RC Circuit

10 Key excerpts on "RC Circuit"

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  eBook - PDF

  Cutnell & Johnson Physics, P-eBK

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  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. To help set the stage for the present discussion, recall from section 20.5 that the rms voltage V rms across the resistor in a purely resistive ac circuit is related to the rms current I rms by V rms = I rms R (equation 20.14). The resistance R has the same value for any frequency of the ac voltage or current. Figure 23.1 emphasises this fact by showing that a graph of resistance versus frequency is a horizontal straight line. For the rms voltage across a capacitor the following expression applies, which is analogous to V rms = I rms R: V rms = I rms X C (23.1) FIGURE 23.2 The capacitive reactance X C is inversely proportional to the frequency f according to X C = 1/(2fC). C Frequency, f (Hz) Capacitive reactance, X C (ohms) V 0 sin 2 ft π The term X C appears in place of the resis- tance R and is called the capacitive reactance. The capacitive reactance, like resistance, is measured in ohms and determines how much rms current exists in a capacitor in response to a given rms voltage across the capacitor. It is found experimentally that the capacitive reac- tance X C is inversely proportional to both the frequency f and the capacitance C, according to the following equation: X C = 1 2fC (23.2) For a fixed value of the capacitance C, figure 23.2 gives a plot of X C versus fre- quency, according to equation 23.2. A com- parison of this drawing with figure 23.1 reveals that a capacitor and a resistor behave differently. As the frequency becomes very large, figure 23.2 shows that X C approaches zero, signifying that a capacitor offers only a negligibly small opposition to the alternating current.

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  eBook - ePub

  Basic AC Circuits

  • Clay Rawlins(Author)
  • 2000(Publication Date)
  • Newnes

    (Publisher)

  Figure 7.2 , is called a series RC Circuit. In this circuit, like in any series circuit, the current flowing through all components is the same value.

  Figure 7.2 Simple Series RC AC Circuit

  (7–1).

  However, the algebraic sum of the voltage drop across the resistor and the voltage drop across the capacitor does not equal the applied voltage as it would in either a purely resistive or purely capacitive circuit. Therefore,

  (7–2).

  This can be demonstrated with a specific example. Suppose the resistance of Figure 7.2 is 40 ohms and the capacitive reactance, XC , is 30 ohms. With an applied voltage of 10 volts, the voltage drops across R and C are calculated to be 8 volts and 6 volts respectively. The algebraic sum of the voltage drops is 14 volts and does not equal the applied voltage of 10 volts.

  This is true because the phase relationship between the voltage across and the current through each component is different.

  Voltage and Current Relationships

  The voltage across a resistor, ER , is in phase with the current through it as shown in Figure 7.3 . For a capacitor, however, recall from Chapter 6 that the current leads the voltage by 90 degrees, as shown in Figure 7.4 . Since the resistor and capacitor are in series with one another, the common factor in both phase relationships is the current.

  Figure 7.3 Phase Relationship of Resistor Voltage and Current

  Figure 7.4 Phase Relationship of Capacitor Voltage and Current

  Phasor diagrams can be drawn for the voltage and current waveforms of Figure 7.3 and Figure 7.4 . The voltage ER , equal to 8 volts, across the resistor is in phase with the current, I, in the series circuit as shown in Figure 7.5 . The same series current, I, leads the voltage EC across the capacitor by 90 degrees as shown in Figure 7.6 . EC

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  eBook - ePub

  Introductory Electrical Engineering With Math Explained in Accessible Language

  • Magno Urbano(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  C is the capacitance, in Farads. RC is time constant.

  25.3 RC Time Constant

  The RC time constant is the time required to charge a capacitor through a resistor, from an initial charge of 0–63.2% of the value of an applied DC voltage or for a discharging capacitor to lose 63.2% of its initial charge.

  This value is derived from the mathematical formulas of charge and discharge for the capacitor. The RC time constant, also called tau (τ) and measured in seconds, is equal to the product of the circuit resistance in Ohms and the circuit capacitance in Farads.

  RC TIME CONSTANT

  • τ is the RC time constant, in seconds.
  • R is the resistance, in Ohms.
  • C is the capacitance, in Farads.

  25.3.1.1 Transient and Steady States

  The period of time in which an RC Circuit exhibits variation of current and voltage, after a voltage pulse is applied to it, is called transient phase. After the transient phase is over, the circuit stabilizes and will not exhibit any other variations. The stable phase is called steady state.

  25.3.1.1.1 How Long Does the Transient Phase Last?

  When a capacitor charges, the amount of charge increases following an exponential curve. For this reason, the ratio of charge decreases with time. Hence, an infinite amount of time is required to reach the maximum charge.

  In practice, engineers cannot wait indefinitely for a capacitor to charge. In electrical engineering, a capacitor is considered charged after a time equal to five time constants.

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  eBook - PDF

  Delmar's Standard Textbook of Electricity

  • Stephen Herman(Author)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The effect this condition has on other circuit quantities is explored. 22–1 Operation of RC Parallel Circuits When resistance and capacitance are connected in parallel, the voltage across all the devices will be in phase and will have the same value. The current flow through the capacitor, however, will lead both the resistive current and applied voltage by 90° (Figure 22–1) . The amount of phase angle shift between the total circuit current and voltage is determined by the ratio of the amount of resistance to the amount of capacitance. The circuit power factor is still determined by the ratio of resistance and capacitance. FIGURE 22–1 Current flow through the capacitor is 90 8 out of phase with both the current flow through the resistor and the applied voltage. 90 ° Capacitive current Resistive current Applied voltage 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. UNIT 22 Resistive-Capacitive Parallel Circuits 523 22–2 Calculating Circuit Values EXAMPLE 22–1 FIGURE 22–2 Resistive-capacitive parallel circuit. E T 240 V I T Z VA PF E R 240 V I R R 30 P E C 240 V I C X C 20 VARs C C o In the RC parallel circuit shown in Figure 22–2 , assume that a resistance of 30 V is connected in parallel with a capacitive reactance of 20 V. The circuit is connected to a voltage of 240 VAC and a frequency of 60 Hz.

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  eBook - ePub

  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Section 20.13 , charge flows in a dc circuit only for the brief period after the battery voltage 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 battery. However, suppose that the battery connections 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.

  To help set the stage for the present discussion, recall from Section 20.5 that the rms voltage Vrms across the resistor in a purely resistive ac circuit is related to the rms current Irms by

  Vrms = Irms R

  (Equation 20.14 ). The resistance R has the same value for any frequency of the ac voltage or current. Figure 23.1 emphasizes this fact by showing that a graph of resistance versus frequency is a horizontal straight line.

  Figure 23.1

  The resistance in a purely resistive circuit has the same value at all frequencies. The maximum emf of the generator is V0 .

  For the rms voltage across a capacitor the following expression applies, which is analogous to

  Vrms = Irms R

  :

  (23.1)

  V rms

  =

  I rms

  X C

  The term XC appears in place of the resistance R and is called the capacitive reactance . The capacitive reactance, like resistance, is measured in ohms and determines how much rms current exists in a capacitor in response to a given rms voltage across the capacitor. It is found experimentally that the capacitive reactance XC is inversely proportional to both the frequency f and the capacitance C, according to the following equation:

  (23.2)

  X C

  =

  1

  2 π f C

  For a fixed value of the capacitance C, Figure 23.2 gives a plot of XC versus frequency, according to Equation 23.2 . A comparison of this drawing with Figure 23.1 reveals that a capacitor and a resistor behave differently. As the frequency becomes very large, Figure 23.2 shows that XC approaches zero, signifying that a capacitor offers only a negligibly small opposition to the alternating current. In contrast, in the limit of zero frequency (i.e., direct current), XC

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  eBook - ePub

  Complete Electronics Self-Teaching Guide with Projects

  • Earl Boysen, Harry Kybett(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 6 Filters

  Certain types of circuits are found in most electronic devices used to process alternating current (AC) signals. One of the most common of these, filter circuits, is covered in this chapter. Filter circuits are formed by resistors and capacitors (RC), or resistors and inductors (RL). These circuits (and their effect on AC signals) play a major part in communications, consumer electronics, and industrial controls.

  When you complete this chapter, you will be able to do the following:

  • Calculate the output voltage of an AC signal after it passes through a high-pass RC filter circuit.
  • Calculate the output voltage of an AC signal after it passes through a low-pass RC Circuit.
  • Calculate the output voltage of an AC signal after it passes through a high-pass RL circuit.
  • Calculate the output voltage of an AC signal after it passes through a low-pass RL circuit.
  • Draw the output waveform of an AC or combined AC-DC signal after it passes through a filter circuit.
  • Calculate simple phase angles and phase differences.

  Capacitors in AC Circuits

   1  An AC signal is continually changing, whether it is a pure sine wave or a complex signal made up of many sine waves. If such a signal is applied to one plate of a capacitor, it will be induced on the other plate. To express this another way, a capacitor will “pass” an AC signal, as illustrated in Figure 6.1 .

  Figure 6.1

  Note Unlike an AC signal, a DC signal is blocked by a capacitor. Equally important is that a capacitor is not a short circuit to an AC signal.

  Questions

  A. What is the main difference in the effect of a capacitor upon an AC signal versus a DC signal? _____

  B. Does a capacitor appear as a short or an open circuit to an AC signal? _____

  Answers

  A. A capacitor will pass an AC signal, whereas it will not pass a DC voltage level.

  B. Neither.

   2  In general, a capacitor will oppose the flow of an AC current to some degree. As you saw in Chapter 5, “AC Pre-Test and Review,” this opposition to current flow is called the reactance

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  eBook - ePub

  Electronics for Scientists

  • Daniel Santavicca, Daniel F. Santavicca(Authors)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  2 Linear AC Circuits

  DOI: 10.1201/9781003408499-2

  AC stands for alternating current and refers to circuits whose currents and voltages are not constant in time. When we talk about AC circuits, we often assume that the currents and voltages oscillate as sinusoidal functions of time, although this is not always the case. A linear AC circuit component is one whose properties do not depend on the amplitude of the current through the component. In this chapter, we will assume that our components are linear. This assumption will be relaxed in later chapters.

  2.1 CAPACITANCE AND INDUCTANCE

  If we have two oppositely-charged conductors that are not in contact and that each contain a net charge magnitude Q, there will be an electric field pointing from the positively-charged conductor to the negatively charged conductor. If we integrate the electric field along some path that goes from the surface of one conductor to the surface of the other conductor, we will find the voltage V between the two. We find that Q and V are linearly proportional, and so we introduce a constant of proportionality that we call the capacitance C:

  Q = C V

  (2.1)

  This is our defining equation for capacitance, in the same way, that Ohm's law is our defining equation for resistance. Capacitance has SI units of farads (F).

  Equation 2.1 is true for any capacitor, regardless of geometry. For a particular geometry known as a parallel-plate capacitor, we can arrive at a convenient expression for the capacitance in terms of the physical properties of the capacitor. A parallel plate capacitor has two identical metal plates, each with surface area A, separated by a uniform distance d. Assuming that d

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  eBook - ePub

  Electrical Engineering 101

  Everything You Should Have Learned in School...but Probably Didn't

  • Darren Ashby(Author)
  • 2011(Publication Date)
  • Newnes

    (Publisher)

  Figure 2.6 .

  Figure 2.6 Step input is applied to a simple RC Circuit.

  Using your intuitive understanding of resistors and capacitors, let's analyze what is going to happen in this circuit. We'll do this by applying a step input. A step input is by definition a fast change in voltage. The resistor doesn't care about the change in voltage, but the cap does. This fast change in voltage can be thought of as high frequencies, 6 and how does the cap respond to high frequencies? That's right, it has low impedance. So, now we apply the voltage divider rule. If the impedance of Rg is low (as compared to Ri), the voltage at Vo is low. As frequency drops, the impedance goes up; as the impedance goes up, based on the voltage divider, the output voltage goes up. Where does it all stop?

  6 This is something a man named Fourier thought of long ago. The more harmonic frequencies you sum together, the faster the rise time of said step input.

  Think about it a moment. Based on what you know about a cap, it resists a change in voltage. A quick change in voltage is what happened initially. After that our step input remained at 5 V, not changing anymore. Doesn't it make sense that the cap will eventually charge to 5 V and stay there? This phenomenon is known as the transient response of an RC Circuit. The change in voltage on the output of this circuit has a characteristic curve. It is described by this equation (note t = time):

  (Eq. 2.4)

  The graph of this output looks like Figure 2.7 . The value of R times C in this equation is also known as tau, or the time constant, often referred to by the Greek letter τ.

  Figure 2.7 Voltage change over time.

  (Eq. 2.5)

  For a step input, this curve is always the same for an RC Circuit. The only thing that changes is the amount of time it takes to get to the final value. The shape of the curve is always the same, but the time it takes to happen depends on the value of the time constant7 τ. You can normalize this curve in terms of the time constant and the final value of the voltage. Let's redraw the curve with multiples of τ along the time axis, as shown in Figure 2.8

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  eBook - ePub

  MSP430-based Robot Applications

  A Guide to Developing Embedded Systems

  • Dan Harres(Author)
  • 2013(Publication Date)
  • Newnes

    (Publisher)

  Figure 3.12 ).

  Figure 3.12 Charging a capacitor.

  The capacitor of Figure 3.12 responds to current flowing into it by increasing its voltage (with respect to the negative terminal). Its approximate behavior is given by:

  (3.14)

  Given this equation, it’s easy to see one of the unique characteristics of the capacitor – a large change in voltage over a short period of time requires a huge amount of current. To state it another way, a capacitor resists changes in voltage .

  To see how Eq. 3.14 works in a real circuit, let’s combine a resistor and a capacitor, as shown in Figure 3.13 . At first glance, this circuit might appear to have a constant voltage source, V B , and therefore no current should be flowing through the capacitor. However, note that there is a switch in the circuit, which is closed at time t =0. Closing that switch causes the voltage across the resistor/capacitor branch to jump from 0 V to V B (assuming the initial capacitor voltage was zero). Even though V B is constant, the voltage across the resistor and capacitor changes, and therefore current will flow through the capacitor. So, let’s analyze the circuit.

  Figure 3.13 A simple RC filter.

  To avoid the confusion of what happens at exactly t =0, we’ll talk about the current, I , at some moment just after 0. We’ll call this time t= 0+ . We’ll also, at times, refer to the circuit current and voltages as functions of time, for example, V C (t= 0+ ), or just V C (0+ ).

  Continuing on, apply Kirchoff’s voltage law clockwise around the circuit and use the convention that, if we encounter the negative terminal of the voltage first, it is treated as a negative voltage. The equation becomes:

  (3.15)

  or

  (3.16)

  The initial capacitor voltage, V C (0+ ), is zero. Therefore, at time, t =0+ , Eq. 3.16

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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)

  Figure 10.3 shows a series RC Circuit driven by a 1 V step function starting at time equals 20 ns. The 25 nF capacitor has the initial voltage v C (0) set to 0. The resistor voltage is defined as V S minus VC, establishing a polarity of pos- itive current flowing clockwise around the loop. With no initial conditions in the schematic in Figure 10.3, this circuit shows step response, for comparison and contrast to the following two examples of complete response. Source voltage 1 0.5 Voltage 0 0 5e-07 1e-06 1.5e-06 Time 2e-06 3e-06 2.5e-06 Capacitor voltage Figure 10.4 Series RC step response simulation plot of capacitor voltage. Figure 10.4 shows the 1 V step function as a dashed line. The starting time of 20 ns is chosen to be able to see the signal not overlapped with the vertical axis. The capacitor voltage, VC in the schematic, builds up toward the source voltage V S . 0.06 Current –0.06 0 5e-07 1e-06 1.5e-06 Time 2e-06 3e-06 2.5e-06 –0.05 –0.04 –0.03 –0.02 –0.01 0 0.02 0.01 0.03 0.05 0.04 Resistor current Source current Figure 10.5 Series RC step response simulation plot of currents. Figure 10.5 shows the resistor current, defined by V 1 (V S ) minus VC, which is positive in the clockwise direction. It jumps up and then decays to zero exponentially over time. The negative source current indicates that the current polarity for the voltage source is positive coming out the bottom of the source and flowing in the counter- clockwise direction. 194 10 Complete Response of First-Order RC and RL Circuits 10.2.2 Second RC Example – Plus 0.5 V Initial Capacitor Voltage V S R1 R = 20 Ohm + – VC C1 C = 25 nF V = 0.5 V1 T2 = 20 us T1 = 20 ns U2 = 1 V U1 = 0 V Figure 10.6 Series RC Circuit simulation schematic with 0.5 V initial condition. Figure 10.6 shows a series RC Circuit driven by a 1 V step function start- ing at time equals 20 ns. The 25 nF capacitor has the initial voltage v C (0) set to 0.5 V.

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