The RC time constant, denoted τ (lowercase tau), the time constant of a resistor–capacitor circuit (RC circuit), is equal to the product of the circuit resistance and the circuit capacitance:
τ = R C . {\displaystyle \tau =RC\,.}
It is the time required to charge the capacitor, through the resistor, from an initial charge voltage of zero to approximately 63.2% of the value of an applied DC voltage, or to discharge the capacitor through the same resistor to approximately 36.8% of its initial charge voltage. These values are derived from the mathematical constant e, where 63.2 % ≈ 1 − e − 1 {\displaystyle 63.2\%\approx 1{-}e^{-1}} and 36.8 % ≈ e − 1 {\displaystyle 36.8\%\approx e^{-1}} . When using the International System of Units, R is in ohms, C is in farads, and τ is in seconds. Discharging a capacitor through a series resistor to zero volts from an initial voltage of V0 results in the capacitor having the following exponentially-decaying voltage curve:
V C ( t ) = V 0 ⋅ ( e − t / τ ) {\displaystyle V_{\text{C}}(t)=V_{0}\cdot (e^{-t/\tau })}
Charging an uncharged capacitor through a series resistor to an applied constant input voltage V0 results in the capacitor having the following voltage curve over time:
V C ( t ) = V 0 ⋅ ( 1 − e − t / τ ) {\displaystyle V_{\text{C}}(t)=V_{0}\cdot (1-e^{-t/\tau })}
which is a vertical mirror image of the charging curve.
Cutoff frequency The time constant τ {\displaystyle \tau } is related to the RC circuit's cutoff frequency fc, by
τ = R C = 1 2 π f c ≈ 0.159 f c , {\displaystyle \tau =RC={\frac {1}{2\pi f_{c}}}\approx {\frac {0.159}{f_{c}}},}
or, equivalently,
f c = 1 2 π R C = 1 2 π τ ≈ 0.159 τ . {\displaystyle f_{c}={\frac {1}{2\pi RC}}={\frac {1}{2\pi \tau }}\approx {\frac {0.159}{\tau }}.}
Using resistance in ohms and capacitance in farads yields a time constant in seconds and cutoff frequency in hertz (Hz). The cutoff frequency when expressed as an angular frequency ( ω c = 2 π f c ) {\displaystyle (\omega _{c}{=}2\pi f_{c})} is simply the reciprocal of the time constant. In more complicated circuits consisting of more than one resistor and/or capacitor, the open-circuit time constant method provides a way of approximating the cutoff frequency by computing a sum of several RC time constants. A rise time that depends primarily on an RC circuit will be proportional to the time constant:
rise time (20% to 80%) t r ≈ 1.4 τ ≈ 0.22 f c {\displaystyle t_{r}\approx 1.4\tau \approx {\frac {0.22}{f_{c}}}}
rise time (10% to 90%) t r ≈ 2.2 τ ≈ 0.35 f c {\displaystyle t_{r}\approx 2.2\tau \approx {\frac {0.35}{f_{c}}}}
Calculator.00000110000001111111.36836.810.3681110.1591111 For instance, 1 of resistance with 1 of capacitance produces a time constant of approximately 1 seconds. This τ corresponds to a cutoff frequency of approximately 159 millihertz or 1 radians per second. If the capacitor has an initial voltage V0 of 1 , then after 1 τ (approximately 1 seconds or 1.443 half-lives), the capacitor's voltage will discharge to approximately 368 millivolts:
The tangent of the voltage V ( t ) {\displaystyle V(t)} hits the zero axis at a time t + τ {\displaystyle t+\tau } .
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