Bohn 6-03 Rane Corporation Active Filters, EQs & Crossovers Dennis Bohn Rane Corporation.

Bohn 6-03 Rane Corporation

Active Filters, EQs & Crossovers

Dennis BohnRane Corporation


It’s All About the Mathematics

Electronic filters are all about the mathematics.

You cannot escape the math.

We will study the math;

… you will love the math.


Simplified Laplace Transforms

• Represents complex (frequency dependent) impedance, i.e., magnitude & phase

• Uses the Laplace Operator, s, wheres = complex frequency variable = jω = j2πf – Resistor Impedance = R (freq.

independent)– Capacitor Reactance = 1/sC– Inductor Reactance = sL

• Allows writing a circuit’s transfer function by summing circuit currents using Kirchoff’s Law


Transfer Functions (TF)

• Transfer functions mathematically describe the frequency domain behavior of filters.

• TF = ratio of Laplace Transforms of a circuit’s input and output voltages:

T(s) = Vout(s) / Vin(s)

FilterVin(s) Vout(s)


Filter Transfer Functions

• General filter transfer function is the ratio of two polynomials:

222

2

112

1)(Tcsbsa

csbsas


TF Poles & Zeros

• “Zeros” = values that make numerator equal zero, i.e., the roots of the numerator.– Makes amplitude response rolloff 6

dB/oct.– Shifts phase +90°/zero (+45° @ fc)

• “Poles” = values that make denominator equal zero, i.e., the roots of the denominator.– Makes amplitude response rise 6

dB/oct.– Shifts phase –90°/zero (–45° @ fc)

222

2

112

1)(Tcsbsa

csbsas


Audio Filter Order

• The order or degree (equivalent terms) is the highest power of s in the transfer function.

• For analog circuits usually equals the number of capacitors (or inductors) in the circuit.

• 2nd-order most common.• For common audio filters the order equals the

rolloff rate divided by 6dB/oct, e.g. 24 dB/oct rolloff = 4th order (24 6 = 4)


Audio Filter Order (cont.)

Rule: 6 dB/oct & 90° per orderExamples:

1st-order = 6 dB/oct; θ = 90° ( 45° @ fc)2nd-order = 12 dB/oct; θ = 180° ( 90° @ fc)3rd-order = 18 dB/oct; θ = 270° (135° @ fc) 4th-order = 24 dB/oct; θ = 360° (180° @ fc) … etc.


Why 6 dB/octave Slope?

The impedance of a capacitor is half with

twice the frequency, i.e., XC = 1/sC = 1/2fC

The impedance of an inductor is twice when

frequency doubles, i.e., XL = sL = 2fL

Twice or Half Impedance = 6 dB change

Twice or Half Frequency = One Octave change


Why Phase Shift?

• Phase shift is the flip side of time• It takes time to build up a charge on a

capacitor -- that’s why you cannot change the voltage on a capacitor instantaneously.

• It takes time to build up a magnetic field (flux) in an inductor -- that’s why you cannot change the current through an inductor instantaneously.

• All this time = phase shift


Why 2nd-Order?

• Maximum phase shift is 180 degrees• Guarantees circuit is unconditionally stable• No oscillation problems under any conditions• Get higher order circuits by cascading 2nd-

order sections … or• Design 4th-order section to mathematically

emulate two cascaded 2nd-order (Rane’s L-R)


Normalized Transfer Function

Low-Pass (LP) =(2 poles)

1

12 ss

2 poles = -12 dB/oct

Frequency

Amplitude



• Bandpass (BP) =(1 zero, 2 poles) 12 ss

s

1 pole = -6 dB/oct

Frequency

Amplitude

1 pole = -6 dB/oct

1 zero = +6 dB/oct



High-Pass (HP) =(2 zeros, 2 poles) 12

2

ss

s

2 zeros = +12 dB/oct

Frequency

Amplitude

2 poles = -12 dB/oct


Coefficients Determine Performance

• Butterworth: maximally flat passband s2 + 1.414s + 1

• Chebyshev: steeper rolloff w/magnitude ripples s2 + 1.43s + 1.51

• Bessel: best step response, but gentle rolloff s2 + 3s + 3

BAss

K

22o

o2 ωQω ss

KLP = =


Response Comparison


Q Effects

Butterworth Q = 0.707 Bessel Q = 0.5


Group Delay Comparison


Step Responses

BesselButterworth


Active or Passive?

• There exists no sound quality attributable to active or passive circuits per se.

• TF determines the overshoot, ringing and phase shift regardless of implementation.

• A transfer function is a transfer function is a transfer function … no matter how it is implemented -- all produce the same fundamental results as long as the circuit stays linear: same magnitude response, same phase response, same time response; however there are secondary differences.


Active vs. Passive

Passive• Less noise• No power supply• More reliable• Less EMI susceptible• Better at RF frequency• No oscillations• No on/off transients• No hard clipping• Handles large V & I

Active• Gain & adjustable• No loading effects• Parameters adjustable• Smaller Cs• No inductors• Smaller, lighter &

cheaper• No magnetic coupling• High Q circuits easy


Creating An Equalizer

1

BP Filter

fc

Input Signal

BP

OutIn


Boost = Original + Bandpass

1 + BP

1

fc

BP

+ OutIn

Boost (Lift)


Cut = Reciprocal

11+BP

1

fc

BP

+ OutInCut (Dip)


Why 1/3-Octave Centers?

• 1/3-Octave (21/3 oct = x1.26) approximately represents the smallest region humans reliably detect change.

• Relates to Critical Bands: a range of frequencies where interaction occurs; an auditory filter.

• About 1/3-octave wide above 500Hz (latest info says more like ~1/6-oct); 100 Hz below 500 Hz


Creating A Crossover:Use LP & HP To Split Signal

HP1

HP2

High Out

Mid Out

Input

LP1Low Out

LP2


1st-Order & Butterworth Crossovers

1st-order plus 2nd through 4th-order Butterworth vector diagrams


Linkwitz-Riley Crossover

• Two Cascaded Butterworth Filters• Outputs Down 6 dB at Crossover

Frequency• Both Outputs Always in Phase• No Peaking or Lobing Error at Crossover

Frequency


Creating A LR CrossoverCascaded Butterworth

BW-HP

BW-LP

High Out

Low Out

Input BW-HP

BW-LP


Linkwitz-Riley Crossovers

LR-2LR-4

LR-8


Ray Miller (Rane)Bessel Crossover


Successfully Crossing-Over

• Must know the exact amplitude and phase characteristics of the loudspeakers.

• Driver response strongly interacts with active crossover response.

• True response = loudspeaker + crossover

• DSP multiprocessors à la Drag Net allow custom tailoring the total response.


Accelerated-Slope Tone Controls


Stop Kidding Yourself (Rick Chinn Request)

Why low-cut and high-cut filters are a must for sound system bandwidth control;

or, Why cutting the end sliders on your EQ

doesn’t do diddly-squat.


Analog vs. Digital Filters

Digital• Very complex filters• Full adjustability• Precision vs. cost• Arbitrary magnitude• Total linear phase • EMI & magnetic noise

immunity• Stability (temp &

time)• Repeatability

Analog• Speed 10-100x faster• Dynamic Range

– Amplitude: 140 dB e.g., 12 Vrms & 1 V

noise– Frequency: 8 decades e.g., 0.01 Hz to 1

MHz• Cheap, small, low power• Precision limited by noise

& component tolerances


Digital Filters and DSP

Allow circuit designers to do new things.We can go back and solve old problems ...like the truth-in-slider-position bugaboo of

graphic equalizers:

– Proportional-Q was good– Constant-Q was better– Perfect-Q is best


Truth in Slider PositionProportional-Q


Truth in Slider PositionConstant-Q


Truth in Slider PositionPerfect-Q


PERFECT-Q™ & DEQ 60

Rick JeffsSr. Design Engineer


DEQ 60 Graphic 1/3-Oct EQ


DEQ 60 Features


DEQ 60 Performance

Bohn 6-03 Rane Corporation Active Filters, EQs & Crossovers Dennis Bohn Rane Corporation.

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Transcript of Bohn 6-03 Rane Corporation Active Filters, EQs & Crossovers Dennis Bohn Rane Corporation.