First and second order electric and electronic circuits contain energy storage elements, capacitors and inductors, fundamental to both time and frequency domain circuit response behavior, including exponential decay, overshoot, ringing, and frequency domain resonance.
First and Second Order Circuits and Equations provides an insightful and detailed learning and reference resource for circuit theory and its many perspectives and duals, such as voltage and current, inductance and capacitance, and serial and parallel. Organized and presented to make each information topic immediately accessible, First and Second Order Circuits and Equations offers readers the opportunity to learn circuit theory faster and with greater understanding.
First and Second Order Circuits and Equations readers will also find:
- Root locus charts of second order characteristic equation roots both in terms of damping factor ζ as well as damping constant α.
- Detailed treatment of quality factor Q and its relationship to bandwidth and damping in both frequency and time domains.
- Inductor and capacitor branch relationship step response insights in terms of calculus intuition.
- Derivations of voltage divider and current divider formulae in terms of Kirchhoff’s laws.
First and Second Order Circuits and Equations is an essential tool for electronic industry professionals learning circuits on the job, as well as for electrical engineering, mechanical engineering, and physics students learning circuits and their related differential equations.
Table of Contents
About the Author xvii
Acknowledgments xix
Part 1 Circuit Elements and Resistive Circuits 1
1 Ohm’s Law, Branch Relationships, and Sources 3
1.1 Chapter Summary and Polarity Reference 3
1.2 Branch Relationships and I-V Characteristics 5
1.3 Ohm’s Law, Resistance, and Resistors 8
1.4 Current, Voltage, and Sources Overview 11
1.5 Voltage Sources 12
1.6 Current and Current Sources 14
2 Kirchhoff’s Laws and Resistive Dividers 17
2.1 Kirchhoff’s Laws and Dividers Comparison Summary 17
2.2 Kirchhoff’s Laws Physical Analogies 18
2.3 Source Polarity in KVL - Time and Frequency Domains 19
2.4 Formulae Summary for Resistors in Series and Parallel 26
2.5 Resistors in Series 27
2.6 Voltage Dividers 32
2.7 Parallel Circuit Element Formulae 33
2.8 Current Dividers 36
2.9 Current and Voltage Intuitions 37
3 Opamp Models and Resistive Circuits 39
3.1 Introduction and Ideal Opamp Model Results Overview 39
3.2 Ideal Opamp Resistive Amplifier Circuits 41
4 Reactive Circuit Elements 45
4.1 Capacitor and Inductor Comparison Summary 45
4.2 Capacitors 47
4.3 Inductors 51
Part 2 First-Order Circuits 57
5 First-Order RC and RL Circuits Introduction 59
5.1 What are First-Order Circuits? 59
5.2 Intuitive First-Order Circuit Frequency Domain Examples 61
5.3 First-Order Natural and Step Response Overview 62
6 First-Order Frequency Domain Response 65
6.1 First-Order Frequency Response Overview 65
6.2 Series RC High-pass Filter Frequency Response 71
6.3 Series RL Low-pass Filter Frequency Response 89
6.4 Series RC Low-pass Filter Frequency Response 108
6.6 Parallel RL Low-pass Filter Frequency Response 128
6.7 Parallel RC High-pass Filter Frequency Response 139
7 Discharging and Charging First-Order RC and RL Circuits 149
7.1 Discharging RC and RL Circuits - Natural Response 149
7.2 Charging RC and RL Circuits - Step Response 153
7.3 The Exponential Time Constant τ (Tau) 155
7.4 Pulse Train Time Constants Simulation Example 156
8 Natural Response of RC and RL Circuits 159
8.1 RC and RL Circuits Natural Response Summary 159
8.2 RC and RL Natural Response Derivation 160
8.3 RC Natural Response (ZIR) Time Constants and Initial Current 166
8.4 Natural Response of Series RL with Voltage Source 167
8.5 First-Order RC and RL Natural Response Summary 171
9 First-Order Step Response of RC and RL Circuits 173
9.1 First-Order Step Response Summary Overview 173
9.2 Intuitive Analysis of RC and RL Step Response 177
9.3 Series RC Step Response Solution Using a Particular Solution 181
9.4 Series RL Step Response Solution Using a Particular Solution 184
9.5 Series RL Step Response with Voltage Source 188
9.6 First-Order Step Response Summary 190
10 Complete Response of First-Order RC and RL Circuits 191
10.1 First-Order Complete Response Summary Overview 191
10.2 Series RC Complete Response Examples 192
10.3 RL Complete Response Example and Intuitive Analysis 195
10.4 Complete Response with Switches 199
10.5 Complete Response General Derivation and Formulae 202
11 First-Order Opamp Integrator and Differentiator Circuits 207
11.1 RC Integrator Circuit Step Response 207
11.2 Opamp Integrator Circuit 208
11.3 Opamp Inverting Differentiator Circuit 210
Part 3 Second-Order Circuits 211
12 Second-Order RLC Circuits Overview 213
12.1 What are Second-Order Circuits? 213
12.2 Resonance in the Frequency Domain 215
12.3 Second-Order RLC Transfer Functions and Q 216
12.4 Two Time Domain Responses 217
13 Second-Order RLC Frequency Response 219
13.1 Series and Parallel RLC Impedance 219
13.2 Second-Order RLC Frequency Response 229
13.3 Second-Order RLC Bandwidth and Quality Factor 238
14 Second-Order RLC Circuit Natural Response 251
14.1 Second-Order Natural Response Introduction 251
14.2 Second-order Natural Response in Terms of R, L, and c 254
14.3 Second-order Damping Variables α and ω 0 283
14.4 Second-order Damping Ratio ζ - Zeta 291
15 Second-Order RLC Step and Complete Response 299
15.1 RLC Step Response Intuitive Overview 299
15.2 RLC Step Response Detailed Analyses 301
15.3 Parallel RLC Intuitive Step Response Example 303
15.4 Complete RLC Time Domain Response 305
Part 4 Technical Background Topics 307
16 Complex Numbers, Exponentials, and Phasors 309
16.1 Imaginary and Complex Numbers 309
16.2 Exponentials, Complex Numbers, and Trigonometry 311
16.3 Phasors and Sinusoidal Steady State 314
Index 319
