Let a 1st order low pass RC Circuit is excited with a unit step function. The voltage response across the resistor is?

Question

Answer

Response of a First Order Low Pass RC Circuit to a Unit Step Function

A first order low pass RC circuit consists of a resistor and a capacitor connected in series. When excited with a unit step function, the circuit undergoes a transient response before reaching steady state. The voltage response across the resistor can be calculated using the following steps:

Step 1: Derive the Transfer Function

The transfer function of a first order low pass RC circuit can be derived using Kirchhoff's laws and Ohm's law. The voltage across the capacitor is given by:

VC = V0(1 - e^(-t/RC))

Where VC is the voltage across the capacitor, V0 is the initial voltage across the circuit, t is time, R is the resistance of the resistor and C is the capacitance of the capacitor.

The voltage across the resistor can be calculated using Ohm's law:

VR = IR x R

Where IR is the current flowing through the circuit.

The transfer function of the circuit is given by:

H(s) = VR/V0 = R/(RCs + 1)

Step 2: Determine the Time Constant

The time constant of the circuit is given by:

τ = RC

The time constant represents the time taken for the circuit to reach 63.2% of its steady state value.

Step 3: Determine the Steady State Value

The steady state value of the circuit can be determined by taking the limit as t approaches infinity:

VC(∞) = V0

This means that the voltage across the capacitor will eventually reach the same value as the initial voltage across the circuit.

Step 4: Determine the Transient Response

The transient response of the circuit can be determined using the following equation:

VC(t) = V0(1 - e^(-t/RC))

This equation represents the voltage across the capacitor as a function of time. The voltage across the resistor can be calculated using Ohm's law:

VR(t) = IR(t) x R

The current flowing through the circuit can be calculated using Kirchhoff's laws:

IR(t) = C(dVC/dt)

Substituting the equation for VC(t) into the equation for IR(t) and then into the equation for VR(t), we get:

VR(t) = V0e^(-t/RC)

This equation represents the voltage across the resistor as a function of time. The voltage across the resistor will decay exponentially from its initial value to zero as the capacitor charges up.

Step 5: Plot the Voltage Response

The voltage response across the resistor can be plotted as a function of time. The plot will consist of a transient response followed by a steady state response. The time constant of the circuit determines

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