Supplementary Notes on Circuit Analysis

David A. Torrey
Clarkson University, Capital Region Campus
80 Nott Terrace
Schenectady, NY 12308
dtorrey@clarkson.edu

Circuit Components

Capacitor

Inductor

Transformer

General Information

Circuit Analysis

Additional Info

Overview

The following notes are offered to help you refresh your memory on some fundamental circuit analysis techniques. In particular, a review of reactive components and transformers is offered, followed by a brief review of time domain circuit analysis. The notes conclude with a discussion of some basic power concepts.

Circuit Analysis

This semester we are going to be analyzing circuits in the time domain. There are three reasons for this. First, it is a nuisance and time-consuming to be working with Laplace transforms. Second, Laplace transforms provide limited insight into how a circuit behaves. Third, we are not going to deal with anything more complex than second-order circuits, so the analysis is rather straight-forward.

Because one element of solving a differential equation involves determining a particular solution, we are also going to briefly discuss phasor analysis.

Time Domain Transient Analysis

This section provides a brief review of circuit analysis in the time domain. Specifically, we are going to walk through the process of analyzing the snubber circuit shown in Figure 5 as the circuit moves through two of its topologies.

The snubber circuit is comprised of the resistor RR, the diode D1, the capacitor C, and the inductor L. The function of these elements is to shape the current through the switch SS at turn-on and to shape the voltage across the switch at turn-off.

LL and CC Ringing

Define t=0 to be the time when diode D2 turns on, thereby setting LL and CC free to ring. The active circuit topology is shown in Figure 6.

Applying Kirchhoff's voltage law gives:


	L (\frac{di_L}{dt}) + (\frac{1}{C}) (\int i_L dt) = V_{dc}.	(27)
	(\frac{d^2 i_L}{dt^2} + \frac{1}{LC} i_L = 0.)	(28)

The general response for this form of second-order constant-coefficient differential equation is:


	(i_L = A \cos(\omega_0 t) + B \sin(\omega_0 t); \omega_0 = \frac{1}{\sqrt{LC}}.)	(29)

Steady State (Phasor) Analysis

Consider the simple rectifier circuit shown in Figure 8.

During the positive half cycle of the source voltage, the load current is dictated by the differential equation:


	L (\frac{di_d}{dt}) + Ri_d = V_s \sin(\omega t).	(46)

The homogeneous (natural) solution to Eq. 46 is (i_{d,h}=Ae^{-Rt/L}); the constant A is chosen to satisfy the initial condition on (i_d). The average power is determined in the same way that the average voltage or current is determined:


	(\langle p_1(t) \rangle = P_1 = \frac{1}{T}\int p_1(t)dt. )	(55)

Summary

These notes have presented a brief discussion of some of the circuit components we will be using this semester. A brief review of time domain circuit analysis techniques was presented, followed by a discussion of some basic power concepts. These notes were not intended to be a substitute for a circuit analysis course; they were intended to help orient you in the way a power electronics engineer approaches the analysis of a circuit.