First Order Circuits

9 Key excerpts on "First Order Circuits"

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  eBook - PDF

  Introduction to Electric Circuits

  • Richard C. Dorf, James A. Svoboda(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  The order of the differential equation is usually equal to the number of capacitors plus the number of inductors in the circuit. Circuits that contain only one inductor and no capacitors or only one capacitor and no inductors can be represented by a first-order differential equation. These circuits are called first-order circuits. 308 Th evenin and Norton equivalent circuits simplify the analysis of first-order circuits by showing that all first-order circuits are equivalent to one of two simple first-order circuits. Figure 8.2-1 shows how this is accomplished. In Figure 8.2-1a, a first-order circuit is partitioned into two parts. One part is the single capacitor or inductor that we expect to find in a first-order circuit. The other part is the rest of the circuit—everything except that capacitor or inductor. The next step, shown in Figure 8.2-1b, depends on whether the energy storage element is a capacitor or an inductor. If it is a capacitor, then the rest of the circuit is replaced by its Th evenin equivalent circuit. The result is a simple first-order circuit—a series circuit consisting of a voltage source, a resistor, and a capacitor. On the other hand, if the energy storage element is an inductor, then the rest of the circuit is replaced by its Norton equivalent circuit. The result is another simple first-order circuit—a parallel circuit consisting of a current source, a resistor, and an inductor. Indeed, all first-order circuits are equivalent to one of these two simple first- order circuits. Consider the first-order circuit shown in Figure 8.2-2a. The input to this circuit is the voltage v s (t). The output, or response, of this circuit is the voltage across the capacitor. This circuit is at steady state before the switch is closed at time t ¼ 0. Closing the switch disturbs this circuit. Eventually, the disturbance dies out and the circuit is again at steady state.

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  First and Second Order Circuits and Equations

  Technical Background and Insights

  • Robert O'Rourke(Author)
  • 2024(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  57 Part 2 First-Order Circuits 59 5 First-Order RC and RL Circuits Introduction This chapter introduces first-order circuits in both time and frequency domain. The time domain response from initial conditions is called natural or zero-input response (ZIR). The time domain response from a step func- tion input is a form of zero-state response (ZSR). The frequency domain response analyzed here is the sinusoidal steady-state response. 5.1 What are First-Order Circuits? 5.1.1 A Capacitor or an Inductor Makes a First-Order Circuit First-order circuits have one reactive circuit element, such as a capacitor or inductor. Capacitors and inductors can store energy. First-order circuits can make frequency selective filters in the frequency domain. First-order circuits are described by first-order differential equations. In the time domain, first-order circuit responses are often rising or decaying exponential functions. (Other responses, like integrators or delta functions, are possible for some topologies and inputs.) 5.1.2 First-Order Circuits Can Filter in the Frequency Domain V S V out Figure 5.1a Series RC circuit driven by AC voltage. I S V Figure 5.1b Parallel RL circuit driven by AC current. Frequency 0 V out Figure 5.2a RC low-pass filter response. Frequency 0 V Figure 5.2b RL high-pass filter response. Figure 5.1a shows a shunt capacitor low-pass fil- ter schematic. The one-pole low-pass frequency response is shown in Figure 5.2a. Figure 5.1b shows a shunt inductor high-pass fil- ter schematic. The one-pole high-pass frequency response is shown in Figure 5.2b. First and Second Order Circuits and Equations: Technical Background and Insights, First Edition. Robert O’Rourke. © 2024 The Institute of Electrical and Electronics Engineers, Inc. Published 2024 by John Wiley & Sons, Inc.

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  Basic Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  We can contrast two very different systems to ob- tain a feel for the parameter. For example, consider the model for a room air-conditioning system and the model for a single-transistor stage of amplification in a computer chip. If we change the setting for the air conditioner from 70° to 60°, the unit will come on and the room will begin to cool. However, the temperature measured by a thermometer in the room will fall very slowly and, thus, the time required to reach the desired temperature is long. However, if we send a trigger signal to a transistor to change state, the action may take only a few nanoseconds. These two systems will have vastly different time constants. Our analysis of first-order circuits begins with the presentation of two techniques for performing a transient analysis: the differential equation approach, in which a differential equation is written and solved for each network, and a step-by-step approach, which takes advantage of the known form of the solution in every case. In the second-order case, both an inductor and a capacitor are present simultaneously, and the network is described by a second-order differential equation. Although the RLC circuits are more complicated than the first-order single storage circuits, we will follow a development similar to that used in the first-order case. 216 CHAPTER 7 First- and Second-Order Transient Circuits Our presentation will deal only with very simple circuits, since the analysis can quickly become complicated for networks that contain more than one loop or one nonreference node. Furthermore, we will demonstrate a much simpler method for handling these circuits when we cover the Laplace transform later in this book. We will analyze several networks in which the parameters have been chosen to illustrate the different types of circuit response. We begin our discussion by recalling that in Chapter 6 we found that capacitors and inductors were capable of storing electric energy.

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  The Analysis and Design of Linear Circuits

  • Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  C H A P T E R 7 FIRST- AND SECOND- ORDER CIRCUITS When a mathematician engaged in investigating physical actions and results has arrived at his own conclusions, may they not be expressed in common language as fully, clearly and definitely as in mathematical formula? If so, would it not be a great boon to such as we to express them so—translating them out of their hieroglyphics that we also might work upon them by experiment. Michael Faraday, 1857, British Physicist Some History Behind This Chapter Michael Faraday (1791–1867) was appointed a Fellow in the Royal Society at age 32 and was a lecturer at the Royal Insti- tution in London for more than 50 years. During this time, he published over 150 papers on chemistry and electricity. The most important of these papers was the series Experimental Researches in Electricity, which included a description of his discovery of magnetic induction. A gifted experimental- ist, Faraday apparently felt that mathematics obscured the physical truths he discovered through experimentation. Why This Chapter Is Important Today OK, this is a tough chapter. It concentrates on the classical methods of finding the transient response of circuits contain- ing resistors, capacitors, and inductors. Mathematically this requires us to solve first- and second-order differential equa- tions. These solutions help us understand applications such as timing circuits and digital gate delays. It is important to understand first- and second-order transients because we will revisit these concepts frequently in subsequent chapters.

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  Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  However, the temperature measured by a thermometer in the room will fall very slowly and, thus, the time required to reach the desired temperature is long. However, if we send a trigger signal to a transistor to change state, the action may take only a few nanoseconds. These two systems will have vastly different time constants. Our analysis of first-order circuits begins with the presentation of two techniques for performing a transient analysis: the differential equation approach, in which a differential equation is written and solved for each network, and a step-by-step approach, which takes advantage of the known form of the solution in every case. In the second-order case, both an inductor and a capacitor are present simultaneously, and the network is described by a second-order differential equation. Although the RLC circuits are more complicated than the first-order single storage circuits, we will follow a development similar to that used in the first-order case. First- and Second-Order Transient Circuits LEARNING OBJECTIVES The Learning goals for this chapter are that students should be able to: ❯ Calculate the initial values for inductor currents and capacitor voltages in transient circuits. ❯ Determine the voltages and currents in first-order transient circuits. ❯ Determine the voltages and currents in second-order transient circuits. ❯ Use PSpice to determine the voltages and currents in first-order and second-order transient circuits. 282 Chapter 6 First- and Second-Order Transient Circuits Our presentation will deal only with very simple circuits, since the analysis can quickly become complicated for networks that contain more than one loop or one nonreference node. Furthermore, we will demonstrate a much simpler method for handling these circuits when we cover the Laplace transform later in this book. We will analyze several networks in which the parameters have been chosen to illustrate the different types of circuit response.

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  Introduction to Energy, Renewable Energy and Electrical Engineering

  Essentials for Engineering Science (STEM) Professionals and Students

  • Ewald F. Fuchs, Heidi A. Fuchs(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  3 DC Circuit Transient Analysis

  In this chapter, we examine the behavior of a DC network as a function of time after a sudden change in the network occurs due to opening or closing of switches. The DC network may contain resistors, inductors, and capacitors. First‐ and second‐order ordinary differential equations will be analytically and numerically solved by well‐known solution techniques and numerical methods such as PSPICE [1 –4 ], MATLAB/Simulink [5 ], and Mathematica [6 ]. Note that there are standard resistor [7 ], capacitor [8 ], and inductor [9 ] values. In electric circuit designs, any (certain) resistor, capacitor, and inductor values might be required based on the desired design performance. These certain (e.g. resistance) values are not necessarily manufactured and sold, simplifying the manufacturing process of resistors: only well‐defined resistance values of resistors are manufactured. For example, a design requires the resistance value of 25 Ω; then a series combination of two resistors with standard resistance values of 10 and 15 Ω must be used because 10 Ω + 15 Ω = 25 Ω. For a 6 Ω resistor, the two resistors with standard values 10 and 15 Ω can be connected in parallel such that .

  3.1 Capacitors

  3.1.1 Energy Stored in a Capacitor

  Ideal lossless (with no losses) capacitors consist of two conducting plates separated by a nonconducting, or dielectric, material. The simplest capacitor consists of two parallel plates as shown in Figure 3.1 . Note the capacitor is the name of a device having the capacitance C. Any conducting or nonconducting material – which is electrically isolated – can be charged with endowed (see Chapter 1 , Appendix 1.B) electrons and has the property of capacitance C

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  Electric Circuits and Signals

  • Nassir H. Sabah(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  13 Switched Circuits Overview After considering the impulse responses of first-order and second-order circuits in the preceding chapter, we turn our attention in this chapter to switched circuits; that is, circuits in which the movement of a switch applies an input to the circuit or changes the circuit configuration. A practically important case is the application of a step input to a circuit. The responses of prototypical first-order and second-order circuits to a step input are analyzed in this chapter and related to the impulse response. We begin by answering a fundamental question as to the need for having a transient component in the response of a circuit. Considering first-order circuits, it turns out that a general procedure can be readily developed that expresses the time variation of any variable in the circuit in terms of the initial value of the variable, its final value, and the effective time constant. When it comes to second-order circuits, the situation is more complicated. The general case of switched circuits that have relatively complicated connections of resistors, inductors, and capacitors, and which may include linear transformers and dependent sources, is best handled using the Laplace transform, as will be discussed in Chapter 15. The discussion is therefore restricted to prototypical second-order circuits that can be reduced to series or parallel connections of resistors, inductors, and capacitors

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  The Fundamentals of Electrical Engineering

  for Mechatronics

  • Felix Hüning(Author)
  • 2014(Publication Date)
  • De Gruyter Oldenbourg

    (Publisher)

  7 Time domain circuit analysis

  In previous chapters some concepts for the analysis of electric circuits like mesh or nodal analysis were introduced. So far only DC circuits have been considered, i.e. circuits with time-independent sources (DC sources) and after initial disturbances (e.g. switching and transients) were settled. Even the few examples were sources were time-dependent transient behavior was not taken into account. If time-dependent parameters like current and voltage are considered, lower case symbols are used to describe these parameters, e.g. u(t), i(t).

  Time domain circuit analysis will be split into two parts:

  1. Transient effects (switching events)
  2. AC circuits

  We will start with the introduction of two new elements in electrical circuits: capacitors and inductors.

  7.1 Capacitor

  A capacitor is an electric element that is able to store electrical energy. In a simplified image an ideal capacitor is built of two plates (electrodes). The electrodes are separated by a non-conducting space (dielectric) and each electrode is connected to one terminal of the capacitor. A current through a capacitor means that positive charges are accumulated inside the capacitor on one electrode and negative charges on the other electrode.

  Fig. 7.1: A simple image of a capacitor; current i(t) causes positive charges to accumulate on one electrode and negative on the other; circuit symbol of a capacitor (center) and adjustable capacitor (right).

  A separation of charges means there is an electric field generated inside the capacitor storing electrical energy. The difference of potentials due to the electric field can be measured as voltage u(t) at the terminals. The ratio of accumulated charges q(t) to created voltage u(t) is called the capacitance of a capacitor:

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  Dorf's Introduction to Electric Circuits

  • Richard C. Dorf, James A. Svoboda(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  334 CHAPTER 8 The Complete Response of RL and RC Circuits 8.5 S t a b i l i t y o f F i r s t - O r d e r C i r c u i t s We have shown that the natural response of a first-order circuit is x n t Ke t t and that the complete response is the sum of the natural and forced responses: x t x n t x f t When t 0, the natural response vanishes as t 0, leaving the forced response. In this case, the circuit is said to be stable. When t 0, the natural response grows without bound as t 0. The forced response becomes negligible, compared to the natural response. The circuit is said to be unstable. When a circuit is stable, the forced response depends on the input to the circuit. That means that the forced response contains information about the input. When the circuit is unstable, the forced response is negligible, and this information is lost. In practice, the natural response of an unstable circuit is not unbounded. This response will grow until something happens to change the circuit. Perhaps that change will be saturation of an op amp or of a dependent source. Perhaps that change will be the destruction of a circuit element. In most applications, the behavior of unstable circuits is undesirable and is to be avoided. How can we design first-order circuits to be stable? Recalling that t R t C or t L R t , we see that R t 0 is required to make a first-order circuit stable. This condition will always be satisfied whenever the part of the circuit connected to the capacitor or inductor consists of only resistors and independent sources. Such circuits are guaranteed to be stable. In contrast, a first-order circuit that contains op amps or dependent sources may be unstable. E X A M P L E 8 . 5 - 1 Response of an Unstable First-Order Circuit The first-order circuit shown in Figure 8.5-1a is at steady state before the switch closes at t 0. This circuit contains a dependent source and so may be unstable.

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