Ahmet Bindal - Electronics for Embedded Systems

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Springer International Publishing Switzerland 2017

Ahmet Bindal Electronics for Embedded Systems 10.1007/978-3-319-39439-8_1

1. Fundamentals of Passive Circuit Analysis

Ahmet Bindal 1

(1)

Computer Engineering Department, San Jose State University, San Jose, California, USA

Ahmet Bindal

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1.1 Laplace Transform

Laplace transform transforms a function, f(t), in time-domain to a function, F(s), in frequency domain . We will use Laplace transform to determine the impedance of a passive element in this section.

It is defined as:

(1.1)

1.2 Definitions of Passive Elements

Current-voltage relationship across a resistor , R, is defined as shown in Fig..

Fig. 1.1

Current-voltage relationship across a resistor, R

This relationship is called Ohms law and mathematically expressed as follows:

(1.2)

Taking Laplace transform of both sides of Eq. yields:

Thus, the impedance of the resistor becomes:

(1.3)

Current-voltage relationship across a capacitor , C, is defined as shown in Fig..

Fig. 1.2

Current-voltage relationship across a capacitor , C

This relationship originates from the Coulombs law and mathematically expressed as follows:

(1.4)

Taking Laplace transform of both sides of Eq. yields:

where VC(0) is assumed 0 V.

Thus, the impedance of the capacitor becomes:

(1.5)

Current-voltage relationship across an inductor , L, is defined as shown in Fig..

Fig. 1.3

Current-voltage relationship across an inductor , L

This relationship is mathematically expressed as follows:

(1.6)

Taking Laplace transform of both sides of Eq. yields:

where IL(0) is assumed 0 A.

Thus, the impedance of the capacitor becomes:

(1.7)

1.3 Time-Domain Analysis of First Order Passive Circuits

1.3.1 RC Circuits

In this section, we will examine the circuit behavior of simple RC circuits. A circuit composed of a series combination of a resistor and a capacitor is shown in Fig.. The switch is assumed to close at t = 0.

Fig. 1.4

A simple RC circuit where the switch closes at t = 0

Applying Kirchoffs voltage law to the circuit yields:

(1.8)

But,

(1.9)

In order to use Eq..

(1.10)

Substituting Eq. yields:

(1.11)

(1.12)

In Eq. , the right side is equal to zero because V is a constant voltage and its derivative becomes zero.

We know I(t) is composed of a general and a particular solution for this first-order constant coefficient differential equation . Thus,

But, IPART(t) = 0 since the right hand side of the differential equation in Eq. is zero .

IGEN(t), on the other hand, is expressed as follows:

(1.13)

Thus,

(1.14)

To find s, we need to take the Laplace transform of both sides of the differential equation in Eq..

Thus,

(1.15)

Since I(s) 0, then Eq. reveals (characteristic equation).

Therefore, , and I(t) becomes:

(1.16)

To find A, we need to use the initial voltage across the capacitor . In this case, we assume VOUT(0) = 0.

But, at t = 0, we have:

(1.17)

Thus,

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