Signals and Systems - Electrical Engineering

336 CHAPTER 5: Frequency Analysis: The Fourier Transform

the capacitor is an open circuit (its impedance would be infinite), so that the voltage in the

capacitor is equal to the voltage in the source. On the other hand, if the frequency of the input

source is very high, then the inductor is an open circuit and the capacitor a short circuit (its

impedance is zero) so that the capacitor voltage is zero. This is a low-pass filter. Notice that this

filter has no finite zeros, and complex conjugate poles.

n High-pass filter:Suppose then that we let the output be the voltage across the inductor. Then

again by voltage division the transfer function

Hhp(s)=

VL(s)

Vi(s)

=

s^2

s^2 +s+ 1

is that of a high-pass filter. Indeed, for a dc input (frequency zero) the impedance in the induc-

tor is zero, so that the inductor voltage is zero, and for very high frequency the impedance

of the inductor is very large so that it can be considered open circuit and the voltage in the

inductor equals that of the source. This filter has the same poles of the low-pass filter (this is

determined by the overall impedance of the circuit, which has not changed) and double zeros

at zero. It is these zeros that make the frequency response for low frequencies be close to zero.

n Band-pass filter:Letting the output be the voltage across the resistor, its transfer function is

Hbp(s)=

VR(s)

Vi(s)

=

s

s^2 +s+ 1

or the transfer function of a band-pass filter. For zero frequency, the capacitor is an open cir-

cuit so the current is zero and the voltage across the resistor is zero. Similarly, for very high

frequency the impedance of the inductor is very large, or an open circuit, making the voltage

across the resistor zero because again the current is zero. For some middle frequency the serial

combination of the inductor and the capacitor resonates and will have zero impedance. At the

resonance frequency, the current achieves its largest value and the voltage across the resistor

does too. This behavior is that of a band-pass filter. This filter again has the same poles as the

other two, but only one zero at zero.

n Band-stop filter:Finally, suppose we consider as output the voltage across the connection of the

inductor and the capacitor. At low and high frequencies, the impedance of the LC connection

is very high, or open circuit, and so the output voltage is the input voltage. At the resonance

frequencyr=1 the impedance of the LC connection is zero, so the output voltage is zero.

The resulting filter is a band-stop filter with the transfer function

Hbs(s)=

s^2 + 1

s^2 +s+ 1

Second-order filters can then be easily identified by the numerator of their transfer functions.

Second-order low-pass filters have no zeros, and the numerator isN(s)=1; band-pass filters

have a zero ats=0 soN(s)=s, and so on. We will see next that such a behavior can be easily

seen from a geometric approach.

n