# How to Calculate Steady-State Voltage

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Voltage levels in time-varying circuits change over time. Time-varying means that the voltage ramps up exponentially until it reaches the steady-state voltage. For this reason, a circuit is said to be in a steady steady when the voltage ceases to change over time. In a simple resistor-capacitor (RC) circuit, consisting of a source voltage (Vs), a resistor (R) and a capacitor (C), the time it takes to reach a steady-state condition is determined by the value of R and C. Therefore, engineers can design circuits to reach steady state at a time of their choosing by adjusting the values of R and C.

Determine the source voltage, or "Vs," as a power supply to your circuit. As an example, choose Vs to be 100 volts.

Pick the value of the resistor, R, and the capacitor, C, for your circuit. R is in units of ohms and C is in units of microfarads. As an example, assume R is 10 ohms and C is 6 microfarads.

Calculate the steady state voltage using the formula: V = Vs (1-e^-t/RC) where e^-t/RC is the exponent e to the negative power of t divided by RC. The variable t represents the elapsed time since Vs was turned on. For example:

at t = 0 seconds RC = 10 x 0.000006 = 0.00006 t/RC = 0/ 0.00006 = 0 e^-t/RC = e^-0 = 1 V = 100 (1-1) = 100 (0) = 0 volts

at t = 5 microseconds RC = 10 x 0.000006 = 0.00006 t/RC = 0.000005/ 0.00006 = 0.083 e^-t/RC = e^-0.083 = 0.92 V = 100 (1- 0.92) = 8 volts

at t = 1 second RC = 10 x 0.000006 = 0.00006 t/RC = 1/ 0.00006 = 16666.7 e^-t/RC = e^-16666.7 = 0 (effectively) V = 100 (1-0) = 100 volts (steady state)

In this example, the voltage increase from 0 at t = 0 to 100 volts at t = 1 second and it will remain at 100 as t increases. As a consequence, 100 volts is the steady-state voltage.

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