0195136047.pdf

156 TIME-DEPENDENT CIRCUIT ANALYSIS

−

+

−

+

(a)

Vin

jωL

jωC

R V^1

out

−

+

−

+

(b)

Vin

R

Vout

Figure 3.4.3Low-pass simple filter circuits.(a)RLlow-pass filter.(b)RClow-pass filter.

−

+

−

+

(a)

Vin jωL jωC

R 1

Vout

−

+

−

+

(b)

Vin R Vout

Figure 3.4.4High-pass simple filter circuits(a)RLhighpass filter.(b)RChigh-pass filter.

The low-pass transfer function, given by Equation (3.4.1), can be written as

H(ω)=

1

√

1 +(ω/ωCO)^2

; θ(ω)=−tan−^1

ω

ωCO

(3.4.3)

which also applies to theRLlow-pass filter withωCO=R/L.

The high-pass transfer function for either of the circuits of Figure 3.4.4 is given by

H(ω)=

(ω/ωCO)

√

1 +(ω/ωCO)^2

; θ (ω)=90°−tan−^1

ω

ωCO

(3.4.4)

whereωCO=R/Lor 1/RC. The ideal response characteristics of low pass, high pass, bandpass,

and band reject are shown in Figure 3.4.5.

Due to the greater availability and lower cost of capacitors and the undesirable resistance

usually associated with inductors, virtually all simple filter designs employ capacitors rather than

inductors, despite the similarity ofRLandRCcircuits. More sophisticated filters use two or more

reactive elements in order to have essentially constant or flatH(ω) over the passband and narrow

transition between the passband and the stopband.

One of the most common forms of display of the network functionH ̄(jω)is theBode

diagram. In this diagram the magnitude and phase ofH ̄(jω)are plotted separately as functions

of the frequency variableω. A logarithmic scale is used for the frequency variable in order

to accommodate wide frequency ranges, and the magnitude of the network functionH(ω)is

expressed indecibels(dB) as

H(dB)=20 log 10 H (3.4.5)

The two portions of the Bode plot,H(ω) andθ(ω), are graphed on semilog axes. The advantage

of this technique is that rather than plotting the network characteristic point by point, one can

employ straight-line approximations to obtain the characteristic quite efficiently.

Asymptotic Bode plotslead to simply drawn approximate characteristics which are quite

adequate for many purposes and reduce the calculational complications greatly. The frequency

response is found by substitutings=jωin the network functionH(s). Let