Is Current Gradually Collected in a Capacitor R Does It Continue to Loop

RC Circuit

THE FOKKER–PLANCK EQUATION

N.G. VAN KAMPEN , in Stochastic Processes in Physics and Chemistry (Third Edition), 2007

Exercise

In an RC circuit with non-Ohmic resistance R(V) the Fokker–Planck equation for the voltage V would be, according to the above phenomenological argument,

(1.8a) $\frac{\partial P\left(V,t\right)}{\partial t}=\frac{\partial }{\partial V}\frac{V}{CR\left(V\right)}P+\frac{1}{2}\frac{{\partial }^2}{\partial V^2}B\left(V\right)P,$

(1.8b) $B\left(V\right)=e^{C{V}^2/2lkT}\left[\mathrm{const}.-2{\displaystyle {\int }_0^V\frac{V^{\prime }}{CR\left(V^{\prime }\right)}{e}^{-C{V^{\prime }}^2/2kT}}d{V}^{\prime }\right].$

See, however, the discussion in X.2.

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Alternating Current Phenomena: Signals and Power

Wayne M. Saslow , in Electricity, Magnetism, and Light, 2002

14.6.3 Phasor Method

In terms of the phasor representation in Figure 14.9(b), by (14.55) we must add the voltage phasors ΔV _R and ΔV _C across the resistor and the capacitor to obtain the total voltage phasor ΔV _g across the generator. Since ΔV _R and ΔV _C are 90° out of phase, they add vectorially. By the Pythagorean theorem, the resultant phasor has a length squared $I_m^2{R}^2+I_m^2{X}_C^2,$ which must equal ${\mathcal{E}}_m^2$ . As expected, this is the same as (14.60). We obtain the phase angle ϕ from the fact that its tangent is the ratio of the phasor voltage component along the local (rotating) y′-axis to the phasor voltage component along the local (rotating) x′-axis. By definition, the voltage I _m R is along the local (rotating) x′-axis, so the voltage –I _m /ωC= –I _m X _c is along the local (rotating) y′-axis. The ratio, by (14.43′), gives (14.59). From (14.59) for ϕ and (14.61) for Z, the total current at any time can be obtained using (14.43).

Example 14.8

An RC circuit

Consider the values R = 10 ohms and C= 2 µF, for which Example 14.7 applies. Which circuit element dominates at low frequencies, and which dominates at high frequencies?

Solution: Since (14.52) gave X _C = 5000 ohms at ω = 10² s⁻¹, and X _C = 0.005 ohm at ω = 10⁸ s⁻¹, the capacitor dominates at the lower frequency, and the resistor dominates at the higher frequency. The reasons are simple: at low frequencies the capacitor has time to charge up, thus blocking current from flowing; at high frequencies the capacitor does not have enough time to charge up, so the resistor dominates.

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Batteries, Kirchhoff's Rules, and Complex Circuits

Wayne M. Saslow , in Electricity, Magnetism, and Light, 2002

8.9.5 Applications

Optional

Another application of the RC circuit is as a filter. Since the voltage ΔV _C = Q/C across the capacitor takes time (on the order of RC) to build up, it is sensitive to emfs that vary slowly relative to RC (low frequencies), but it is insensitive to emfs that vary quickly relative to RC (high frequencies). On the other hand, the voltage ΔV _R = IR across the resistor builds up immediately, but it dies out after a time on the order of RC. Hence it is sensitive to emfs at high frequencies, but it is insensitive to emfs at low frequencies. As a consequence, in an RC circuit the capacitor can be used to pass low frequencies but filter out high frequencies, and the resistor can be used to filter out low frequencies but pass high frequencies. See Figure 8.22, which shows the output leads in each case, where it is assumed that the output resistance (also called the impedance) is so high that it draws no current (as for an ideal voltmeter). The emf is circled to indicate that it might be something other than a battery—a power supply or signal generator, for example. Filters are discussed in more detail in Chapter 14.

Figure 8.22. RC circuit with emf ε: (a) Voltage across capacitor read by high resistance voltmeter. (b) Voltage across resistor read by high resistance voltmeter.

Related to this are two more applications of an RC circuit. If the voltage ΔV _C = Q/C across the capacitor is thought of as the input, then the voltage ΔV _R across the resistor can be thought of as a differentiator because $\Delta V_R=RdQ/dt=RCd\left(\Delta V_C\right)/dt.$ . Similarly, if the voltage across the resistor is thought of as the input, then the voltage across the capacitor can be thought of as an integrator because $\Delta V_C=Q/C={\displaystyle \int Idt/C}={\displaystyle \int \Delta V_{R}dt/RC}.$ .

Note that when a capacitor discharges, its electrical energy goes into heat. We can see how this occurs at each instant. Consider a resistor and capacitor in series. In a time dt the charge dQ = Idt passes through the circuit. The capacitor loses energy –(Q/C)dQ(note that dQ < 0). The resistor heats up by I ² Rdt. Equating the loss in electrical energy to the heat loss gives $-\left(Q/C\right)dQ=I^{2}Rdt=IRdQ,$ which leads to $I=\left(-Q/C\right)/R,$ as in (8.45).

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Circuit Fundamentals

Martin Plonus , in Electronics and Communications for Scientists and Engineers (Second Edition), 2020

1.8.2 Time Constant

A time constant τ is associated with an exponential process such as e ^{−  t/τ}. It is defined as the time that it takes for the process to decrease to 1/e or 37% of its initial value (1/e  =   1/2.71   =   0.37). Hence, when the process is 63% complete a time t  =   τ has elapsed. Time constants provide us with a convenient measure of the speed with which transients in circuits occur. By the time 1τ, 2τ, 3τ, 4τ, and 5τ have elapsed, 37%, 13%, 5%, 1.8%, and 0.67% of the transient remain to be completed. We can then state that for most practical purposes a transient will be completed at the end of, say, five time constants, as only two-thirds of 1% of the original transient then remains. Consequently, a knowledge of the time constant allows us to estimate rapidly the length of time that a transient process will require for completion.

Referring to Eq. (1.50), we see that in a capacitor-charging circuit, the current will have decayed to 1/e or to 37% of its initial value in a time of t  = RC . Hence, the time constant τ for an RC circuit is RC. We could have also examined voltage and come to the same conclusion. For example, using Eq. (1.51), which gives the capacitor voltage, we conclude that the time that it takes to charge an initially uncharged capacitor to 63% (1   −   1/e  =   1   −   1/2.71   =   0.63) of the battery voltage V is the time constant τ.

We can now make an important observation: the time constant is a characteristic of a circuit. Therefore, in an RC circuit, the time constant τ   = RC is the same for charge or discharge—which can be easily seen by looking at the charging voltage (Eq. 1.51) and comparing it to the discharge voltage (Eq. 1.52).

There is another aspect of time constants that should be understood. We observe that a transient would be complete in the time of one time constant τ if the current in Fig. 1.25b were to decrease at the same slope that it began. We can differentiate Eq. (1.50), evaluate it at t  =   0, and obtain di/dt  =   −  i(0)/τ. This gives us the slope of a straight line, which if it starts at i(0)   = V/R, intersects the t-axis at τ   = RC. This curve is shown as a dashed line in Fig. 1.25b. In summary, we can state the following: The time constant is the time in which the current in the circuit would reach its final value if it continued to change at the rate at which it initially started to change.

RC Circuits and Speed Limitation in Digital Electronics

Another very important aspect of RC circuits is that their characteristics are a major limitation to the speed of digital computers. In a microprocessor, gates which are interconnected transistor circuits, process signals which are a stream of "0" and "I." One can view interconnects between gates as two copper wires with capacitance between the wires (as well as the capacitance at the wire input and wire output), essentially RC circuits as shown in Fig. 1.25a. A stream of zero and ones that flows from gate to gate can be mimicked by a voltage rise to V in Fig. 1.25c (capacitor C fills with electrons) followed by a voltage drop to zero in Fig. 1.25f (C drains off electrons). This fill-drain process takes time. Toggling the switch in Fig. 1.25a between positions 1 and 2 would create such a digital or bit stream of ones and zeros. As just pointed out, it takes time to charge C, starting from zero volts, to reach state "1" represented by voltage V. If we toggle the switch at a speed, say, every few time constants τ, where τ   =   RC, the capacitor will charge to practically V and discharge to practically zero volts, hence we should be able to recognize the resulting wave shape as a stream of square pulses, that is as "1"s followed by a "0"s. A time longer than a few time constants would be even better, as it would further decrease the error rate in recognizing the ones and zeros in the bit stream (similarly toggling less than a time constant, capacitor will not be able to fully charge to V and not be able to fully discharge to zero, greatly increasing the error rate and possibly making the bit stream unrecognizable as "0" and "1"s). However, remember that we are trying to send signals as fast as possible, but yet with an acceptable error rate. This usually means that, for example, in a computer with a 1   V power supply, we recognize logic "1" as a voltage between 0.7 and 1   V and logic "0" as a voltage between 0 and 0.3   V with a gap of 0.4   V between the two states which should result in an acceptable error rate. So, for example, if a connection between logic gates has a resistance R of ≈   1   kΩ (gate and wire) and capacitance C of ≈   10   fF (f  =   10^{−   15}) (gate and wire), resulting in a time constant (or single gate delay) of τ   ≈   10   ps. A microprocessor logic path has n gates, where n is typically 20–40 gates, making the total delay T about T  =   nτ   ≈   400   ps. If we use τ as the shortest time that we can change logic gates, then we can state that the maximum computer operating frequency is given by f_max  =   (T)^{−   1}  =   2.5   GHz which is typical for a laptop computer. Clearly reducing R and C would enable to perform calculations faster. Besides R and C, we have transistor on-off switching speed, heat dissipation in processors with multi-billion transistors as additional speed limitations.

RC circuits also characterize the heat generated in microprocessors. Transistors in processors are basically on-off switches similar to the two position switch in Fig. 1.25a. With switch in position 1, voltage across C will build to V (we are generating logic "1") and C will store energy $w_C=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{CV}}^2$ (see Eq. 1.18). During this process the charging current flows through R and generates heat given by $w_R={\int }_0^{\infty }{\mathrm{i}}^{2}R\mathrm{dt}={\int }_0^{\infty }\frac{{\mathrm{v}}^2}{\mathrm{R}}{e}^{-\frac{2t}{\mathit{RC}}}\mathit{dt}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\mathrm{C}{V}^2$ , where Eq. (1.50) was used for i. Adding these two energies, we see that the battery or power supply must supply energy of CV ². Switching to position 2 (now generating a "0"), the capacitor discharges its energy in R, which is again given by $w_R={\int }_0^{\infty }{\mathrm{i}}^2\mathit{Rdt}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\mathrm{C}{V}^2$ where the discharge current shown in Fig. 1.25e was used. We now see that in a single on-off cycle, an amount of CV ² heat energy in Joules must be dissipated. Given computer speed as f_max, then in 1   s, energy f_maxCV² in Joules must be dissipated. To reduce generated heat, the power supply for a microprocessor is nowadays roughly between 0.8 (14   nm CMOS) and 1.0   V (28   nm CMOS). Whereas in older, slower computers, voltages of up to 5   V were used.

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INDUCTANCE AND INDUCTORS

George B. Arfken , ... Joseph Priest , in International Edition University Physics, 1984

LC Circuit

We discussed the behavior of an RC circuit in Section 31.6 and that of an LR circuit in Section 35.2. Now let's consider the LC circuit. A charged capacitor connected to an inductor is an interesting and useful configuration. The inductor provides a conducting path, which allows the capacitor to discharge. By discharging, the capacitor converts the electric energy stored in its electric field to magnetic energy in the inductor. Once this transfer of energy is complete, the magnetic field begins to diminish, and the magnetic energy is transformed back to electric energy in the capacitor. In the absence of a resistor, this back-and-forth transfer of energy continues indefinitely. In principle, an LC circuit behaves precisely like the spring-mass version of the simple harmonic oscillator (Sections 14.1). The initial potential energy of a compressed spring is transferred into kinetic energy of an attached mass. In the absence of energy-dissipating mechanisms (friction), the back-and-forth transfer of kinetic and potential energy continues indefinitely.

We can show the equivalence between a simple harmonic oscillator and an LC circuit formally by analyzing Kirchhoff's equation for the LC circuit. Let us connect a charged capacitor through a switch to an inductor having zero electrical resistance (Figure 35.17). Charge flows from the capacitor when the switch is closed. The resulting current produces in the inductor a magnetic field and an induced emf that opposes the action of the capacitor. Applying Kirchhoff's loop law, we have

Figure 35.17. A charged capacitor is connected to an inductor and a switch. There is no current as long as the switch is open. Closing the switch allows charges to flow through the inductor.

(35.24) $\frac{q}{\text{C}}-L\frac{dI}{dt}=0$

The charge, on the capacitor decreases so that

$I=-\frac{dq}{dt}$

Therefore the equation governing the charge q is

(35.25) $\frac{q}{\text{C}}+L\frac{d^{2}q}{d{t}^2}=0$

We arrive at a differential equation identical in form to that deduced for a mass connected to a spring (Sections 14.1).

$\begin{array}{ll}kx+m\frac{d^{2}x}{d{t}^2}=0& \text{spring-mass}\\ \frac{1}{\text{C}}q+L\frac{d^{2}q}{d{t}^2}=0& \text{capacitor-inductor}\end{array}$

Charge q corresponds to displacement x, 1/C corresponds to the spring constant k, and L corresponds to mass m. Calling q₀ the initial charge on the capacitor and measuring time from the instant the switch is closed, we have as the solution of Eq. 35.25

(35.26) $q=q_0\cos \omega t$

where ω = 1/√LC. (You can check Eq. 35.26 by differentiating q₀ cos ωt twice with respect to time and substituting back into Eq. 35.25.) The frequency ν = ω/2π is called the natural frequency of the oscillations. This is the oscillation frequency when there is no electrical resistance in the circuit. It is analogous to the oscillation frequency of a mechanical simple harmonic oscillator when no frictional forces are present. Differentiating Eq. 35.26 to obtain an expression for the current, we have

(35.27) $I=-\frac{dq}{dt}=\omega q_0\sin \omega t$

Equations 35.26 and 35.27 indicate that charge and current will vary with time as shown in Figure 35.18. In a time π/2ω all the initial charge has been removed from the capacitor. The capacitor then begins recharging, but with the polarity reversed. In a time π/ω the plate that was initially charged positive has achieved a negative charge −q₀ . Continuing, the capacitor discharges, recharges, and regains its initial charge and polarity after a time 2π/Ω has elapsed. An actual oscilloscope recording of the potential difference across the capacitor in an LC circuit is shown in Figure 35.19.

Figure 35.18. The time variation of charge stored by the capacitor, and the associated current, in the circuit shown in Figure 35.17. The charge stored is a maximum when the current is zero. We say that the charge stored and current are π/2 radians out of phase.

Figure 35.19. Oscilloscope recording of the potential difference across a capacitor for a circuit like that shown in Figure 35.17.

Initially the capacitor has energy $U_{\text{o}}=\mathrm{½}\left(q_0^2/C\right)$ . At any instant thereafter its electric energy (U_E ) is

$\begin{array}{l}{U}_E=\frac{1}{2}\frac{q^2}{\text{C}}=\left(\frac{1}{2}\frac{q^2}{\text{C}}\right){\cos }^2\omega t\\ =U_{\mathrm{o}}{\cos }^2\omega t\end{array}$

If we start with zero magnetic energy, the magnetic energy (U_M ) stored by the inductor at any instant is

$\begin{array}{l}{U}_M=\frac{1}{2}L{I}^2\\ =\frac{1}{2}L{\omega }^2{q}_0^2{\sin }^2\omega t\end{array}$

Since

${\omega }^2=\frac{1}{LC}$

then

$\begin{array}{l}{U}_M=\left(\frac{1}{2}\frac{{\text{q}}_0^2}{C}\right){\sin }^2\omega t\\ =U_{\mathrm{o}}{\sin }^2\omega t\end{array}$

The total energy at any instant is

$\begin{array}{l}{U}_{\text{total}}=U_E+U_M\\ =U_{\mathrm{o}}({\cos }^2\omega t+{\sin }^2\omega t)\\ =U_{\mathrm{o}}\end{array}$

Thus, analogous to a mechanical simple harmonic oscillator, the total energy is conserved.

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Electrophysiology

S.E. Pagnotta , in Encyclopedia of Condensed Matter Physics, 2005

Passive properties

As far as it concerns passive properties, common to all cell types, the cell membrane can be schematized with a simple RC circuit. C _m and R _m, the membrane capacitance and resistance, are independent of the membrane potential and represent the membrane dielectric or insulating properties and the ion permeability through the membrane, respectively (Figure 2a). If a pulse of electric current is passed through the cell membrane, its potential will rise to a maximal and sustained value and then will decay, following in both cases an exponential law:

Figure 2. (a) Equivalent circuit of membrane passive properties. (b) Membrane voltage recorded in response to a current pulse injection. (c) Current–voltage relationship for a nonexcitable membrane.

[3a] $V_{\text{m}}=V_0\left[1-{\text{e}}^{-t/R_{\text{m}}{C}_{\text{m}}}\right]=V_0\left[1-{\text{e}}^{-t/{\tau }_{\text{m}}}\right]$

for the rising phase and

[3b] $V_{\text{m}}=V_0{\text{e}}^{-t/R_{\text{m}}{C}_{\text{m}}}=V_0{\text{e}}^{-t/{\tau }_{\text{m}}}$

for the decaying one (Figure 2b). τ _m is the membrane time constant. For the sake of the following discussion, a change in the membrane potential toward more positive values is referred to as a depolarization, while a change toward more negative values will be referred to as a hyperpolarization. The R _m values being voltage and time independent, the current–voltage relationship for the maximal sustained potential during current pulse application will be linear (Figure 2c), and if the membrane potential is kept at a particular voltage level V _m, the total current flow will be

[4] $J_{\text{m}}=J_{\text{c}}+J_{\text{i}}=C_{\text{m}}\left(\text{d}{V}_{\text{m}}/\text{d}t\right)+G_{\text{m}}\left(V_{\text{m}}-V_{\text{rev}}\right)$

where $G_{\text{m}}=1/R_{\text{m}}$ is the membrane conductance and V _rev is the previously defined reversal potential.

Actually, for excitable membranes C _m and R _m values are voltage-dependent (as voltage-gated ion channels are present on the membrane), and the previous schematic description holds only at resting condition or in a range of membrane potential below a particular threshold. Indeed, in these conditions the values of C _m and R _m become virtually constant, while in any other condition active properties arise.

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Molecular and Polymer Semiconductors, Transport Properties of

P.W.M. Blom , in Encyclopedia of Materials: Science and Technology, 2001

2.2 Impedance Spectroscopy

The dynamics of the charge transport in PLEDs may be further investigated by impedance spectroscopy, which is a powerful technique to investigate relaxation processes and charge transport kinetics in solid state devices. Previous a.c. studies performed on PLEDs have been interpreted in terms of equivalent RC-circuit models. A possible frequency dependence of the circuit components had not been taken into account in the analysis of the a.c. measurements. In Fig. 9 the a.c. response of the capacitance, defined as C=Im(Y/ω), with Y the complex admittance i _ac/v _ac, of an ITO/PPV/Cu hole-only device is shown for different bias voltages. The capacitance curves are offset by 0.5   nF for clarity. It appears that the capacitance C exhibits a distinct frequency dependence; at low frequencies C decreases and the frequency at which this decrease stops shifts to higher frequencies with increasing voltage. For the interpretation of these results it is important to realize that, since the device is space-charge limited, v _ac directly probes the number of charge carriers in the device. However, the timescale for the buildup of charge carriers is given by the transit time τ_t of the injected holes. At low frequencies ω<τ_t ⁻¹ the buildup of space charge is fast enough to follow the modulation in v _ac and will result in an additional contribution to the capacitance. For high frequencies ω>τ_t ⁻¹ the space charge cannot be redistributed in a period of the applied voltage v _ac and the measured capacitance equals the geometrical value, C=εA/L. The frequency dependent behavior is modeled using a time-dependent SCLC model. Since the hole transport in PPV is dispersive due to disorder, the distribution of transit times will be reflected in the frequency-dependent response of the device. Model calculations show that the observed frequency behavior can be explained by taking into account a broad distribution of transit times, characterized by a dispersion parameter α=0.50±0.05 as defined in the SM theory, which is in agreement with the reported value of α=0.45 from delayed EL measurements. The d.c. mobility can be obtained by calculating the average of the distribution in transit times. The d.c. mobility at room temperature of various hole-only devices is shown in Fig. 10, as a function of the electric field. As expected, the obtained mobility is independent of the sample thickness L and thus reflects a genuine material parameter. The magnitude and field dependence of the d.c. mobility, characterized by exp(γ√E), is in excellent agreement with the result from earlier d.c. (J–V) measurements.

Figure 9. Frequency dependent capacitance C of an Au/PPV/Cu hole-only device at various applied voltages. The capacitance traces are off-set by 0.5   nF for clarity. The solid lines are fit to a transient SCLC model including a distribution in transit times.

Figure 10. Mobilities obtained from several Au/PPV/Au (open symbols) and Au/PPV/Cu (closed symbols) hole-only devices with various thickness as a function of electric field. The mobility follows the exp(γ√E) dependence in agreement with the results obtained from the J–V characteristics.

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Impedance Spectroscopy and Mobility Spectra

R.A. Gerhardt , in Encyclopedia of Condensed Matter Physics, 2005

Equivalent Circuits

It is customary to analyze the measured impedance response in terms of the possible equivalent circuit components that can give rise to the frequency response observed. The circuit elements of interest are capacitors represented by C, resistors represented by R, and inductors represented by L . Most dielectric materials can be represented by a parallel RC circuit, whereas others are better represented by a series RC circuit. The presence of magnetic components introduces inductive components, and one may expect that series or parallel RL circuits may represent such materials.

It is useful to remember that if elements are in series, it is easier to write the impedance response. On the other hand, if the circuit elements are in parallel, it is more convenient to write the admittance and then convert it to the impedance. For example, the impedance of a series RC circuit is written as

[2] $Z*=R+\left[1/j\omega C\right]$

whereas the impedance of a parallel RC circuit is given by

[3] $Z*=1/Y*$

where $Y*=\left(1/R\right)+\text{j}\omega C$ .

Most modern equipment include some version of software that can be used to fit the most appropriate equivalent circuit to the experimentally measured results. However, the reader is urged to consider the physical mechanisms that may have contributed to the experimental data before concluding that a derived R or C is representative of the bulk of the material or device being measured (see the section "Data analysis" for more details).

Table 1 lists the simplest equivalent circuits and their corresponding impedance equations. Data acquired can be plotted in terms of frequency-explicit and frequency-implicit impedance plots. Frequency-explicit plots are often referred to as Bode plots. They are just simply real (Z′) or imaginary impedance (Z″) plotted versus frequency. The magnitude of the impedance vector, which is equal to

Table 1. Simplest equivalent circuits and their impedance equations

Resistor		Z*=R1
Capacitor		Z*=1/jωC1
Inductor		Z*=jωL1
RC parallel		Z*=1/[1/R1+jωC1]
RC series		Z*=R1+(1/jωC1)
RL parallel		Z*=1/[1/R1+1/jωL1]
RL series		Z*=R1+jωL1
LC series		Z*=jωL1+1/jωC1
LC parallel		Z*=1/[1/jωL1+jωC1]
LCR series		Z*=jωL1+(1/jωC1)+R

[4] $|Z*|={\left[Z^{\prime 2}+Z^{″2}\right]}^{1/2}$

and the phase angle θ are also plotted versus the measured frequency. The magnitude of the impedance vector is useful for quickly assessing the presence of resistive components (flat with frequency), capacitive components (decrease with increasing frequency), and inductive components (increase with increasing frequency) (see Figure 1 for representative frequency-explicit plots for a perfect resistor, a perfect capacitor, and a perfect inductor).

Figure 1. Representative impedance vector response for (a) a perfect resistor, (b) a perfect capacitor, and (c) a perfect inductor as a function of frequency.

Frequency-implicit plots are better known as complex plots because the imaginary impedance is plotted versus the real impedance. These plots are often referred to as Nyquist plots or Cole–Cole plots and result in the formation of semicircles only under certain conditions. As shown in Figure 2, a semicircle will only appear on the complex impedance plots when the measured object can be represented by a parallel RC circuit or a parallel RL circuit. It should be noted that the imaginary part of these two circuits carries opposite signs. By established convention, a parallel RC circuit will display a semicircle in the first quadrant if the imaginary axis is labeled as negative (Figure 2a). In contrast, a parallel RL circuit will display an upside down semicircle in the fourth quadrant (Figure 2d). It may be added that a series RC circuit or a series RL circuit do not display semicircles in the complex impedance plane (Figures 2b and 2c).

Figure 2. Equivalent circuits and representative complex impedance plots for (a) a parallel RC circuit, (b) a series RC circuit, (c) a series RL circuit, and (d) a parallel RL circuit.

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Transient analysis

Eur Ing RG Powell , in Introduction to Electric Circuits, 1995

Example 8.14

An inductor of 250 μH inductance is energized from a 1 kV d.c. supply via two thyristors connected in series. The thyristors are identical except that one of them has a delay at turn-on of a few microseconds longer than the other. Voltage sharing is assisted by placing an RC circuit in parallel with each thyristor. Obtain an expression for the current in the RC circuit connected across the 'slow' thyristor after the other one has turned on.

Solution

The circuit is shown in Fig. 8.37 and it should be noted that the thyristors Th₁ and Th₂ may be considered to be simply switches which are either closed (when they are turned on) or open (before they are turned on).

Figure 8.37.

At the instant of closing the switch the current in the circuit is zero and the 1 kV is shared equally between the capacitors. When thyristor Th₁ turns on it short circuits the RC circuit in parallel with it and the circuit will then be as shown in Fig. 8.38, the R and the C being those in parallel with the still open switch Th₂. The transform circuit takes the form shown in Fig. 8.39. From the transform circuit we have that

Figure 8.38.

Figure 8.39.

$\begin{array}{ll}i\left(s\right)& =\left[\right(1000/s)-(500/s\left)\right]/(R+1/Cs+Ls)\hfill \\ & =500/s(R+1/Cs+Ls)\hfill \\ & =500/L{s}^2+Rs+1/C\hfill \\ & =(500/L)/[s^2+(R/L)s+1/CL]\hfill \end{array}$

Completing the square of the denominator we have

i(s) = (500/L)/[{s + (R/2L)}² + {(1/CL) − (R/2L)²)}]

Putting L = 250 μH, R/2L = α, and {(1/CL) − (R/2L)²} = ω², we have

i(s) = 2/[(s + α)² + ω²] μs

Multiplying numerator and denominator by ω we have

i(s) = (2/ω){ω/[(s + α)² + ω²]}

From transform pair number 7 in Table 8.1 we see that ω/[(s + α)² + ω²] is the transform of exp (−αt) sin ωt so that, as a function of time we have for the current,

i(t) = (2/ω) exp (−αt) sin ωt

This is of the form shown in Fig. 8.40 which is an exponentially decaying sine wave. It is said to be underdamped. This means that the current undergoes a period of oscillation before reaching its new required value (zero in this case).

Figure 8.40.

This result was obtained assuming that ω² is positive (i.e. (1/CL) > (R/2L)²). There are other possibilities (ω² could be negative or zero) leading to other results associated with overdamping and critical damping, respectively, in which the current reaches zero more or less rapidly and without oscillation.

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Cognitive Computing: Theory and Applications

A.S. Maida , in Handbook of Statistics, 2016

5.1 Abstract Neurons

All network models discussed in this section are spiking neural networks in contrast to traditional ANNs that use a sigmoidal or rectified linear output function. In abstract models, there are two common model neurons in use. The simplest is the leaky integrate-and-fire (LIF) neuron. The LIF neuron is modeled as a parallel RC circuit that charges in response to its input.

It has a voltage threshold and when this is reached, the circuit emits a spike and then resets to its resting level, which is usually zero. The LIF neuron has a time constant that is associated with its RC circuit. If the resistance in the circuit is high, so that leakage is negligible, then it is said to be an IF neuron instead of an LIF neuron. The capacitor model is motivated by the fact the cell membrane is a good insulator and is very thin, so has properties of a parallel plate capacitor. For increased biological realism, an absolute refractory period can be implemented by simply constraining the LIF neuron not to emit another spike for 2 or 3   ms whenever it emits a spike. The LIF model neuron, in its simplest form, is deterministic and it gives the same output on repeated trials with the same input. Under some conditions, the temporal precision of the spike times of an isolated neuron can be extremely high (Jolivet et al., 2006). However, biological neurons in vivo generally show trial-to-trial variability. A more sophisticated model that captures this variability is the Poisson spiking neuron, explained below.

In a Poisson spiking model neuron, the neuron stochastically emits discrete spikes as its output. A spiking neuron simulation usually advances in 1-ms time steps. On a particular time step, the neuron either emits a spike (1) or not (0). The spiking activity of a neuron exhibits trial-to-trial variability and baseline spiking activity. For modeling purposes, spike events can be generated by a Poisson process. The firing rate of the neuron is controlled by the rate parameter of an inhomogeneous Poisson process (Ermentrout and Terman, 2010). These models are classified "as simplified conceptual models amenable to mathematical analysis" (Gurstner et al., 2012).

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