For the transfer function
Tv(s)=[10(s+5)(s+20)]/[(s+1)(s+100)]
is this a low-pass, high-pass, bandpass or bandstop function?
estimate the corner frequency(ies)
estimate the passband gain
what is the output if v1(t)=10cos(1000t+45deg) V??
To determine whether the given transfer function is a low-pass, high-pass, bandpass, or bandstop function, we need to analyze the poles and zeros of the transfer function.
In this case, the transfer function is:
Tv(s) = [10(s+5)(s+20)]/[(s+1)(s+100)]
We can determine the type of the transfer function by looking at the poles. Poles are the values of 's' that make the denominator equal to zero.
In the denominator, we have two poles: s = -1 and s = -100.
A low-pass filter allows low-frequency signals to pass through, while attenuating high-frequency signals. The corner frequency indicates the frequency at which the magnitude starts to decrease significantly.
For this transfer function, since the poles with the negative real parts are closer to the origin (-1), it indicates that this is a low-pass filter.
The corner frequency can be estimated by taking the square root of the product of the two closest poles' magnitudes. In this case, the magnitude of the poles -1 and -100 is 1 and 100, respectively.
So, the corner frequency would be approximately √(1 * 100) = 10.
To estimate the passband gain, we can evaluate the transfer function at low frequencies where the signal is not significantly attenuated. For low frequencies s ≈ 0, we have:
Tv(s) ≈ [10 * (0+5)(0+20)]/[(0+1)(0+100)]
Simplifying the expression, we get:
Tv(s) ≈ 100/100 = 1
Therefore, the passband gain is approximately equal to 1.
Now, let's determine the output if v1(t) = 10cos(1000t+45deg) V.
To find the output, we substitute s = jω into the transfer function, where j is the imaginary unit, and ω is the frequency.
v1(t) = 10cos(1000t+45deg) V
v1(s) = 10/(s + jw)
Replacing s with jω, we get:
v1(jω) = 10/(jω)
Now, substitute this expression into the transfer function:
Tv(jω) = [10(jω+5)(jω+20)]/[(jω+1)(jω+100)]
We can simplify further by multiplying and dividing by -j (to eliminate j in the denominator):
Tv(jω) = [10(jω+5)(jω+20)]/[(-j)(-j)(jω+1)(jω+100)]
Expanding the terms, we get:
Tv(jω) = [10(jω+5)(jω+20)]/[(-j)(j^3ω^2 + 101jω + 100)]
Simplifying the expression, we get:
Tv(jω) = [10(jω+5)(jω+20)]/[(-j)(101jω - ω^2 - 100)]
Finally, we can multiply the numerator and denominator by (-1) to make the denominator real:
Tv(jω) = -[10(jω+5)(jω+20)]/[(101jω - ω^2 - 100)]
By evaluating this expression, we can calculate the output.