|G(w) 3 dB 1/√√2 u(t) y(t) 20 dB/decade 1+ jan 1 + jωτ 1/τ W Figure 3: Bode plot of a low-pass filter Figure 4: Block diagram of a cascaded low-pass filter 3. Consider a low-pass filter with frequency response function G₁(w) 1 1+ jwT (3) where is a time constant, from which a cutoff frequency 1/7 is defined. Moreover, magnitude |G1(w) is plotted in Fig. 3. When w<1/7, the magnitude |G₁(w) is roughly 1. When w>1/7, the magnitude |G₁(w)| rolls off with 20 dB/decade. The bandwidth is defined when G₁(w) is dropped from 1 to 1/√2. Engineer X wants to have a filter with a faster 2 rolloff. Therefore, Engineer X designs a new filter by cascading two low-pass filters as shown in Fig. 4. Therefore, the new filter has a frequency response function 1 G₂(w)= (1+jwT)² Answer the following questions. (4) (a) Derive the magnitude and phase of G2(w) as functions of w. (b) Let us focus on |G2(w), the magnitude of G₂(w). Plot |G2(w)| by conducting the fol- lowing asymptotic analysis. When w<1/7, what does the magnitude |G2(w)| approach to? When w>1/7, how fast does the magnitude |G2(w)| rolls off? (c) What is the bandwidth of the new filter G2(w)? [Hint: Recall the definition of band- width above.] What is the phase angle LG2(w), when the driving frequency reaches the bandwidth of G2(w)? (d) Does Engineer X achieve his goal, i.e., a faster rolloff when w>1/7? Does Engineer X sacrifice any performance of the new filter? Please justify your answer instead of simply stating yes or no.
|G(w) 3 dB 1/√√2 u(t) y(t) 20 dB/decade 1+ jan 1 + jωτ 1/τ W Figure 3: Bode plot of a low-pass filter Figure 4: Block diagram of a cascaded low-pass filter 3. Consider a low-pass filter with frequency response function G₁(w) 1 1+ jwT (3) where is a time constant, from which a cutoff frequency 1/7 is defined. Moreover, magnitude |G1(w) is plotted in Fig. 3. When w<1/7, the magnitude |G₁(w) is roughly 1. When w>1/7, the magnitude |G₁(w)| rolls off with 20 dB/decade. The bandwidth is defined when G₁(w) is dropped from 1 to 1/√2. Engineer X wants to have a filter with a faster 2 rolloff. Therefore, Engineer X designs a new filter by cascading two low-pass filters as shown in Fig. 4. Therefore, the new filter has a frequency response function 1 G₂(w)= (1+jwT)² Answer the following questions. (4) (a) Derive the magnitude and phase of G2(w) as functions of w. (b) Let us focus on |G2(w), the magnitude of G₂(w). Plot |G2(w)| by conducting the fol- lowing asymptotic analysis. When w<1/7, what does the magnitude |G2(w)| approach to? When w>1/7, how fast does the magnitude |G2(w)| rolls off? (c) What is the bandwidth of the new filter G2(w)? [Hint: Recall the definition of band- width above.] What is the phase angle LG2(w), when the driving frequency reaches the bandwidth of G2(w)? (d) Does Engineer X achieve his goal, i.e., a faster rolloff when w>1/7? Does Engineer X sacrifice any performance of the new filter? Please justify your answer instead of simply stating yes or no.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
![|G(w)
3 dB
1/√√2
u(t)
y(t)
20 dB/decade
1+ jan
1 + jωτ
1/τ
W
Figure 3: Bode plot of a low-pass filter
Figure 4: Block diagram of a cascaded
low-pass filter
3. Consider a low-pass filter with frequency response function
G₁(w)
1
1+ jwT
(3)
where is a time constant, from which a cutoff frequency 1/7 is defined. Moreover, magnitude
|G1(w) is plotted in Fig. 3. When w<1/7, the magnitude |G₁(w) is roughly 1. When
w>1/7, the magnitude |G₁(w)| rolls off with 20 dB/decade. The bandwidth is defined
when G₁(w) is dropped from 1 to 1/√2. Engineer X wants to have a filter with a faster
2
rolloff. Therefore, Engineer X designs a new filter by cascading two low-pass filters as shown
in Fig. 4. Therefore, the new filter has a frequency response function
1
G₂(w)=
(1+jwT)²
Answer the following questions.
(4)
(a) Derive the magnitude and phase of G2(w) as functions of w.
(b) Let us focus on |G2(w), the magnitude of G₂(w). Plot |G2(w)| by conducting the fol-
lowing asymptotic analysis. When w<1/7, what does the magnitude |G2(w)| approach
to? When w>1/7, how fast does the magnitude |G2(w)| rolls off?
(c) What is the bandwidth of the new filter G2(w)? [Hint: Recall the definition of band-
width above.] What is the phase angle LG2(w), when the driving frequency reaches the
bandwidth of G2(w)?
(d) Does Engineer X achieve his goal, i.e., a faster rolloff when w>1/7? Does Engineer X
sacrifice any performance of the new filter? Please justify your answer instead of simply
stating yes or no.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1720cd77-2294-45a6-8b70-43447dc02920%2F1f3d49eb-2755-4112-937c-aeba9e772d59%2Fnzqnjh8_processed.png&w=3840&q=75)
Transcribed Image Text:|G(w)
3 dB
1/√√2
u(t)
y(t)
20 dB/decade
1+ jan
1 + jωτ
1/τ
W
Figure 3: Bode plot of a low-pass filter
Figure 4: Block diagram of a cascaded
low-pass filter
3. Consider a low-pass filter with frequency response function
G₁(w)
1
1+ jwT
(3)
where is a time constant, from which a cutoff frequency 1/7 is defined. Moreover, magnitude
|G1(w) is plotted in Fig. 3. When w<1/7, the magnitude |G₁(w) is roughly 1. When
w>1/7, the magnitude |G₁(w)| rolls off with 20 dB/decade. The bandwidth is defined
when G₁(w) is dropped from 1 to 1/√2. Engineer X wants to have a filter with a faster
2
rolloff. Therefore, Engineer X designs a new filter by cascading two low-pass filters as shown
in Fig. 4. Therefore, the new filter has a frequency response function
1
G₂(w)=
(1+jwT)²
Answer the following questions.
(4)
(a) Derive the magnitude and phase of G2(w) as functions of w.
(b) Let us focus on |G2(w), the magnitude of G₂(w). Plot |G2(w)| by conducting the fol-
lowing asymptotic analysis. When w<1/7, what does the magnitude |G2(w)| approach
to? When w>1/7, how fast does the magnitude |G2(w)| rolls off?
(c) What is the bandwidth of the new filter G2(w)? [Hint: Recall the definition of band-
width above.] What is the phase angle LG2(w), when the driving frequency reaches the
bandwidth of G2(w)?
(d) Does Engineer X achieve his goal, i.e., a faster rolloff when w>1/7? Does Engineer X
sacrifice any performance of the new filter? Please justify your answer instead of simply
stating yes or no.
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