An R C filter is shown. The filter resistance R is variable between 180 Ωand 2200 Ωand the filte

An R C filter is shown. The filter resistance R is variable...

An $R C$ filter is shown. The filter resistance $R$ is variable between $180 \Omega$ and $2200 \Omega$ and the filter capacitance is $C=0.086 \mu \mathrm{F}$. At what frequency is the output amplitude equal to $1 / \sqrt{2}$ times the input amplitude if $R=$ (a) $180 \Omega$ ? (b) $2200 \Omega$ ? (c) Is this a low-pass or highpass filter? Explain.

An $R C$ filter is shown.
The filter resistance $R$
is variable between $180 \Omega$ and $2200 \Omega$ and
the filter capacitance
is $C=0.086 \mu \mathrm{F}$. At what frequency is the output amplitude equal to $1 / \sqrt{2}$ times the input amplitude if $R=$
(a) $180 \Omega$ ? (b) $2200 \Omega$ ?
(c) Is this a low-pass or highpass filter? Explain.

Key Concepts

Frequency Response

Frequency response characterizes how a filter's output amplitude and phase vary with input signal frequency. For an RC circuit, this analysis is crucial for understanding how different frequency components of a signal are affected, which underpins the practical application of the circuit in filtering noise or isolating specific frequency bands.

Low-pass and High-pass Filters

Low-pass filters allow signals with frequencies below the cutoff frequency to pass while attenuating higher frequencies. Conversely, high-pass filters allow higher frequencies to pass through while attenuating lower frequencies. Distinguishing between these relies on analyzing the circuit configuration and the phase relationships between the resistor and capacitor.

RC Filter

An RC filter is an electronic circuit composed of resistors and capacitors that is used to control the frequency content of signals. It is a fundamental building block in electronics for tasks such as smoothing, timing, and signal processing.

Cutoff Frequency

The cutoff frequency is the point at which the filter begins to significantly attenuate the input signal, specifically where the output amplitude is 1/?2 of the input amplitude. In RC circuits, this is typically calculated using the formula f_c = 1/(2?RC), which is essential for designing and understanding filter behavior.

Transcript

00:01 Inductors and capacitors can be built into simple frequency filter circuits.

00:07 They kind of behave opposite each other.

00:12 And you can use either one for either what's called a low pass, which means preferentially giving a high voltage for low frequencies or high pass, which means preferentially providing a high voltage for high frequencies.

00:30 And if we can see in this circuit then, because the capacitive reactants is equal to 1 over omega -c, and we can think about the r and the c as being in a series rc circuit with the ac driver, the current will be the same, current amplitude, will be given by the amplitude of the drive, divided by the impedance.

01:14 Okay, and of course we can write the impedance.

01:17 We'll do that in a little bit.

01:19 But the output voltage, v .c, and this is really the amplitude, it will be wiggling in time.

01:33 That is going to be the amplitude of the current times the capacitive reactants.

01:41 This is the generalized form of oms law.

01:45 And what we can tell is, that because the omega is in the denominator, xc becomes larger at low frequencies.

02:01 It is inversely proportional to the frequency.

02:06 And this means the output vc amplitude goes up at low frequency.

02:16 And so this is a low pass filter.

02:19 Now we can actually determine the output from the input voltage.

02:33 The i amplitude, of course, we can put in.

02:38 Okay, determine vc over the amplitude, and we're going to use vc is equal to v amp over z times xc.

02:57 So this is kind of like a generalized voltage, relationship, simply putting in for the current because everything's in series.

03:09 And so the ratio of the voltage on the capacitor to the amplitude, these are amplitudes only, is going to be equal to the capacitive reactants divided by the full impedance of the series circuit.

03:26 And we can play around with that a little bit more algebraically, 1 over omega -c, divided by the square root of the resistance squared, plus 1 over omega -c squared.

03:45 And a little bit more algebra.

03:47 We can see that this ratio, we're going to kind of try to simplify the denominator a little bit, by pulling out a factor of 1 over omega -c, and then that simplifies down...

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