You have a resistor and a capacitor of unknown values. First, you charge the capacitor and discharge it through the resistor. By monitoring the capacitor voltage on an oscilloscope, you see that the voltage decays to half its initial value in 2.5 ms. You then use the resistor and capacitor to make a low-pass filter. What is the crossover frequency f_c?

검증된 단계별 안내

Determine the time constant (τ) of the RC circuit. The voltage across a discharging capacitor decays according to the equation: \( V(t) = V_0 e^{-t/\tau} \). Given that the voltage decays to half its initial value in 2.5 ms, use the relationship \( \frac{1}{2} = e^{-t/\tau} \) to solve for \( \tau \). Take the natural logarithm of both sides to find \( \tau = \frac{t}{\ln(2)} \).

Substitute the given time \( t = 2.5 \; \text{ms} = 2.5 \times 10^{-3} \; \text{s} \) into the equation \( \tau = \frac{t}{\ln(2)} \) to calculate the time constant \( \tau \).

Recall that the time constant \( \tau \) for an RC circuit is defined as \( \tau = R C \), where \( R \) is the resistance and \( C \) is the capacitance. Since \( \tau \) is already determined, this relationship can be used to understand the RC circuit's behavior, though the individual values of \( R \) and \( C \) are not required for the next step.

The crossover frequency (also called the cutoff frequency) \( f_c \) for a low-pass RC filter is given by the formula: \( f_c = \frac{1}{2 \pi \tau} \). Use the previously calculated value of \( \tau \) to find \( f_c \).

Substitute the value of \( \tau \) into the formula \( f_c = \frac{1}{2 \pi \tau} \) to calculate the crossover frequency. This will give the frequency at which the output signal is attenuated to \( \frac{1}{\sqrt{2}} \) of its maximum value.

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주요 개념

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RC Time Constant

The RC time constant, denoted as τ (tau), is a measure of the time it takes for the voltage across a capacitor to charge or discharge to approximately 63.2% of its maximum value. It is calculated as τ = R × C, where R is the resistance in ohms and C is the capacitance in farads. In this scenario, the time it takes for the voltage to decay to half its initial value can be used to determine the time constant, which is crucial for understanding the behavior of the RC circuit.

Crossover Frequency

The crossover frequency (fc) in a low-pass filter is the frequency at which the output voltage is reduced to 70.7% of the input voltage, corresponding to a -3 dB point in the frequency response. It is calculated using the formula fc = 1/(2πRC), where R is the resistance and C is the capacitance. This frequency is significant because it defines the boundary between the passband and the stopband of the filter, determining how the circuit responds to different frequencies.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of the capacitor discharging through the resistor, the voltage across the capacitor decreases exponentially over time, following the equation V(t) = V0 * e^(-t/τ), where V0 is the initial voltage, t is time, and τ is the time constant. Understanding this behavior is essential for analyzing the voltage decay observed on the oscilloscope and for calculating the time constant needed for further analysis.