Principles of Analog Electronics

223

Frequency Domain

7.1 INTRODUCTION

We saw in Section 4.6 of Chapter 4 that the equations for voltage and current for

capacitors are

)

)

(

(

=it C

dv t

dt

;

1

0

∫

() ()

=+

vt

C

itdt v

and in Section 4.6 for inductors

()

()

=vt L

di t

dt

;

1

0

∫

() ()

=+

it

L

vtdt i

Such equations depend on time, which is the domain of our functions.

However, thanks to a mathematical tool, an integral transform, it is possible

to change the domain of functions and operate not in time but in frequency or,

more generically, in the domain of the complex argument s by the Laplace trans-

form, denoted

Lft

{}

()

. In such a way, the equations of capacitors and inductors

are no longer given by derivatives and integrals, which can be difcult to treat

mathematically, but become easy expressions of the type v = Xi, in which X is

named reactance.

In particular, it is called capacitive reactance

X

C

, if it concerns the capacitor,

and inductive reactance

X

L

, if it concerns the inductor.

7

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Principles of Analog Electronics

224

Observation

The behavior of resistances in a circuit is described by Ohm’s law. Given that

this law does not depend on the time, the results are not varied by the integral

transform.

The pure sinusoidal function we are dealing with in the time domain now becomes

a Dirac delta function in the frequency domain, as represented in Figure 7.1.

7.2 RESISTANCE, REACTANCE, IMPEDANCE

It is useful to refer to the Laplace transform expressed in the domain of the com-

plex argument s = σ + jω.

In the new domain s, the voltage to current ratio might not be a simple real

number, and it is not dened as resistance as done previously but, more generi-

cally, impedance.

For reasons of dimensional coherence, impedance is measured in ohm

[

Ω

]

and

it is generically a complex number. However it can be reduced to the real num-

ber already known as resistance (if it lacks the imaginary part), or an imaginary

number (if it lacks the real part) called reactance.

Impedence = resistance + j * reactance

In symbols:

Z = R + jX

The inverse of the impedance is named admittance

 

1

YS

or mhoor

[]

Ω

−

.

v

s

(t)

t

(a)

v

s

( f )

f

(b)

v

s

(t)

t

(c)

v

s

(f )

f

(d)

FIGURE 7.1 (a) A pure sinusoidal wave in the time domain becomes a (b) Dirac delta

function in the frequency domain. (c) A periodic function (blue) in the time domain made

from two basic sine functions (green and gray), becomes two (d) Dirac delta functions in

the frequency domain.

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Frequency Domain

225

In analyzing a circuit, it is often preferred to work in the frequency domain

respect to the time one, and we will see that the frequency f is highly correlated

to the variable s. This is why we need to have equations with the circuit elements

directly in the domain of s.

7.2.1 Capacitive Reactance

In the case of a capacitor the relations expressed in s assume the following forms

(the demonstrations are beyond the scope of our discussion):

it C

dv t

dt

()

()

=

transform

⇒

0Is sCVs Cv

()

() ()

=−

0

1

0

'

vt v

C

itdt

t

∫

() () ()

=+

transform

⇒

01

Vs

v

ssC

Is

()

()

()

=+

or, if the initial conditions can be considered null, more simply they are

it C

dv t

dt

()

()

=

transform

⇒

Is sC

Vs

() ()

=

1

0

'

vt

C

itdt

t

∫

() ()

=

transform

⇒

1

Vs

sC

Is

()

()

=

The impedance of the capacitor is therefore expressed as

1

Z

sC

C

=

In the case of a steady state (i.e., when the transient described in Section 4.6.4

in Chapter 4 has run out) the real part of the variable s is null (σ = 0), so s = jω and

the term

=ωXC

C

1

is called capacitive reactance, so we can write

==−=−

ω

11

V

j

XI jX Ij

C

I

CC

(with

=−

jj

1

) This equation is reminiscent of Ohm’s law, whose characteristics

it must maintain: dimensionally a voltage

V

[]

is equal to a current

A

[]

multiplied

by a resistance

Ω

[]

, consequently

X

C

has to be measured in Ohm too. But Ohm

is tied to the concept of electrical resistance and the capacitor, differing from the

resistor, does not present power consumption, so the term

X

C

is called reactance

(i.e., “reactive resistance”).

If the voltage applied at the terminals of the capacitor is sinusoidal,

vt Vt

M

sin

() ()

=ω

, the current becomes 90° phase shifted,

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