Consider the filter circuit shown in Active Figure 33.23 a . (a) Show that the ratio of the outp

step 1 I'm going to begin by drawing active figure 33 .23. Just like this. There's a resistor up here.

Consider the filter circuit shown in Active Figure 33.23 a ....

Consider the filter circuit shown in Active Figure 33.23 $\mathrm{a}$ . (a) Show that the ratio of the output voltage to the input voltage is $$ \frac{\Delta v_{\mathrm{out}}}{\Delta v_{\mathrm{in}}}=\frac{1 / \omega C}{\sqrt{R^{2}+\left(\frac{1}{\omega C}\right)^{2}}} $$ (b) What value does this ratio approach as the frequency decreases toward zero? What value does this ratio approach as the frequency increases without limit? (c) At what frequency is the ratio equal to one-half?

Consider the filter circuit shown in Active Figure 33.23 $\mathrm{a}$ . (a) Show that the ratio of the output voltage to the input voltage is
$$
\frac{\Delta v_{\mathrm{out}}}{\Delta v_{\mathrm{in}}}=\frac{1 / \omega C}{\sqrt{R^{2}+\left(\frac{1}{\omega C}\right)^{2}}}
$$
(b) What value does this ratio approach as the frequency decreases toward zero? What value does this ratio approach as the frequency increases without limit? (c) At what frequency is the ratio equal to one-half?

Key Concepts

Transfer Function Magnitude

The transfer function in a circuit defines the ratio of the output to input signal in the frequency domain. Its magnitude provides insights into how effectively a circuit attenuates or passes signals of different frequencies. Evaluating the magnitude of the transfer function, including its behavior at the limits of low and high frequencies, helps in characterizing filter performance.

Voltage Divider Rule in AC Circuits

The voltage divider rule is a fundamental concept used to determine the voltage across components connected in series. In AC circuits, the rule is applied using complex impedances, meaning that the output voltage is a fraction of the input voltage determined by the ratio of a component’s impedance to the sum of all series impedances. This approach is key to deriving the transfer function for filter circuits.

Frequency Response of RC Circuits

The frequency response of an RC circuit describes how the output voltage amplitude changes with frequency. By analyzing the transfer function, one can determine the behavior of the circuit as frequency varies, such as approaching certain limits when frequency tends to zero or infinity, and identifying specific frequencies at which the output is significantly attenuated or amplified.

Phasor Analysis

Phasor analysis is employed to convert circuit elements operating with sinusoidal signals from the time domain into a frequency-domain representation. This technique simplifies the analysis by turning differential equations into algebraic ones using complex numbers, which is critical when dealing with AC circuit behavior and determining the relationships between voltage and current.

Impedance of Reactive Components

In AC circuits, components such as capacitors and inductors are represented by their impedance rather than simply resistance. For capacitors, the impedance is inversely proportional to frequency and is given by 1/(j?C). This concept is essential for understanding how the phase and magnitude of the output signal in a filter circuit vary with frequency.