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Sinusoidal Steady-State Analysis

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Sinusoidal Steady-State Analysis

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A three‑dimensional printing of the output of an AutoCAD model of a NASA flywheel. © Ansoft Corporation

Learning Objectives

By using the information and exercises in this chapter you will be able to:

    1. Analyze electrical circuits in the frequency domain using nodal analysis.
    1. Analyze electrical circuits in the frequency domain using mesh analysis.
    1. Apply the superposition principle to frequency domain electrical circuits.
    1. Apply source transformation in frequency domain circuits.
    1. Understand how Thevenin and Norton equivalent circuits can be used in the frequency domain.
    1. Analyze electrical circuits with op amps.

10.1 Introduction

In Chapter 9, we learned that the forced or steady ‑state response of cir‑ cuits to sinusoidal inputs can be obtained by using phasors. We also know that Ohm’s and Kirchhoff’s laws are applicable to ac circuits. In this chapter, we want to see ho w nodal analysis, mesh analysis, Thevenin’s theorem, Norton’s theorem, superposition, and source transformations are applied in analyzing ac circuits. Since these techniques were already introduced for dc circuits, our major effort here will be to illustrate with examples.

Analyzing ac circuits usually requires three steps.

Steps to Analyze AC Circuits:

    1. Transform the circuit to the phasor or frequency domain.
    1. Solve the problem using circuit techniques (nodal analysis, mesh analysis, superposition, etc.).
    1. Transform the resulting phasor to the time domain.

Step 1 is not necessary if the problem is specified in the frequency domain. In step 2, the analysis is performed in the same manner as dc circuit analysis except that complex numbers are involved. Having read Chapter 9, we are adept at handling step 3.

Toward the end of the chapter , we learn ho w to apply PSpice in solving ac circuit problems. We finally apply ac circuit analysis to two practical ac circuits: oscillators and ac transistor circuits.

10.2 Nodal Analysis

The basis of nodal analysis is Kirchhof f’s current la w. Since KCL is valid for phasors, as demonstrated in Section 9.6, we can analyze ac cir‑ cuits by nodal analysis. The following examples illustrate this.

Frequency domain analysis of an ac circuit via phasors is much easier than analysis of the circuit in the time domain.

Find ix in the circuit of Fig. 10.1 using nodal analysis. Example 10.1

For Example 10.1.

Solution:

We first convert the circuit to the frequency domain:

20cos⁡4t⇒20/0∘‾,Ω=4 rad/s20 \cos 4t \Rightarrow 20 \underline{/0^{\circ}}, \qquad \Omega = 4 \text{ rad/s}

\n

1 H⇒jΩL=j41 \text{ H} \Rightarrow j \Omega L = j4

\n