Logarithmic Plot and Approximations Quiz

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Questions and Answers

Which function has the form A sin ωt, where A is its amplitude and ω is its frequency in radians per unit time?

  • Sine function (correct)
  • Tangent function
  • Cosine function
  • Exponential function

What is the complex conjugate of N = x + jy?

  • x + jy
  • -x - jy
  • x - jy (correct)
  • -x + jy

What is the magnitude of the complex number N = -2 - 5j / 3 + 4j?

  • |N| = 5
  • |N| = 195
  • |N| = 29 (correct)
  • |N| = 3

What is the angle of the complex number N = -2 - 5j / 3 + 4j?

<p>φ = 248° (A)</p> Signup and view all the answers

Which of the following is true about complex numbers as functions of frequency?

<p>Complex numbers as functions of frequency have a magnitude and an angle (B)</p> Signup and view all the answers

What is the magnitude of the frequency transfer function of a stable LTI system?

<p>The magnitude is the ratio of the sinusoidal steady-state output amplitude over a sinusoidal input amplitude (D)</p> Signup and view all the answers

What is the phase shift of the steady-state output relative to the input in a stable LTI system?

<p>The phase shift is the angle of the frequency transfer function (C)</p> Signup and view all the answers

What is the steady-state response of a linear system to a sinusoidal input?

<p>The steady-state response is sinusoidal with the same frequency as the input, but with a different amplitude and phase shift (B)</p> Signup and view all the answers

According to the text, what is the steady-state response of the system with a frequency of 80 rad/s?

<p>$v2(t) = 5 + 1.59 cos(80t - 1.012)$ (B)</p> Signup and view all the answers

What happens to the steady-state amplitude of the output as the frequency of the input increases?

<p>It decreases (C)</p> Signup and view all the answers

What is the relationship between the magnitude ratio M and its decibel equivalent m?

<p>$m = 10 log M^2 = 20 log M dB$ (B)</p> Signup and view all the answers

What does a negative value of m correspond to in terms of the magnitude ratio M?

<p>$M &lt; 1$, the system attenuates the input (B)</p> Signup and view all the answers

According to equation (9.1.20), what is the phase angle φ(ω) equal to?

<p>K + tan−1 (τ1 ω) − tan−1 (τ2 ω) (D)</p> Signup and view all the answers

What are the low-frequency, corner frequency, and high-frequency values of φ(ω) in equation (9.1.20)?

<p>0°, 45°, and 90° (B)</p> Signup and view all the answers

In equation (9.2.3), what is the expression for the magnitude m(ω) in decibel units?

<p>20 log |K | + 20 log |N1 ( jω)| + 20 log |N2 ( jω)| - 20 log |D1 ( jω)| - 20 log |D2 ( jω)| (D)</p> Signup and view all the answers

Which of the following is the transfer function for a first order system with numerator dynamics?

Signup and view all the answers

What is the phase angle of the transfer function $T(j\omega) = \frac{4}{5+j\omega}$?

<p>$0.253$ radians (A)</p> Signup and view all the answers

What is the steady-state response of the system $ẏ + 5y = 4ġ + 12g$ with input $g(t) = 20\sin(4t)$?

<p>$y_{ss}(t) = 62.46\sin(4t + 0.253)$ (C)</p> Signup and view all the answers

What is the plot of $m(\omega)$ for the transfer function $T(j\omega) = \frac{K(\tau_1\omega j + 1)}{\tau_2\omega j + 1}$?

<p>The plot of $m(\omega)$ is obtained by subtracting the plot of $\tau_2s + 1$ from that of $\tau_1s + 1$, with a scale adjustment of $20\log|K|$. (C)</p> Signup and view all the answers

According to the text, what is the steady-state response of a system with transfer function $T(j\omega) = \frac{1},{1+j\omega\tau}$?

<p>$y_{ss}(t) = A\sin(\omega t + \phi)$ (D)</p> Signup and view all the answers

According to the text, what is the phase shift of a system with transfer function $T(j\omega) = \frac{1},{1+j\omega\tau}$?

<p>$\phi = -\tan^{-1}(\omega\tau)$ (C)</p> Signup and view all the answers

According to the text, what is the magnitude of the transfer function $T(j\omega) = \frac{1},{1+j\omega\tau}$?

<p>$|T(j\omega)| = \frac{1},{\sqrt{1+\omega^2\tau^2}}$ (D)</p> Signup and view all the answers

According to the text, what is the steady-state response of a system with transfer function $T(j\omega) = \frac{1},{1+j\omega\tau}$ to a sinusoidal input $f(t) = A\sin(\omega t)$?

<p>$y_{ss}(t) = B\sin(\omega t + \phi)$ (B)</p> Signup and view all the answers

Which of the following is true about periodic inputs?

<p>Periodic inputs are commonly found in many applications. (B)</p> Signup and view all the answers

What does the term 'frequency response' refer to?

<p>How a system responds to a periodic input. (B)</p> Signup and view all the answers

What can be determined from the frequency response plots or transfer function of a system?

<p>The steady-state response to a sinusoidal input. (D)</p> Signup and view all the answers

Which term refers to how a system responds to a periodic input, such as a sinusoid?

<p>Frequency response (D)</p> Signup and view all the answers

What is the period of a sinusoidal input with a frequency of 60 Hz?

<p>1/60 s (B)</p> Signup and view all the answers

What can be determined from the frequency response plots or the transfer function of a system?

<p>All of the above (D)</p> Signup and view all the answers

Which of the following is true about the frequency response of a system?

<p>All of the above. (D)</p> Signup and view all the answers

What is the period of a sinusoidal input with a frequency of 60 Hz?

<p>1/60 s (C)</p> Signup and view all the answers

What does the steady-state response of a system depend on?

<p>All of the above (D)</p> Signup and view all the answers

Which of the following statements is true about the magnitude of the frequency transfer function of a stable LTI system?

<p>It is always positive (D)</p> Signup and view all the answers

What is the range of possible values for the magnitude of the frequency transfer function of a stable LTI system?

<p>$[0, \infty)$ (B)</p> Signup and view all the answers

Which of the following can be a possible value for the magnitude of the frequency transfer function of a stable LTI system?

<p>$3.5$ (A)</p> Signup and view all the answers

What does a magnitude of zero for the frequency transfer function of a stable LTI system indicate?

<p>The system is at rest (D)</p> Signup and view all the answers

According to equation (9.1.20), what is the phase angle φ(ω) equal to?

<p>$-\arctan(\omega\tau)$ (D)</p> Signup and view all the answers

What is the relationship between the magnitude ratio M and its decibel equivalent m?

<p>$m = 20\log_{10}(M)$ (A)</p> Signup and view all the answers

What is the steady-state response of a linear system to a sinusoidal input?

<p>A sinusoidal output with the same frequency but possibly different amplitude and phase (D)</p> Signup and view all the answers

What happens to the steady-state amplitude of the output as the frequency of the input increases?

<p>It decreases (A)</p> Signup and view all the answers

Which of the following is the equation of a line in slope-intercept form?

<p>$y = 2x + 3$ (B)</p> Signup and view all the answers

Which of the following is the equation of a quadratic function?

<p>$y = x^2 - 4x + 3$ (A)</p> Signup and view all the answers

Which of the following is the equation of a circle?

<p>$(x - 2)^2 + (y + 3)^2 = 4$ (C)</p> Signup and view all the answers

Which of the following is the equation of a parabola?

<p>$y = x^2 - 4x + 3$ (B)</p> Signup and view all the answers

Which of the following is the equation of an exponential function?

<p>$y = 2^x$ (A)</p> Signup and view all the answers

Which of the following is the equation of a logarithmic function?

<p>$y = \log_2{x}$ (A)</p> Signup and view all the answers

According to equation (9.1.20), what is the phase angle φ(ω) equal to?

<p>K + tan−1 (τ1 ω) − tan−1 (τ2 ω) (B)</p> Signup and view all the answers

What is the magnitude of the frequency transfer function of a stable LTI system?

<p>|K ||N1 ( jω)||N2 ( jω)|. |D1 ( jω)||D2 ( jω)| (B)</p> Signup and view all the answers

What is the expression for the magnitude m(ω) in decibel units?

<p>20 log |T ( jω)| = 20 log |K | + 20 log |N1 ( jω)| + 20 log |N2 ( jω)| + · · · − 20 log |D1 ( jω)| − 20 log |D2 ( jω)| − · · · (B)</p> Signup and view all the answers

According to equation (9.1.20), what is the phase angle φ(ω) equal to?

<p>\arctan(\frac{\tau_2\omega},{1}) (A)</p> Signup and view all the answers

What is the magnitude of the complex number N = -2 - 5j / 3 + 4j?

<p>\frac{5},{3} (D)</p> Signup and view all the answers

Which of the following is the transfer function for a first order system with numerator dynamics?

<p>\frac{K(s + \tau_1)},{s(\tau_2 s + 1)} (B)</p> Signup and view all the answers

Which function has the form A sin ωt, where A is its amplitude and ω is its frequency in radians per unit time?

<p>Trigonometric function (B)</p> Signup and view all the answers

What is the steady-state response of the system $ẏ + 5y = 4ġ + 12g$ with input $g(t) = 20\sin(4t)$?

<p>\frac{12},{\sqrt{145}}\sin(4t + \arctan(\frac{4},{3})) (A)</p> Signup and view all the answers

Which of the following best describes the magnitude of the frequency transfer function of a stable LTI system?

<p>The absolute value of the transfer function (A)</p> Signup and view all the answers

What is the complex conjugate of N = x + jy?

<p>N = x - jy (C)</p> Signup and view all the answers

Which term refers to how a system responds to a periodic input, such as a sinusoid?

<p>Steady-state response (D)</p> Signup and view all the answers

What is the relationship between the magnitude ratio M and its decibel equivalent m?

<p>m = 20 \log_{10}(M) (D)</p> Signup and view all the answers

What is the phase shift of the steady-state output relative to the input in a stable LTI system?

<p>0 degrees (C)</p> Signup and view all the answers

Which one of these is the phase shift of a system with transfer function $T(j\omega) = \frac{1},{1+j\omega\tau}$?

<p>$-\arctan(\omega\tau)$ (C)</p> Signup and view all the answers

Which one of these is the steady-state response of a system with transfer function $T(j\omega) = \frac{1},{1+j\omega\tau}$ to a sinusoidal input $f(t) = A\sin(\omega t)$?

<p>$|T(j\omega)|A\sin(\omega t - \arctan(\omega\tau))$ (D)</p> Signup and view all the answers

Which of the following best describes a Bode plot?

<p>A graph of the frequency response of a system (B)</p> Signup and view all the answers

What is the purpose of a Bode plot?

<p>To graph the gain and phase-shift plots of a system (A)</p> Signup and view all the answers

What does a Bode magnitude plot show?

<p>The magnitude (usually in decibels) of the frequency response (C)</p> Signup and view all the answers

Which of the following is true about the Bode phase plot?

<p>The phase is plotted on a linear vertical axis. (A)</p> Signup and view all the answers

What is the magnitude of the response of a linear, time-invariant system subjected to an input with frequency ω?

<p>|H(jω)| (D)</p> Signup and view all the answers

What is the phase shift of the response of a linear, time-invariant system subjected to an input with frequency ω?

<p>arg(H(jω)) (D)</p> Signup and view all the answers

What is the premise of a Bode plot?

<p>One can consider the log of a function as a sum of the logs of its zeros and poles. (B)</p> Signup and view all the answers

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