0195136047.pdf

3.4 FREQUENCY RESPONSE 155

the input signal changes, a graphical display of the frequency-response characteristics of networks
is often employed.
Afilteris a network used to select one range of frequencies, while rejecting all other
frequencies. Let us now consider simple networks which function as filters. For theRCnetwork
shown in Figure 3.4.1, the voltage transfer functionV ̄out/V ̄inis given by

H ̄(jω)=

V ̄out

V ̄in

=

1

1 +jωRC

=

1

√

1 +(ωRC)^2

−tan−^1 (ωRC)=H(ω)θ(ω) (3.4.1)

whereH(ω)istheamplitude ratioandθ(ω)is thephase shift. Defining thecutoff frequencyas
the frequencyωat whichH(ω)is reduced to 1/

√
2, or 0.707 of its maximum value (which is 1.0
atω=0), it follows that

ωCO=

1

RC

(3.4.2)

and the phase shift becomes−45° at that cutoff frequency, as shown in Figure 3.4.2. The power
delivered to the circuit, at this frequency, is one-half the maximum power. As a consequence,ωCO
is also called thehalf-power point,or half-power angular frequency. We see that this filter circuit
of Figure 3.4.1 is indeed alow-passfilter because any input signal components atω<<ωCOare
passed to the output with virtually unchanged amplitude and phase, whereas any components
atω>>ωCOhave greatly reduced output amplitudes.ωCOroughly divides thepassband
(region of transmitted frequencies) from thestopband(region of high attenuation). The range
of transmitted frequencies is known as thebandwidth. Note that in Figure 3.4.1 the low-pass
filtering properties come from the fact that the capacitor is an open circuit forω=0 but becomes
a short circuit forω→∞and “shorts out” any high-frequency voltage components across the
output.
The low-pass simple filter circuits are shown in Figure 3.4.3 and Figure 3.4.4 gives the high-
pass simple filter circuits. Note the interchange of resistance and reactance, which converts a
low-pass filter into a high-pass filter, or vice versa.

Vin^1 Ω Vout

jωc

R Ω

+

−

+

−

−− −−

Figure 3.4.1RCnetwork.

− 45 °

− 90 °

ωco = 1/RC ω

ω

Phase shift θ (ω)

Pass

band

Amplitude ratio H (ω)

1.0

0.707

0

Figure 3.4.2Frequency response of

RCnetwork of Figure 3.4.1.