2A.6 The s-Plane

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We have noted that the Laplace variable s is a complex variable, with a real part and an imaginary part.

Hence, to represent it we will need two axes at right angles to each other—the real axis and the imaginary

axis. These two axes form a plane, which is called the s-plane. Any general value of s (or any variation or

trace of s) may be marked on the s-plane.

2A.6.1 An Interpretation of Laplace and Fourier Transforms

In the Laplace transformation of a function, f ðtÞ; we multiply the function by e2st and integrate with

respect to t. This process may be interpreted as determining the “components” FðsÞ of f ðtÞ in the

“direction” e2st ; where s is a complex variable. All such components FðsÞ should be equivalent to the

original function, f ðtÞ:

In the Fourier transformation of f ðtÞ we multiply it by e2jvt and integrate with respect to t.

This is the same as setting s ¼ jv: Hence, the Fourier transform of f ðtÞ is FðjvÞ: Furthermore, FðjvÞ

represents the components of f ðtÞ that are in the direction of e2jvt : Since e2jvt ¼ cos vt 2 j sin vt; in

the Fourier transformation we determine the sinusoidal components of frequency v, of a time

Frequency-Domain Analysis 2-51

© 2005 by Taylor & Francis Group, LLC

function f ðtÞ: Since s is complex, FðsÞ is also complex and so is Fð jvÞ: Hence, they all will have a real part

and an imaginary part.

2A.6.2 Application in Circuit Analysis

The fact that sin vt and cos vt are 908 out of phase is further confirmed in view of

ejvt ¼ cos vt þ j sin vt ð2A:21Þ

Consider the R– L – C circuit shown in Figure 2A.2. For the capacitor, the current, i, and the voltage, v, are

related through

i ¼ C

dv

dt ð2A:22Þ

If the voltage v ¼ v0 sin vt; the current i ¼ v0vC cos vt: Note that the magnitude of v=i is 1=vC

(or 1=2p fC where v ¼ 2p f ; f is the cyclic frequency and v is the angular frequency). But v and i are out

of phase by 908. In fact, in the case of a capacitor, i leads v by 908. The equivalent circuit resistance of

a capacitance is called reactance, and is given by

XC ¼

1

2p fC ð2A:23Þ

¼

1

vC ð2A:24Þ

Note that this parameter changes with the frequency.

We cannot add the reactance of the capacitor and the resistance of the resistor algebraically; we must

add them vectorially because the voltages across a capacitor and resistor in series are not in phase, unlike

in the case of a resistor. Also, the resistance in a resistor does not change with frequency. In a series circuit,

as in Figure 2A.2, the current is identical in each element, but the voltages differ in both amplitude and

phase; in a parallel circuit, the voltages are identical, but the currents differ in amplitude and phase.

Similarly, for an inductor,

v ¼ L

di

dt ð2A:25Þ

The corresponding reactance is

XL ¼ vL ¼ 2p f L ð2A:26Þ

If the voltage (E) across R in Figure 2A.2(a) is in the direction shown in Figure 2A.2(b) (i.e., pointing to

the right), then the voltage across the inductor, L, must point up (908 leading) and the voltage across the

capacitor, C, must point down (908 lagging). Since the current (I) is identical in each component of a

E

R

L

C

(a)

EL or IXL

ER or IR

EC or IXC

XL − XC

R

q

Z

(b)

(c)

FIGURE 2A.2 (a) Series RLC circuit; (b) phases of voltage drops; (c) impedance triangle.

2-52 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

series circuit, we see the directions of IR, IXL and IXC as in Figure 2A.2(b), giving the impedance triangle

shown in Figure 2A.2(c).

To express these reactances in the s-domain, we simply substitute s for jv :

2jXC ¼

1

sC

jXL ¼ sL

The series impedance of the RLC circuit can be expressed as

Z ¼ R þ jXL 2 jXC ¼ R þ sL þ

1

sC

In this discussion, note the use of

ffiffiffiffi

21 p or j to indicate a 908 phase change.

Frequency-Domain Analysis 2-53

© 2005 by Taylor & Francis Group, LLC

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