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How to synthesise variable resistors with hyperbolic taper

01 Feb 2016  | T Flaig

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In adjustable, frequency-selective RC networks, the corner frequencies of the network is determined by the reciprocal of an RC product, ωC=1/RC. If the adjustable elements are potentiometers with a linear-control characteristic—that is, taper—R(α)=αRP, where α is the normalized wiper position, 0&α&1, and RP is the potentiometer's end-to-end resistance, then the corner frequencies are reciprocal functions of the potentiometer's wiper position, and the frequency scale compresses at the high end of the adjustment range. This situation is usually undesirable because it complicates adjustment of the network at the high end. To make the frequency scale linear requires a control element with a hyperbolic taper—that is, something in the form R(α)=RP/(A+αB). Such variable resistances are not generally available from manufacturers, but you can synthesise them using a linear taper potentiometer and a few other components.


Figure 1: This simple circuit synthesises a grounded variable resistance with a hyperbolic-control characteristic.


Figure 1 shows a simple circuit for producing a ground-referenced variable resistance having the desired hyperbolic-control characteristic. Analysis of this circuit yields the following relationship between the control setting and the resistance from Node 1 to ground: R1-0(α)=R1R2RP/(R1R2+R1RP+αR2RP)0&α&1. If you use this resistance in series or in parallel with a capacitor, the resulting corner frequency will be a linear function of α: ωC=(R1R2+R1RP+αR2RP)/R1R2RPC. The minimum and maximum values for R1-0 are R1-0MIN=R1R2RP/(R1R2+R1RP+R2RP) and R1-0MAX=R2RP/(R2+RP).

To design this circuit for specific values of R1-0MIN and R1-0MAX, choose RP>R1-0MAX and then compute R1=R1-0MAX R1-0MIN/(R1-0MAX–R1-0MIN) and R2=RPR1-0MAX/(RP–R1-0MAX).


Figure 2: You can realise a floating variable resistance, with hyperbolic taper, with this circuit. Note that fixed resistors with the same number are matched pairs.



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