Question

A.C. Fundamentals - Step Response of R-C Circuit

Answer 1

The step response of an R-C (resistor-capacitor) circuit refers to the behavior of the circuit when a sudden change, or step, in input voltage is applied. This behavior is especially important in understanding how the circuit responds to changes in voltage, such as switching it on or off.

Let's consider a simple first-order RC circuit consisting of a resistor (R) and a capacitor (C) connected in series. When a step voltage is applied to this circuit, the capacitor begins to charge or discharge based on the time constant of the circuit.

The time constant (τ) of an RC circuit is given by the product of the resistance (R) and the capacitance (C):

τ = R * C

The step response of an RC circuit is characterized by the charging or discharging of the capacitor. Let's consider two scenarios:

Charging:
When a step voltage is applied to the circuit (for example, switching from 0V to a higher voltage), the capacitor starts to charge. The voltage across the capacitor increases exponentially towards the final voltage level. The charging behavior can be described by the following equation:

Vc(t) = V_final * (1 - e^(-t/τ))

Where:

Vc(t) is the voltage across the capacitor at time 't'
V_final is the final voltage level
e is the base of the natural logarithm (approximately 2.71828)
τ is the time constant
Discharging:
When a step voltage is removed from the circuit (for example, switching from a higher voltage to 0V), the capacitor starts to discharge. The voltage across the capacitor decreases exponentially towards 0V. The discharging behavior can be described by a similar equation:

Vc(t) = V_initial * e^(-t/τ)

Where:

V_initial is the initial voltage across the capacitor

In both charging and discharging scenarios, it's important to note that the capacitor's voltage doesn't change instantaneously. Instead, it changes gradually over time based on the time constant of the circuit.

The time constant (τ) determines how quickly the voltage across the capacitor changes. A larger time constant means slower changes, while a smaller time constant leads to faster changes. The time constant is typically used to define the behavior of the circuit's response.

In summary, the step response of an R-C circuit involves the exponential charging or discharging of the capacitor, and this behavior is characterized by the time constant of the circuit.