Filter Characteristics 179
Figure 7.35: The amplitude response of a 5th-order Inverse Chebyshev lowpass filter. Note how the passband is maximally flat (no ripples) before the very steep roll-off. Cutoff frequency 1 kHz.
Elliptical Filters (Cauer Filters)
Elliptical filters (also called Cauer filters) are basically a combination of Chebyshev and Inverse Chebyshev filters; amplitude ripples in the passband are accepted as the price of a faster roll-off, and there are also one or notches (zeros) in the stopband to steepen the roll-off rate. They have a sharp cutoff, high group delay, and greatest stopband attenuation. They are sometimes called complete-
Chebyshev filters or Zolotarev filters.
As for the Chebyshev filter, the definition of an elliptic filter cutoff frequency depends on the passband ripple amplitude. In most filter design software any value of attenuation can be defined as the cutoff point.
The design of elliptical filters is not simple, and even the authors of filter textbooks that are a morass of foot-long complex equations are inclined to say things like “it is rather involved . . .” and recommend you use published tables to derive component values. Regrettably, the use of these tables is in itself rather hard going, so here I am just going to give one example of how these filters are put together.
This can be easily scaled for different frequencies. There is another example in Chapter 12 on notch crossovers.
The amount of passband ripple in an elliptic filter is sometimes quoted as a “reflection coefficient” percentage ρ, which as you might imagine is a hangover from transmission line theory, and not in my opinion a very helpful way of putting it.

180 Lowpass and Highpass Filter Characteristics
Wilhelm Cauer (June 24, 1900–April 22, 1945) was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters, [16] and his work marked the beginning of the field of network synthesis. He was shot dead in his garden in Berlin-Marienfelde in Berlin by Soviet soldiers during the capture of the city in 1945.
Elliptical filters are commonly implemented by combining a notch filter with an all-pole filter such as a Butterworth type. The notches used are not in general symmetrical notches that go up to 0 dB either side of the central crevasse—they are usually lowpass or highpass notch filters.Alowpass notch response starts out at 0 dB at low frequencies, plunges into the crevasse, and then comes up again to flatten out at a lower level, often −10 dB. When combined with an all-pole lowpass filter this gives much improved high-frequency attenuation rather than a symmetrical notch. Conversely, a highpass notch has a response that is 0 dB at high frequencies but comes back up to, say, −10 dB at low frequencies. There is more on lowpass and highpass notches in Chapter 12 on bandpass and notch filters.
Figure 7.36 shows a simple 3rd-order elliptical filter with a cutoff frequency of 1 kHz and a reflection coefficient of 20%. The passband ripple is therefore very small at 0.2 dB, and there is only one notch in the stopband. The filter structure is based on an example given by Williams and Taylor [17]. The first stage aroundA1 is a lowpass notch filter, made up of a Twin-T notch filter with its Q (notch sharpness) enhanced by positive feedback through C1, the amount being fixed by R5, R6, which set the closed-loop gain of A1. C4 is added to make a lowpass notch rather than a symmetrical notch. The output of this 2nd-order stage is the upper trace in Figure 7.37, and you can see it has been arrange to peak gently just before the roll-off. When this is combined with the 1st-order lowpass filter R7,
Figure 7.36: A 3rd-order elliptical lowpass filter made up of a lowpass notch filter followed by a 1st-order lowpass. Cutoff frequency 1 kHz.

Lowpass and Highpass Filter Characteristics 181
Figure 7.37: The amplitude response of the 3rd-order elliptical lowpass filter. The lower trace is the final output, and the upper trace is the signal from the first stage. Cutoff frequency 1 kHz.
C5, the final response is the 3rd-order lower trace in Figure 7.37. The 0.2 dB ripple in the passband is just visible.You will observe that as the frequency increases, once the drama of the notch is over the ultimate roll-off slope is only 6 dB/octave because the lowpass notch response is now flat, and so only the final 1st-order filter is contributing to the roll-off. This is the price you pay for implementing a fast roll-off with a filter that is only 3rd-order. Fourth-order filters that have an ultimate roll-off slope of 12 dB/octave are described in Chapter 5 on notch crossovers.
To scale this circuit for different frequencies you can alter all the capacitors, keeping their ratios to each other the same, or the resistors R1–4 and R7.You can of course change both to get appropriate circuit impedances. R5 and R6 can be altered, but their ratio must remain the same.
We have compared the roll-off of the previous filters by looking at the attenuation at 10 kHz, a decade above the 1 kHz cutoff frequency. That is less helpful here because of the way the amplitude response comes back up at high frequencies; the response at 10 kHz is −42 dB, but in my simulation the attenuation at the bottom of the notch (at 2.6 kHz) was about−65 dB. The passband gain is +12.7 dB because of the positive feedback applied to the notch network, and in many cases this will be less than convenient. Elliptical filters can sometimes be very useful for crossover use, for if we have an otherwise good drive unit with some nasty behaviour just outside its intended frequency range, the notch can be dropped right on top of it.
This elliptical filter and the Inverse Chebyshev filter are the only ones in this chapter that are not allpole filters. More details on the design of elliptical filters can be found in Williams and Taylor [18] and Van Valkenburg. [19]