Frequency response

Derives the frequency response of RC low-pass filter using the Laplace transform. Part of a series about the properties of the RC low-pass filter.\(\)

Frequency Response

The frequency response \(y_{ss}(t)\) is defined as the steady state response to a sinusoidal input signal \(u(t)=\sin(\omega t)\,\gamma(t)\). It describes how well the filter can distinguish between different frequencies.

In Evaluating Transfer Functions, we have proven that

$$ y_{ss}(t)=|H(j\omega)|\,\sin(\omega t+\angle H(j\omega))\,\gamma(t) $$

The transfer function \(H(s)\) for this RC Filter is given by \transfer polynominal.

$$ H(s )= K\frac{1}{s-p},\ K = \frac{1}{RC},\ p = -\frac{1}{RC} \label{eq:splane} $$

The system behavior at \(\omega=0\) and at \(\omega\rightarrow \infty\) indicates that this is a low pass filter.

$$ \begin{align} \lim_{s \rightarrow j0} |H(s)| &=-p\frac{1}{0-p} =1 \\ \lim_{s \rightarrow j\infty} |H(s)| &=-p\frac{1}{\infty-p} =0 \end{align} $$

Based on Euler’s formula, we can express \(H(s)\) in polar coordinates

$$ \left\{ \begin{align} H(s) &= |H(s)|\ e^{j\angle{H(s)}}\nonumber \\ |H(s)| &= K \frac{\prod_{i=1}^m\left|(s-z_i)\right|}{\prod_{i=1}^n\left|(s-p_i)\right|} =K \frac{1}{\left|s-p\right|}\nonumber \\ \angle{H(s)}&=\sum_{i=1}^m\angle(s-z_i)-\sum_{i=1}^n\angle(s-p_i) =-\angle(s-p)\nonumber \end{align} \right. $$

This transfer function with pole \(p\), evaluated for \(s=j\omega\) can be visualized with a vector from the pole to \(j\omega\).

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Evaluated for \(s=j\omega\)

The length of the vector corresponds to \(|(H(j\omega)|\), and minus the angle with the real axis corresponds to phase shift \(\angle H(j\omega)\).

$$ \left\{ \begin{align} |H(j\omega)| &=K \frac{1}{\left|j\omega-p\right|}= K \frac{1}{\sqrt{\omega^2+p^2}}\nonumber \\ \angle{H(j\omega)}&=-\angle(j\omega-p)=\mathrm{atan2}(\omega,-p)\nonumber\\ &=-\arctan\frac{\omega}{-p},\ p\lt 0\land p\in\mathbb{R}\nonumber \end{align} \right. \label{eq:polar1} $$

Substitute \(p=-\frac{1}{RC}\) and \(K=\frac{1}{RC}\)

$$ \left\{ \begin{eqnarray} |H(j\omega)| &=&\frac{1}{RC} \frac{1}{\sqrt{\omega^2+\left(\frac{1}{RC}\right)^2}}\nonumber \\ \angle{H(j\omega)}&=&-\arctan\frac{\omega}{\frac{1}{RC}}\nonumber \end{eqnarray} \right. \label{eq:eq101} $$

The output signal \(y_{ss}(t)\) for a sinusoidal input signal \(\sin(\omega t)\,\gamma(t)\)

$$ \shaded{ \begin{aligned} y_{ss}(t)&=|H(j\omega)|\,\sin(\omega t+\angle H(j\omega))\,\gamma(t)\nonumber \\ |H(j\omega)| &=\frac{1}{\sqrt{(1+\omega RC)^2}}\\ \angle{H(j\omega)}&=-\arctan(\omega RC)\nonumber \end{aligned} } \label{eq:frequencyresponse} $$

This frequency response for different frequencies can be visualized in a Bode plot or a Nyquist diagram. Each of these are a topic of the remaining sections.

Effect on Input with Harmonics

As a side step, we examine the effect of the filter on a square wave input signal. The Fourier series of the square wave shows that it consists of a base frequency and odd harmonics.

$$ x(t)=\frac{4}{\pi}\sum_{n=1,3,\dots }^\infty \frac{\sin\left(n\omega t\right) }{n} $$

Substituting \(C=470\ \mathrm{nF}\), \(R=100\ \Omega\) and 20 kHz in \(\eqref{eq:frequencyresponse}\), gives the output signal \(y_{ss}(t)\)

$$ \left\{ \begin{align} y_{ss}(t)&=\frac{4}{\pi}\sum_{n=1,3,\dots }^\infty |H(jn\omega)| \frac{\sin\left(n\omega t+\angle H(jn\omega)\right) }{n}\nonumber\\ |H(jn\omega)| &=\frac{1}{\sqrt{(1+n\omega RC)^2}}\nonumber\\ \angle{H(jn\omega)}&=-\arctan(n\omega RC)\nonumber\\ RC&=47\cdot 10^{-6}\nonumber\\ \omega&=2\pi20\cdot 10^{3}\nonumber \end{align} \right. $$


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Input and output signals

Bode plot

A Bode plots frequency as the horizontal axis and usually consists of two separate plots to that show the magnitude and phase of the frequency response \(y_{tt}\). Since the range of magnitudes may also be large, the amplitude scale is usually expressed in decibels \(20\log_{10}\left|H(j\omega)\right|\) . The frequency axis uses a logarithmic scale as well.

$$ \begin{align} |H(j\omega)|\ &= \frac{1}{\sqrt{1+(\omega RC)^2}} \\ |H_{dB}(j\omega)|\ &= -20\log\sqrt{1+ (\omega RC)^2} \label{eq:polar2} \\ \angle{H(j\omega)}\ &=-\arctan\left(\omega RC\right) \end{align} $$

The magnitude of the frequency response has a relatively shallow drop-off.

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Frequency response

The phase shift depends on the frequency, causing signals composed of multiple frequencies to be distorted.

Angular frequency \(\omega_c=|p|\), is is known as the cutoff, break, -3dB or half-power frequency because the magnitude of the transfer function \(\eqref{eq:polar2}\) equals \(1/\sqrt{2}\)

$$ \begin{align} |H(j\omega_c)|\ &=\dfrac{1}{\sqrt{1+(\omega_c RC)^2}}=\frac{1}{\sqrt{2}},\quad\omega_c=\frac{1}{RC} \\ |H_{dB}(j\omega_c)|\ &= 20\log\frac{1}{\sqrt{2}} \approx-3\rm{\ dB} \end{align} $$

The attenuation slope is calculated by first expressing the magnitude relative to the cutoff angular frequency \(\omega_c\)

$$ \begin{array}{llr} &|H(j\omega)| = \frac{1}{\sqrt{1+\left(\frac{\omega}{\omega_c}\right)^2}}\\ \Rightarrow&|H(j\omega)| =\frac{\omega}{\omega_c} & \forall\ {\omega\gg\omega_c} \end{array} $$

The rate of change of attenuation is usually expressed in dB/decade, where an decade is a factor of 10 in frequency, \(\omega_2=10\omega_1\)

$$ \begin{gather}\begin{aligned} |H_{dB}(j\omega_2)|-|H_{dB}(\omega_1)|\ &= -20\log\left(\frac{10\omega_1}{\omega_c}\right)+20\log\left(\frac{\omega_1}{\omega_c}\right) \\ &=20\log\left(\frac{\omega_1}{\omega_c}\frac{\omega_c}{10\omega_1}\right) \\ &=-20 \mathrm{\ dB/decade} \end{aligned}\end{gather} $$

This single-pole filter gives has a relatively shallow -20 dB/decade drop-off.

In general, the cutoff frequency is equal to the radial distance of the poles or zeros from the origin of the \(s\)-plane. For information on sketching the Bode magnitude plot from the poles and zeros, refer to Understanding Poles and Zeros [MIT 3.1].

Nyquist plot

The Nyquist plots display both amplitude and phase angle on a single plot, using the angular frequency as the parameter. It helps visualize if a system is stable or unstable.

Starting with the transfer function transfer polynominal

$$ \begin{gather}\begin{array}{ccc} H(s) = K\frac{1}{s-p}, &K = \frac{1}{RC}, &p = -\frac{1}{RC} \end{array}\end{gather} $$

Evaluate at \(s=j\omega\) and split into real and imaginary parts

$$ \begin{gather} \begin{aligned} H(j\omega)\ &=K\frac{1}{j\omega-p} \\ &=K\frac{1}{j\omega-p}\frac{j\omega+p}{j\omega+p} \\ &=K\frac{j\omega+p}{-\omega^2-p^2} \\ &=K\frac{-j\omega-p}{\omega^2+p^2} \end{aligned} \\ \Rightarrow \left\{ \begin{aligned} \Re\left\{{H(j\omega)}\right\}=&K\frac{-p}{p^2+\omega^2} \\ \Im\left\{{H(j\omega)}\right\}=&K\frac{-\omega}{p^2+\omega^2} \\ \end{aligned} \right. \end{gather} $$

Plot the frequency transfer function for \(-\infty\lt\omega\gt\infty\), indicating an increase of frequency using an arrow. A dashed line is used for negative frequencies. (The plot was generated using the GNU/Octave as shown in Appendix A.)

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Nyquist diagram

From the plot we see that for \(\omega=0\) the gain is 1, and for \(\omega\to\infty\) the gain becomes 0. The high frequency portion of the plot approaches the origin at an angle of -90 degrees. For more information on Nyquist refer to Determining Stability using the Nyquist Plot [swarthmore].

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