Minimizing RC area for given Time Constant

20 August 2019 Link

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Objective

During IC design it is often required to choose appropriate resistor and capacitor sizes to implement a filter or design networks. Here we derive a formula to decide what resistor and capacitor values can be used for a given time constant so that the area is minimized.

Derivation

$C_□$ = capacitance per unit area for the process
$R_□$ = resistance per square for the process
$τ$ = Time constant of the circuit = $RC$
$$Area=R/{R_□}W^2+C/{C_□}=τ/{CR_□}W^2+C/{C_□}$$

Here W = width of the resistor, this is chosen based on the accuracy needed for the resistor according to the process data
Differentiating with respect to C we have:
$${d{Area} }/{dC} = -τ/{R_□}W^2 1/{C^2}+1/{C_□}$$

Equating that to 0 and calculating C we have:
$$C=√{{τW^2C_□}/{R_□}}$$


Thus using this capacitance value and using $R=τ/C$ would give the optimized area for the RC network