Easy 3 Steps of Laplace Transform Circuit Element Models

Laplace Transform Circuit Element Models

Laplace transform circuit element models is one of the Laplace transform applications. It converts the time-domain variable of the circuit elements into s-domain for Laplace transform analysis purpose.

Having mastered how to obtain the Laplace transform and its inverse, we are now prepared to employ the Laplace transform to analyze circuits.

Laplace Transform Circuit Element Models

This usually involves three steps.

Steps in Applying the Laplace Transform:
1. Transform the circuit from the time domain to the s-domain.
2. Solve the circuit using nodal analysis, mesh analysis, source transformation, superposition, or any circuit analysis technique with which we are familiar.
3. Take the inverse transform of the solution and thus obtain the solution in the time domain.

Only the first step is new and will be discussed here. As we did in phasor analysis, we transform a circuit in the time domain to the frequency or s-domain by Laplace transforming each term in the circuit.

For a resistor, the voltage-current relationship in the time domain is

Laplace Transform Circuit Element Models
(1)

Taking the Laplace transform, we get

Laplace Transform Circuit Element Models
(2)

For an inductor,

Laplace Transform Circuit Element Models
(3)

Taking the Laplace transform of both sides gives

Laplace Transform Circuit Element Models
(4)

or

Laplace Transform Circuit Element Models
(5)

The s-domain equivalents are shown in Figure.(1), where the initial condition is modelled as a voltage or current source.

Laplace Transform Circuit Element Models
Figure 1. Representation of an inductor: (a) time-domain, (b,c) s-domain equivalents.

For a capacitor,

Laplace Transform Circuit Element Models
(6)

which transforms into the s-domain as

Laplace Transform Circuit Element Models
(7)

or

Laplace Transform Circuit Element Models
(8)

The s-domain equivalents are shown in Figure.(2). With the s-domain equivalents, the Laplace transform can be used readily to solve first- and second-order circuits.

Laplace Transform Circuit Element Models
Figure 2. Representation of a capacitor: (a) time-domain, (b,c) s-domain equivalents.

We should observe from Equations.(3) to (8) that the initial conditions are part of the transformation. This is one advantage of using the Laplace transform in circuit analysis. Another advantage is that a complete response—transient and steady-state—of a network is obtained.

If we assume zero initial conditions for the inductor and the capacitor, the above equations reduce to:

Laplace Transform Circuit Element Models
(9)

The s-domain equivalents are shown in Figure.(3).

Laplace Transform Circuit Element Models
Figure 3. Time-domain and s-domain representations of passive elements under zero initial conditions.

We define the impedance in the s-domain as the ratio of the voltage transform to the current transform under zero initial conditions; that is,

Laplace Transform Circuit Element Models
(10)

Thus, the impedances of the three circuit elements are

Laplace Transform Circuit Element Models
(11)

Table.(1) summarizes these. Assuming zero initial conditions

Laplace Transform Circuit Element Models
Table 1. Impedance of an element in the s-domain.

The admittance in the s-domain is the reciprocal of the impedance, or

Laplace Transform Circuit Element Models
(12)

The use of the Laplace transform in circuit analysis facilitates the use of various signal sources such as impulse, step, ramp, exponential, and sinusoidal.

The models for dependent sources and op amps are easy to develop drawing from the simple fact that if the Laplace transform of f(t) is F(s), then the Laplace transform of af(t) is aF(s)—the linearity property.

The dependent source model is a little easier in that we deal with a single value. The dependent source can have only two controlling values, a constant time either a voltage or a current. Thus,

Laplace Transform Circuit Element Models
(13)
Laplace Transform Circuit Element Models
(14)

The ideal op-amp can be treated just like a resistor. Nothing within an op-amp, either real or ideal, does anything more than multiplying a voltage by a constant.

Thus, we only need to write the equations as we always do use the constraint that the input voltage to the op-amp has to be zero and the input current has to be zero.

Laplace Transform Circuit Element Models Examples

Let’s review the Laplace transform circuit element models examples below:

Laplace Transform Circuit Element Models Example 1

Find in the circuit of Figure.(4), assuming zero initial condition

Laplace Transform Circuit Element Models
Figure 4

Solution:
We first transform the circuit from the time domain to the s-domain.

Laplace Transform Circuit Element ModelsThe resulting s-domain circuit is in Figure.(5).

Laplace Transform Circuit Element Models
Figure 5. Mesh analysis of the frequency-domain equivalent of the same circuit

We now apply mesh analysis. For mesh 1,

Laplace Transform Circuit Element Models
(1.1)

For mesh 2,

Laplace Transform Circuit Element Models
(1.2)

Substituting this into Equation.(1.1),

Laplace Transform Circuit Element Models

Multiplying through by 3s gives

Laplace Transform Circuit Element Models

Taking the inverse transform yields

Laplace Transform Circuit Element Models

Laplace Transform Circuit Element Models Example 2

Find vo(t) in the circuit of Figure.(6). Assume vo(0) = 5 V.

Laplace Transform Circuit Element Models
Figure 6

Solution:
We transform the circuit to the s domain as shown in Figure.(7).

Laplace Transform Circuit Element Models
Figure 7. Nodal analysis of the equivalent of the circuit in Figure.(6)

The initial condition is included in the form of the current source
C vo(0) = 0.1(5) = 0.5 A [See Figure.(2c)].

We apply nodal analysis. At the top node,

Laplace Transform Circuit Element Modelsor

Laplace Transform Circuit Element ModelsMultiplying through by 10,

Laplace Transform Circuit Element Models

or

Laplace Transform Circuit Element Modelswhere

Laplace Transform Circuit Element Models

Thus,

Laplace Transform Circuit Element Models

Taking the inverse Laplace transform, we obtain

Laplace Transform Circuit Element Models

Laplace Transform Circuit Element Models Example 3

In the circuit in Figure.(8a), the switch moves from position a to position b at t = 0. Find i(t) for t > 0.

Laplace Transform Circuit Element Models
Figure 8

Solution:
The initial current through the inductor is i(0) = Io. For t > 0, Figure.(8a) shows the circuit transformed to the s domain. The initial condition is incorporated in the form of a voltage source as Li(0) = LIo.

Using mesh analysis,

Laplace Transform Circuit Element Models
(4.1)

or

Laplace Transform Circuit Element Models
(4.2)

Applying partial fraction expansion on the second term on the right-hand side of Equation.(4.2) yields

Laplace Transform Circuit Element Models
(4.3)

The inverse Laplace transform of this gives

Laplace Transform Circuit Element Models
(4.4)

where τ = R/L. The term in fences is the transient response, while the second term is the steady-state response. In other words, the final value is i() = Vo/R, which we could have predicted by applying the final-value theorem on Equations.(4.2) or (4.3); that is,

Laplace Transform Circuit Element Models
(4.5)

Equation.(4.4) may also be written as

Laplace Transform Circuit Element Models
(4.6)

The first term is the natural response, while the second term is the forced response. If the initial condition Io = 0, Equation.(4.6) becomes

Laplace Transform Circuit Element Models
(4.7)

which is the step response, since it is due to the step input Vo with no initial energy.