What is the formula for the impedance of a capacitor in the s-domain?

What is the formula for the impedance of a capacitor in the s-domain?

• A. 1/jωC
• B. 1/sC
• C. sC
• D. jωC

Answer: (B)

The impedance of a capacitor in the s-domain is given by the formula \( Z(s) = \frac{1}{sC} \). In this expression, \( s \) represents a complex frequency variable used in Laplace transforms, where \( C \) is the capacitance in farads. The use of the s-domain is particularly useful in control systems and engineering applications because it allows for the analysis of circuits in a frequency domain where systems can be represented in terms of their dynamics. When we derive this impedance, we start from the time-domain relationship for a capacitor, where the current \( I \) through the capacitor is equal to the derivative of the voltage \( V \) across it multiplied by the capacitance \( C \): \[ I(t) = C \frac{dV(t)}{dt}. \] Applying the Laplace transform to this equation, we switch from the time domain to the s-domain. The Laplace transform of the time derivative introduces the term \( s \) into our expressions. This transformation results in: \[ I(s) = sC V(s), \] which can be rearranged to show that the voltage across the capacitor is related to the current through its impedance: \[ V(s)

The impedance of a capacitor in the s-domain is given by the formula ( Z(s) = \frac{1}{sC} ). In this expression, ( s ) represents a complex frequency variable used in Laplace transforms, where ( C ) is the capacitance in farads. The use of the s-domain is particularly useful in control systems and engineering applications because it allows for the analysis of circuits in a frequency domain where systems can be represented in terms of their dynamics.

When we derive this impedance, we start from the time-domain relationship for a capacitor, where the current ( I ) through the capacitor is equal to the derivative of the voltage ( V ) across it multiplied by the capacitance ( C ):

[ I(t) = C \frac{dV(t)}{dt}. ]

Applying the Laplace transform to this equation, we switch from the time domain to the s-domain. The Laplace transform of the time derivative introduces the term ( s ) into our expressions. This transformation results in:

[ I(s) = sC V(s), ]

which can be rearranged to show that the voltage across the capacitor is related to the current through its impedance:

[ V(s)