Physics
RL Circuit
An RL circuit is a type of electrical circuit that contains a resistor (R) and an inductor (L). When a voltage is applied to the circuit, the inductor resists changes in current flow, causing the current to rise gradually. This behavior is characterized by a time constant, which determines how quickly the current reaches its maximum value.
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8 Key excerpts on "RL Circuit"
eBook - PDF- Stephen Herman(Author)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
536 8 AC Circuits Containing Resistance-Inductance-Capacitance © I. Pilon/Shutterstock.com SECTION Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 537 Why You Need to Know T his unit combines the elements of resistance, inductance, and capac-itance in the same circuit. RLC series circuits are generally employed in a particular application. An RLC series resonant circuit, for example, can produce a very large increase in the voltage across a particular component. This unit ● discusses the values of voltage, current, power, and impedance in series circuits containing resistance, inductance, and capacitance. ● determines the voltage and current phase relationship in this circuit when applied to the resistor, capacitor, and inductor. ● illustrates how to calculate the values needed to produce a resonant circuit at a particular frequency. ● presents another important concept when dealing with resonance, called bandwidth. Key Terms Bandwidth Lagging power factor Leading power factor Resonance Outline 23–1 RLC Series Circuits 23–2 Series Resonant Circuits UNIT 23 Resistive-Inductive-Capacitive Series Circuits Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
- J. A. Brandão Faria(Author)
- 2008(Publication Date)
- Wiley(Publisher)
Despite the presence of inductors and capacitors, the generator feeding the circuit interprets the latter as a pure resistor; the phase shift between the voltage and current at the generator terminals is zero, = 0. Resonance conditions are critically dependent on the working frequency. Just to give you a simple example, if the impedance angle of the RLC circuit in (7.28) is analyzed, and = 0 is enforced, you will at once conclude that the resonance condition for such a particular circuit is L = 1 /C , which implies that Z = Z min = R and I = I max = U/R . In addition, you may note that the capacitor and inductor voltages of the RLC resonant circuit are in phase opposition but have identical amplitudes: U L = − U C . This common amplitude, depending on the circuit parameters, may become much higher than the resistor voltage amplitude; their ratio Q is ordinarily termed the quality factor of the circuit at resonance Q = U L U R = U C U R = res L R = 1 res CR = √ L/C R The fact that resonant circuits behave as pure resistors, = 0, signifies that the inductor and capacitor effects must cancel each other in some way. According to the complex Poynting 264 Electromagnetic Foundations of Electrical Engineering theorem in (7.26), the reactive power is zero, and the time-averaged values of the magnetic and electric energies stored compensate for each other, W m a v = W e a v . The concept of quality factor for an RLC resonant circuit was introduced above as a means to quantify circuit overvoltages. However, a more general definition of the quality factor, which applies to any resonant circuit, is usually given in the form Q = res W m a v + W e a v p J a v = res W em a v p J a v where W em denotes the total electromagnetic energy stored. 7.2.8 Application Example ( RL C Circuit) The RL series circuit with a capacitor C in parallel finds applications in several areas.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Explain. 19. Describe how the currents through R 1 and R 2 shown below vary with time after switch S is closed. 20. Discuss possible practical applications of RL Circuits. 14.5 Oscillations in an LC Circuit 21. Do Kirchhoff’s rules apply to circuits that contain inductors and capacitors? 22. Can a circuit element have both capacitance and inductance? 23. In an LC circuit, what determines the frequency and the amplitude of the energy oscillations in either the inductor or capacitor? 14.6 RLC Series Circuits 24. When a wire is connected between the two ends of a solenoid, the resulting circuit can oscillate like an RLC circuit. Describe what causes the capacitance in this circuit. 25. Describe what effect the resistance of the connecting wires has on an oscillating LC circuit. 26. Suppose you wanted to design an LC circuit with a frequency of 0.01 Hz. What problems might you encounter? 27. A radio receiver uses an RLC circuit to pick out particular frequencies to listen to in your house or car without hearing other unwanted frequencies. How would someone design such a circuit? PROBLEMS 14.1 Mutual Inductance 28. When the current in one coil changes at a rate of 5.6 A/ s, an emf of 6.3 × 10 −3 V is induced in a second, nearby coil. What is the mutual inductance of the two coils? 29. An emf of 9.7 × 10 −3 V is induced in a coil while the current in a nearby coil is decreasing at a rate of 2.7 A/ s. What is the mutual inductance of the two coils? 30. Two coils close to each other have a mutual inductance of 32 mH. If the current in one coil decays according to 654 Chapter 14 | Inductance This OpenStax book is available for free at http://cnx.org/content/col12074/1.3 I = I 0 e −αt , where I 0 = 5.0 A and α = 2.0 × 10 3 s −1 , what is the emf induced in the second coil immediately after the current starts to decay? At t = 1.0 × 10 −3 s? 31. A coil of 40 turns is wrapped around a long solenoid of cross-sectional area 7.5 × 10 −3 m 2 .
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
664 CHAPTER 23 Alternating Current Circuits 23.2 Inductors and Inductive Reactance As Section 22.8 discusses, an inductor is usually a coil of wire, and the basis of its operation is Faraday’s law of electromagnetic induction. According to Faraday’s law, an inductor develops a voltage that opposes a change in the current. This voltage V is given by V = −L(ΔI/Δt) (see Equa- tion 22.9*), where ΔI/Δt is the rate at which the current changes and L is the inductance of the inductor. In an ac circuit the current is always changing, and Faraday’s law can be used to show that the rms voltage across an inductor is V rms = I rms X L (23.3) Equation 23.3 is analogous to V rms = I rms R, with the term X L appearing in place of the resistance R and being called the inductive reactance. The inductive reactance is measured in ohms and determines how much rms current exists in an inductor for a given rms voltage across the induc- tor. It is found experimentally that the inductive reactance X L is directly proportional to the fre- quency f and the inductance L, as indicated by the following equation: X L = 2πfL (23.4) This relation indicates that the larger the inductance, the larger is the inductive reactance. Note that the inductive reactance is directly proportional to the frequency (X L ∝ f ), whereas the capaci- tive reactance is inversely proportional to the frequency (X C ∝ 1/f ). Figure 23.6 shows a graph of the inductive reactance versus frequency for a fixed value of the inductance, according to Equation 23.4. As the frequency becomes very large, X L also becomes very large. In such a situation, an inductor provides a large opposition to the alternating current. In the limit of zero frequency (i.e., direct current), X L becomes zero, indicating that an inductor does not oppose direct current at all. The next example demonstrates the effect of induc- tive reactance on the current in an ac circuit.
eBook - PDF- Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Because of duality, we do not need to study the RC and RL Circuits as independent problems. Everything we learn by solving the RC circuit, for example, can be applied to the RL Circuit as well. We refer to the RC and RL Circuits as first-order circuits because they are described by a first-order differential equation. The first-order differential equations 259 RC AND RL C IRCUITS in Eqs. (7–3) and (7–6) describe general RC and RL Circuits shown in Figure 7–2. Any circuit containing a single uncombinable capacitor or inductor and linear resistors and sources is a first-order circuit. Z E R O - I N P U T R E S P O N S E O F F I R S T - O R D E R C I R C U I T S The response of a first-order circuit is found by solving the circuit differential equa- tion. For the RC circuit, the response υ t must satisfy the differential equation in Eq. (7–3) and the initial condition υ 0 . By examining Eq. (7–3), we see that the response depends on three factors: 1. The inputs driving the circuit υ T t 2. The values of the circuit parameters R T and C 3. The value of υ t at t = 0 (i.e., the initial condition) The first two factors apply to any linear circuit, including resistance circuits. The third factor relates to the initial energy stored in the circuit. The initial energy can cause the circuit to have a nonzero response even when the input υ T t = 0 for t ≥ 0. The existence of a response with no input is something new in our study of linear circuits. To explore this discovery, we find the zero-input response. Setting all independent sources in Figure 7–2 to zero makes υ T t = 0 in Eq. (7–3): R T C dυ t dt + υ t = 0 (7 –7) Mathematically, Eq. (7–7) is a homogeneous equation because the right side is zero. The classical approach to solving a linear homogeneous differential equation is to try a solution in the form of an exponential υ t = Ke st (7 –8) where K and s are constants to be determined.
eBook - PDFFirst and Second Order Circuits and Equations
Technical Background and Insights
- Robert O'Rourke(Author)
- 2024(Publication Date)
- Wiley-IEEE Press(Publisher)
Inductors and capacitors are energy storage elements. Connected together, they can exchange energy back and forth between one another. Sinusoidal current is alternating current (AC). At resonance, the back-and-forth energy exchange between capacitor and inductor matches the back-and-forth alternating amplitude of the sinusoidal source. This match – this resonance – makes the series LC combination transparent to the sinusoidal source. This resonance is connected in series along one current path. The exchange of energy between the inductor and capacitor is in the same path as the current. Energy in the capacitor comes from stored charge which creates a voltage. Energy in the inductor comes from magnetic flux and appears as current. At resonance, the rise and fall of voltage across the capacitor, due to the exchange of energy between the inductor and the capacitor, matches the rise and fall of the sinusoidal source voltage. 13.1.3 Series RLC Impedance Magnitude and Resonant Frequency Example This section explores the magnitude of the input impedance in Equation 13.17 and calculates the resonant fre- quency for an RLC circuit in Figure 13.8. |Z RLC-S | = √ R 2 + ( L − 1 C ) 2 (13.17) Equation 13.17 shows the magnitude of the impedance of a series RLC circuit, as shown in Figures 13.8 and 13.9. R = 3 Ohms L = 10 nH C = 10 nF Z RLC-S Figure 13.8 Series RLC impedance schematic. Impedance is voltage divided by current. To analyze the input impedance Z RLC-S , in the circuit schematic in Figure 13.8, look at the voltage, across the left-hand-side terminal, divided by the current going into the terminal. 224 13 Second-Order RLC Frequency Response V 1 V 2 V 3 U = 1 V Pr1 + – L 2 L = 10 nH C 2 C = 10 nF R 3 R = 3 Ohm + – Figure 13.9 Series RLC simulation schematic with sinusoidal voltage source. Figure 13.9 shows a sinusoidal steady-state circuit sim- ulation schematic for this series RLC input impedance analysis.
eBook - PDFNewnes Circuit Calculations Pocket Book
with Computer Programs
- Thomas J. Davies(Author)
- 2016(Publication Date)
- Newnes(Publisher)
11 A.c. circuits 1 1 . 1 Introduction Chapter 6 looked at the nature of the a.c. waveform. This chapter looks at the behaviour of the resistor, the inductor and the capacitor, as individual components, and then in both series and parallel combinations when driven by a.c. voltages. The aim is to show that an a.c. circuit can involve the calculation of up to 20 parameters. Generally, if four parameters are known, the others can be deduced. The worked examples are chosen to show how the formulas are used and transposed. 11 .2 Resistor, R The treatment of a purely resistive circuit shown in Figure 11.1 is identical to the treatment given in Chapter 3 under d.c. conditions. The formulas are the same for the voltage, current, resistance and power. The units used are the same. The voltage and current are in phase, as shown in Figure 11.2. If a voltage is quoted as 240, say, it is always assumed to be 240 r.m.s. It is quite in order, of course, to carry out the calculations using peak or peak to peak values, but care must be taken not to mix the units. R Figure 11.1 Figure 11.2 11 .3 Inductor, L Example I In the circuit of Figure 11.3, if V L = 100 V, / = 50 Hz, and L = 200 mH, calculate the current I L .
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 23 Alternating current circuits LEARNING OBJECTIVES After reading this module, you should be able to: 23.1 calculate capacitive reactance 23.2 calculate inductive reactance 23.3 calculate impedance in an RCL circuit 23.4 calculate the resonance frequency of an RCL circuit 23.5 describe how semiconductor devices operate. INTRODUCTION This performance by Queen and Adam Lambert was part of Fire Fight Australia, the 2020 concert for national bushfire relief. Without the aid of alternating current (ac) circuits it would not be possible to stage such entertainment spectaculars. Ac circuits lie at the heart of all the audio systems used in the performance. 23.1 Capacitors and capacitive reactance LEARNING OBJECTIVE 23.1 Calculate capacitive reactance. FIGURE 23.1 The resistance in a purely resistive circuit has the same value at all frequencies. The maximum emf of the generator is V 0 . R Frequency, f (Hz) Resistance, R (ohms) V 0 sin 2 π ft Our experience with capacitors so far has been in dc circuits. As we have seen in section 20.13, charge flows in a dc circuit only for the brief period after the battery volt- age is applied across the capacitor. In other words, charge flows only while the capacitor is charging up. After the capacitor becomes fully charged, no more charge leaves the bat- tery. However, suppose that the battery con- nections to the fully charged capacitor were suddenly reversed. Then charge would flow again, but in the reverse direction, until the battery recharges the capacitor according to the new connections. In an ac circuit what happens is similar. The polarity of the voltage applied to the capacitor continually switches back and forth, and, in response, charges flow first one way around the circuit and then the other way. This flow of charge, surging back and forth, constitutes an alternating current. Thus, charge flows continuously in an ac circuit containing a capacitor.
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