General

Scaling is the process of shifting an electronic parameter up or down.  Frequency scaling refers to shifting the frequency breakpoints of a circuit up or down, such as moving the corner frequency of a low pass filter up from 1kHz to 2kHz, or down from 1kHz to 500Hz.  Impedance scaling refers to shifting the impedance breakpoints of a circuit up or down, such as changing a tone control circuit that uses 1Meg pots to one that uses 500k pots, or changing a 75 ohm filter to a 50 ohm filter. Any tone control network or passive filter network can be frequency or impedance scaled very easily, by following a few simple steps.

Impedance Scaling

Impedance scaling is accomplished by first calculating an impedance scaling factor, ZSF, as follows:

ZSF = Znew/Zold

The impedance-scaled values are then calculated using the following formulas:

R'  = R*ZSF
L'  = L*ZSF
C' = C/ZSF

where

R', L', and C' are the resistance, inductance, and capacitance values after impedance scaling

Frequency Scaling

Frequency scaling is accomplished by first calculating a frequency scaling factor, FSF as follows:

FSF = desired frequency/existing frequency

The frequency-scaled values are calculated as follows:

R' = R
L' = L/FSF
C' = C/FSF

Frequency and Impedance Scaling

Both frequency and impedance scaling can be accomplished in one step with the following equations:

R' = R*ZSF
L' = (L*ZSF)/FSF
C' = C/(ZSF*FSF)

Examples

For example, if you have a tone control that has two 250K pots, and a 10K pot, and 250pF, 0.1uF, and 0.047uF caps, and a 100K "slope" resistor (standard Fender values), and you would like to impedance-scale the network to use 500K pots, you would use the following values:

ZSF = 500K/250K = 2

Pot1 and Pot2 =250K*2 = 500K
Pot3  = 10K*2      = 20K
R1     = 100K*2    =200K
Cap1 = 250pF/2    = 125pF
Cap2 = 0.1uF/2     = 0.05uF
Cap3 = 0.047uF/2 = 0.0235uF

(Of course, you have to round the values to the closest available 5% or 10% available components)

If you desired to shift the center frequency of the tone controls from 300Hz to 600Hz, you would calculate a frequency scaling factor as follows:

FSF = 600Hz/300Hz = 2

The frequency-scaled values would be calculated as follows:

Pot1 and Pot2 = 250K (no change)
Pot3  = 10K (no change)
R1     = 100K (no change)
Cap1 = 250pF/2     = 125pF
Cap2 =  0.1uF/2     = 0.05uF
Cap3 = 0.047uF/2  = 0.0235uF

If you desired to both frequency- and impedance-scale the circuit, you would calculate both a frequency scaling factor and an impedance scaling factor as follows:

ZSF = 500K/250K   = 2
FSF = 600Hz/300Hz = 2

The new component values would be as follows:

Pot1 and Pot2 = 250K*2 = 500K
Pot3 = 10K*2  = 20K
R1 = 100K*2   = 200K
Cap1 = 250pF/(2*2)     = 62.5pF
Cap2 = 0.1uF/(2*2)      = 0.025uF
Cap3 = 0.047uF/(2*2)  = 0.01175uF

One more thing to take into consideration is the driving source impedance. A typical Marshall tone stack is driven from a cathode follower, which has a very low output impedance (a few K) in relation to the tone stack impedance. A typical Fender tone stack is driven from the plate of a 12AX7, which has a 100K plate resistor, so the driving source impedance is around 38.5K (the 100K plate resistor in parallel with the 62.5K internal plate resistance of the tube). This source impedance can skew the center frequency and attenuation levels if it is significant in comparison to the tone stack impedance.  This is not a problem with a cathode follower drive, unless the impedance of the tone stack is very low in comparison with the cathode follower output impedance.

These scaling techniques work on any passive RLC circuit, including tone controls, lowpass, bandpass, and highpass filters, crossover networks, etc.