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PDHeng ineer . com Course E-6002

First Order RLC Circuits: Time Domain

Analysis

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First Order RLC Circuits: Time Domain Analysis

First Order RLC Circuits: Time Domain Analysisrev. 1.3 6/11/2007

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Table of Contents

1 Preface ........................................................................................................................... 1-11.1 Purpose and Intended Audience................................................................................................1-11.2 Navigation ..................................................................................................................................1-1

2 Introduction ................................................................................................................... 2-13 Using RLC Devices and Circuits.................................................................................. 3-1

3.1 Device Electrical Characteristics................................................................................................3-13.2 Practical Use of First Order RLC Circuits...................................................................................3-2

3.2.1 Cost of Discreet Devices................................................................................................................... 3-24 Getting the Complete Response .................................................................................. 4-1

4.1 Obtain the Reduced System Equation .......................................................................................4-14.2 Solve for The Transient Term ....................................................................................................4-24.3 Solve for The Steady State Term...............................................................................................4-3

4.3.1 Unit Step Input .................................................................................................................................. 4-34.3.2 Complex Sinusoidal Input ................................................................................................................. 4-34.4 The Complete Response ...........................................................................................................4-64.4.1 Complete Response to Unit Step Function Input ............................................................................. 4-64.4.2 Complete Response to Complex Input ............................................................................................. 4-74.4.3 Complete Response to sin(t) Input ................................................................................................ 4-84.4.4 Complete Response to cos(t) Input ............................................................................................... 4-9

5 First Order RC Circuits ................................................................................................. 5-15.1 RC Low Pass Circuit ..................................................................................................................5-2

5.1.1 The System Equation........................................................................................................................ 5-25.1.2 Complete Responses........................................................................................................................ 5-35.1.3 Graphs of the Complete Responses................................................................................................. 5-4

5.2 RC High-Pass Circuit .................................................................................................................5-75.2.1 The System Equation........................................................................................................................ 5-75.2.2 Complete Responses........................................................................................................................ 5-85.2.3 Graphs of the Complete Responses................................................................................................. 5-9

6 First Order RL Circuits.................................................................................................. 6-16.1 RL Low-Pass Circuit...................................................................................................................6-2

6.1.1 The System Equation........................................................................................................................ 6-26.1.2 Complete Responses........................................................................................................................ 6-36.1.3 Graphs of the Complete Responses................................................................................................. 6-4

6.2 RL High-Pass Circuit..................................................................................................................6-76.2.1 The System Equation........................................................................................................................ 6-76.2.2 Complete Responses........................................................................................................................ 6-86.2.3 Graphs of the Complete Responses................................................................................................. 6-9

Appendix A Trigonometric Properties and Identities........................................................A-1Appendix B Complex Variables and Eulers Identities......................................................B-1Appendix B-2 Solving for Complex Input ...........................................................................................B-2

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Table List

Table 3-1 Ideal R,L,C Voltage and Current Characteristics............................................... 3-1Table 3-2 Some Uses of RLC Circuits................................................................................. 3-2Table 5-1 RC Low-pass Circuit Complete Responses....................................................... 5-3Table 5-2 RC High-pass Circuit Complete Responses...................................................... 5-8Table 6-1 RL Low-pass Circuit Complete Responses ....................................................... 6-3Table 6-2 RL High-pass Circuit Complete Responses ...................................................... 6-8

Figures List

Figure 4-1 RC High-pass circuit example ........................................................................... 4-1Figure 4-2 Unit Step Function.............................................................................................. 4-3Figure 4-3 Typical Voltage Divider ...................................................................................... 4-4Figure 5-1 RC Low Pass Circuit ......................................................................................... 5-2Figure 5-2 RC Low Pass Unit Step Response.................................................................... 5-4Figure 5-3 RC Low Pass Response to sin(t).................................................................... 5-5Figure 5-4 RC Low Pass Response to cos(t)................................................................... 5-6Figure 5-5 RC High Pass Circuit.......................................................................................... 5-7Figure 5-6 RC High Pass Unit Step Response ................................................................... 5-9Figure 5-7 RC High Pass Response to sin(t)................................................................. 5-10Figure 5-8 RC High Pass Response to cos(t)................................................................ 5-11Figure 6-1 RL Low Pass Circuit........................................................................................... 6-2Figure 6-2 RL Low Pass Unit Step Response .................................................................... 6-4Figure 6-3 RL Low Pass Response to sin(

t) .................................................................... 6-5

Figure 6-4 RL Low Pass Response to cos(t) ................................................................... 6-6Figure 6-5 RL High Pass Circuit .......................................................................................... 6-7Figure 6-6 RL High Pass Unit Step Response.................................................................... 6-9Figure 6-7 RL High Pass Response to sin(t) ................................................................. 6-10Figure 6-8 RL High Pass Response to cos(t) ................................................................ 6-11

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1-1

1 Preface

TOC

1 Preface

1.1 Purpose and Intended Audience

This lesson is a refresher in basic network analysis of passive RLC circuits using discreet passive ( R) resistor,(L) inductor and (C) capacitor components. The complete response for each of the 4 basic circuits has transientand steady state responses.

This course covers RC high and low-pass, and RL high and low-pass circuits (first order circuits).Well obtain the complete response using time domain analysis. Time domain network analysis uses linear,integral and differential equations for voltage and current characteristics of the devices and the overall circuit.

The intended audience is practicing electrical and electronics engineers or second year college level electricalengineering students. The focus is not so much on the mathematics but on electrical engineering concepts,having the necessary math skills as a foundation for rapid and accurate circuit analysis.

1.2 Navigation

Suggestions for navigating this course:

The TOC link at the top of each page brings you up to the Table of Contents of this document.

Navigable links are in bold blue-colored font

Use the back button if from Microsoft Word or Adobe Reader

Use the Document Map in Word

Use the Bookmark Pane in Adobe


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2 Introduction

TOC

2 Introduction

Passive discrete electronic components used in signal conditioning / generation are the resistor (R), inductor (L)and capacitor (C). Passive discrete components are stand-alone packaged devices whose physicalcharacteristics and parameters are independent of any sources of voltage or current applied to them.

In this lesson we will not consider device tolerances, environmental parameters (temperature, power or workingvoltage ratings) or (EMI) electromagnetic interference.

The nominal frequency band for our analysis is DC to 10MHz; beyond 10MHz we have to take high frequencyeffects into account. At higher frequencies we have to consider: stray capacitance and inductance, EMI, andphysical construction of the circuit.

Our objective is to come up with the complete response (with transient and steady state terms) for each of 4 firstorder RLC circuits: RC high and low-pass circuits and RL high and low-pass circuits.

2-1


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3 Using RLC Devices and Circuits 3.1 Device Electrical Characteristics

TOC

3 Using RLC Devices and Circuits

3.1 Device Electrical Characteristics

The resistor is a linear device and is characterized by a straight-line equation. It dissipates power as heat,its value in ohms can vary as to the tolerance rating (ohms % of rated value). The resistor cannot store energy.

An inductor or capacitor is an energy storage device; a capacitors current or an inductors voltage does notchange instantaneously. Initial conditions can apply to both of these devices. The ideal capacitor has zeroconductance or infinite resistance and the ideal inductor has zero resistance or infinite conductance. Ideally,neither device dissipates heat (power). The total power consumed or delivered in an RLC is presented as acomplex variable (phasor) with a real (dissipated power by resistors) and imaginary (reactive power ) component.

Although we will not include the following in this lesson, it should be mentioned that, a capacitors conductance (oran inductors resistance) only approaches zero and the rated component value (Farads for capacitors or Henries

for inductors) may also vary. These variants in addition to EMI and environmental effects would require you toalter your design or analysis somewhat, depending on how critical they are to your design or model.

The Ideal voltage vs. current characteristics for the resistor, inductor and capacitor are shown below in Table 1.

Device Voltage and Current Characteristics Comments

Resistor RiV RR = R

Vi RR = straight line equation y = mx

Inductor idtdLVL = dtV

L

1i LL = integro-differential equation

Capacitor CC Vdt

di = dtI

C

1V CC = integro-differential equation

Table 3-1 Ideal R,L,C Voltage and Current Characteristics

3-1


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3 Using RLC Devices and Circuits

3-2

3.2 Practical Use of First Order RLC Circuits

TOC

3.2 Practical Use of First Order RLC Circuits

Some Uses of First Order RLC Circuits are:noise reduction and filtering reduce noise from signals

coupling circuits between subsystems block DC voltages or meet next stage requirements

compensating networks or circuitsphase, delay or distortion compensation or meet nextstage input requirements

high-pass and low-pass filtering

signal generationpre-conditioning for oscillators, pulse and waveshapegenerators

signal shaping pre-conditioning for pulse and waveshape generators

Table 3-2 Some Uses of RLC Circuits

3.2.1 Cost of Discreet Devices

The cost for each of the R, L, C components varies with power rating, tolerance ratings, and market demand andsupply. The inductor can be the most expensive of the three because of the physical construction; ferrite core,copper windings etc. In most instances a capacitor will do the job unless an inductors characteristics arespecifically required.


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4 Getting the Complete Response 4.2 Solve for The Transient Term

TOC

4.2 Solve for The Transient Term

To obtain the transient term, if necessary, differentiate both sides of Eqn 4-1 with respect to t to get the system

equation in differential equation form; in this case Eqn 4-2

[ ]

+= ooi VdtVRC

1

dt

dV

dt

d

+= ooi VVRC

1V Eqn 4 -2

set Eqn 4-2 = 0 and find a solution to the differential equation;

oo

oo

VRC

1V

0VVRC

1

=

=+

So a solution isEqn 4-3

RC

t

o Ae)t(V h

= Eqn 4 -3

Eqn 4-3 is the solution for the transient term. The transient term is the same whether the input is a step function ora sinusoidal function. The constant A is found after the steady state term is found.

4-2


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4 Getting the Complete Response 4.3 Solve for The Steady State Term

TOC

4.3 Solve for The Steady State Term

The steady state term can be found by inspection using our engineering background. Long after the transient termdecays to a negligible value the steady state term remains the only term.

Designate the steady state term )(Vas)t(Vt

o

.

The steady state term will depend on what we have for an input The following subsections will go over

2 types of input; the unit step input and the various sinusoidal inputs. The unit step function input is not requiredfor sinusoidal inputs or for any other reason other than giving us an idea of the envelope of the completeresponse for other inputs. The unit step input is not a periodic function; it is the equivalent of throwing the switchto a DC source at time t = 0 and leaving it onindefinitely.

)(V )t(Vi

4.3.1 Unit Step Input

u(t)V(t)V

unitais)t(VIf

mi

i

=

0tV

0t0(t)V;functionstep

m

i

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