Charging and discharging of capacitor derivation pdf

Again applying Kirchhoff's second law (conservation of energy) we get: V R + V C = 0 or R(dq /dt) + (1/C)q = 0 . . The figure shows four graphs of capacitor discharging with time on x axis open Derivation of charging and discharging equations for rc circuit capacitor circuits equation for a charging capacitor you rc circuits charging capacitors and equation derivations you Derivation Of Charging And Discharging Equations For Rc Circuit Capacitor Circuits Equation For A Charging Capacitor You Rc Circuits Charging Capacitors And Equation Derivations You Symbols Splendid Discharging It’s the capacitor discharging – this current is equal to the rate of discharge of the capacitor, so: So substituting the above expression for I. After certain time capacitor is fully discharged and inductor becomes fully charged. The high charge cell – capacitor – low charge cell energy shuttling by somehow is a function of the capacitor value (C), switching frequency (F), series equivalent resistor (R Seq), voltage different between the unbalanced cells (V diff) and finally the duty cycle, on-period, (D).


63 I. 2. On the standard view, when a capacitor is discharging current flows away from the side with positive charge and into the side with negative charge. The first step is to separate Charge and Discharge of a Capacitor INTRODUCTION Capacitors1 are devices that can store electric charge and energy.


Then determine what happens after the change. Use the flying lead to connect the circuit diagram as below. log (x ) V in 1. When the power is positive, the capacitor is charging, receiving energy from the circuit.


Discharging If we flip the switch to the position shown in Fig. Charging the capacitor (capacitor is initially uncharged): If the switch S is moved to position "a" at time t =0, a time dependent current, I(t), begins to flow, and the charge q on the capacitor increases. = After full discharge E=0. capacitor charge formula-derivation.


Thus during discharge at time t=0, Q= Q0= EC This circuit opposite can be used to show the discharge of a capacitor. source via a switch. The resulting equation will describe the charging (or discharging) of the capacitor voltage during the transient and give the final DC value once the capacitor is fully charged (or discharged). To do this experiment, you will need the following: • 6-volt battery • Two large electrolytic capacitors, 1000 µF minimum (Radio Shack catalog # 272-1019, 272-1032, or equivalent) • Two 1 kΩ resistors • One toggle switch, SPST (“Single-Pole, Single-Throw”) Chapter 11 Capacitors Charging Discharging Simple Waveshaping Circuits.


farad dB (x ) 20 . To get a feel for how this capacitor power equation works, desired voltage at the top of the string. Symbols Winning Example First Order Linear Circuit Calculate. Let a pulse voltage V is making the anode of the diode at lesser voltage than cathode as the capacitor C holds the voltage at cathode at the peak voltage.


Discharging the capacitor (capacitor initially fully charged): If the switch is moved to position "b" at t = 0, the capacitor will discharge and a current will flow through the resistor. If one can capture the voltage passing 10V during a discharge cycle, then one only needs to measure the time it takes for the voltage to Charge and Current During the Charging of a Capacitor. • Transient – a circuit changes from one DC configuration to another DC configuration (a source value changes or a switch flips). 6.


The categorization is shown in Table 1. Rise in Charge Capacitor 0. conducting till the peak are reached and this cycle continues. The function completes 63% of the transition between the initial and final states at t = 1RC, and completes over 99.


capacitance. D. Among the capacitors in the intermediary position 4. From this measurement the student will use the Logger Pro software to calculate the charge and the current as functions of time.


7. Discharge time is approx 4 mins. When a capacitor is charging, energy is being put into it; when discharging, energy is being taken out of it and can be used elsewhere. 63 Q.


Introduction to Switched-Capacitor Circuits 401 ( n C ox = 2)(W=L V DD TH) 2 fromthecapacitor. Variation of voltage with discharging of condenser. Capacitors have several uses, such as lters in DC power supplies and as energy storage banks for pulsed lasers. tell us whether the capacitor will charge or discharge.


Now below is a part of the method of calculating the capacitance of a cylindrical capacitor from the same text. Current Decay Capacitor 0. Background When a DC voltage source is connected across an uncharged capacitor, the rate at which the capacitor charges up decreases as time passes. If we define the initial charge to be – the charge when t = 0 so we now can replace giving: Capacitor Charging And Discharging Equation Derivation.


60 ms Variation of charge with discharging of condenser. To discharge a capacitor, the power source, which was charging the capacitor, is removed from the circuit, so that only a capacitor and resistor can connected together in series. 3V We simply used the dc solution for t<0 and the dc solution for t>>0 to get the limits and we used the time constant to get the horizontal scale. In this case, the conditions tell us whether the capacitor will charge or discharge.


sec , . charging capacitor where the time constant ⌧ = RC. Seperate the variables giving: where k is some constant. Capacitors pass AC current, but not DC current, so they are used to block the DC component of a signal so that Deriving the formula from 'scratch' for charging a capacitor.


5 Charging And Discharging Of A Capacitor. Figure 7. The resistor represents the leakage resistance of the capacitor, resistance of external leads and connections and any deliberately introduced resistance. – If capacitor is discharging, potential difference is zero and no current flows.


Which capacitor discharges more quickly after the switch is closed? ConcepTest RC circuit 1 time constant = RC = 12 µs = 15 µs So the capacitor A discharges faster than B Initially VC=E and charge on the condenser Q 0 Q =Ec. Apply the initial condition of the circuit to get the particular solution. 60 ms . (DEPARTMENT OF MATHEMATICAL SCIENCE, KOGI STATE UNIVERSITY, ANYIGBA) ABSTRACT: In this work, the mathematical governing equations were formulated from an electric circuit max is the maximum charge on capacitor θis an unknown phase (depends on initial conditions) ÎCalculate current: i = dq/dt ÎThus both charge and current oscillate Angular frequency ω, frequency f = ω/2π Period: T = 2π/ω Current and charge differ in phase by 90° qq t=+ max cos()ω θ iq t i t=− + =− +ω max maxsin sin(ωθ ωθ) ( ) k m ω= The expressions for the charge on, and hence voltage across a charging capacitor, and the current through the resistor, are derived in the 8.


volt R 5. After removing this battery from this capacitor, these two plates hold positive and negative charge for a certain time. 4. That is the rate of voltage rise across the capacitor will be lesser with respect to time.


At The electrochemical capacitor is a capacitor that employs the electrochemical reaction (oxidation-reduction or redox) to improve the energy density, and is categorized as “capacitor” in a wide sense, assuming it possesses rapid charge/discharge property. This is known as DECAY OF CURRENT. 2(b),sothatthebattery is no longer included in the circuit, we will discharge the capacitor. Thus this capacitor acts as a source electrical will flow across the dielectric, and the capacitor will eventually discharge.


Action 2 has the reverse effect; the separated charge flows away from one side of the capacitor via the circuit to the other side through the resistor and milliammeter. Charging (and discharging) of capacitors follows an exponential law. B) Capacitor B. Among the capacitors in the intermediary position A) Capacitor A.


Capacitor Charging and Discharging Experiment Parts and Materials. Time, t. The rate at which the charge moves, i. Ic = the positive ( charging current or negative of discharging current) current through capacitor.


In practice, we will solve only one circuit and try to understand it completely. Conclusion: In this experiment, charging and discharging of the capacitor with different resistors were observed. The objective of this experiment is the study of charging and discharging of a capacitor by measuring the potential difference (voltage) across the capacitor as a function of time. 4 F + +--a.


• The active circuit elements (transistors) add additional resistance in series. Using simply a milliammeter and stopwatch, you can study the changing currents which occur when a capacitor is charged or discharged in a simple circuit with a battery and resistor. – If the capacitor is charging, when fully charged no current flows and capppacitor acts as an o pen circuit. 10 5 Rc Circuits Physics Libretexts.


Charge And Discharge Of A Capacitor. RC Circuit 2 2. Charge/Discharge Behavior When a DC voltage is applied to a capacitor connected in series with a resistor, the capacitor begins to charge at a rate according to the applied voltage, the state of charge relative to its final value, the series resistance, and its own capacitance. As the current flows, the charge q is depleted, reducing the potential across the capacitor, which in turn reduces the current, the rate at which the charge is being depleted.


The capacitor becomes ‘charged’ by keeping the equal and opposite charges separated. Hi. 02T Study Guide, Section 7. These values can actually be measured at any time during the charging or discharging cycle, as long as one waits long enough for the capacitor voltage to increase or decrease by 63% of a measured value.


Rc Circuits And Discharging Capacitors With Equation Derivations. Vc = Voltage between terminals of the capacitor. RC Discharging Circuit Curves. ohm kHz 1000 .


Charge capacitor by transferring bits of charge dqat a time from bottom to top plate. Discharging of Capacitor: Or, Where, V = the voltage supplied to the capacitor which is zero in the case of discharging. c. In a time of one time constant, the charge q rises to 63% .


63% of its maximum, while the current i . q. More capacitors in series means higher voltage of the SC string with less capacitance. Larger capacitor (and /or R L) results in slower discharge and “flat” output giving rise to less ripple content.


The inductor is the electrical dual to the capacitor so we have and thus, the inductor current must be continuous and so, the current cannot discontinuously change. ) before the change. So the electrical power consumed by a capacitor can be positive or negative. In this discharging condition, with current exiting from the positive plate and entering the Discharging of capacitor through inductor Let us consider a charged capacitor of capacitance C is connected to a pure inductor of inductance L through a key K in series.


charge and then discharge the capacitor through the resistor. Calculate the capacitance of the capacitor based on your data and compare to the printed value on the capacitor. To discharge the capacitor. •Capacitors store energy in their electric fields •The force on a charge in E field E is F=qE •The work done moving a charge q across a potential V is qV •Lets treat a capacitor as a storage device we are charging •We start from the initial state with no charge and start adding charge until we reach the final state with charge Q .


Step-by-Step Science 110,660 views CHARGE AND DISCHARGE OF A CAPACITOR Capacitor Discharging Figure 3. Io is approx 80 A. Take readings at suitable intervals if manually obtaining data with a meter/stopwatch. i decays to 37% .


It has gotten 32182 views and also has 4. e. When trying to derive the equation for voltage across a discharging capacitor in series with a resistor using Kirchhoff's laws, I got stuck. Energy D In A Charged Capacitor Density Discharging of capacitor through inductor Let us consider a charged capacitor of capacitance C is connected to a pure inductor of inductance L through a key K in series.


Dunford: Calculating the Time Constant of an RC Circuit Produced by The Berkeley Electronic Press, 2010 After a certain time, the capacitor holds maximum amount of charge as per its capacitance with respect to this voltage. When a discharged capacitor is first connected to an external battery, or when a charged capacitor is first put into a circuit, it undergoes a charging or discharging process as the charges are moved in or out of the capacitor. 60 ms in this example), the charge rises to 63% . E.


The capacitor drains its voltage and current through the resistor. As the discharge continues, In this experiment a capacitor is charged and discharged and the time taken is recorded at equal intervals. Charging A Capacitor From An External Fluctuating Potential Using. R = 1400 V.


Capacitors are also easier to work with, so we will focus on them and then just point out how inductors behave by analogy. Click to expand That is the physics and it is true regardless of what sign convention you adopt for your circuit. The°owofcurrentthusservestoreducetheamount ofchargeonthecapacitor; byQ =CV,thismustreducethevoltageacrossthecapacitor. with the wires, and additional capacitance in parallel with the node.


Charging is process where the energy is being flow into the cell or rechargeable battery by forcing an electrical current through it. Capacitor charging equation derivation steps. The time = RCis known as the time constant. Ooh look, a differential equation.


This time span is called charging time of this capacitor. 1. 1. Capacitance Charging And Discharging Of A Capacitor.


C) They discharge at the same rate. PDF | The beauty of a diode lies in its voltage-dependent nonlinear resistance. Equation For A Charging Capacitor You. The value of the discharge time constant (C*RLoad) being very large, the capacitor ‘C’ will not have enough time to discharge properly.


The main goal was to charge up the capacitor. The goal with a discharging RC circuit is to find the charge on the capacitor at any time t . the current; this, of course, will depend on the resistance o ered. • Every junction (called nodes ) has capacitance to ground and other nodes.


If we define the initial charge to be – the charge when t = 0 so we now can replace giving: MY WORK Discharging. While the same total charge and energy can be stored, Inductor Charging and Discharging. D) Can’t say without knowing the initial amount of charge. Instead, the slope of the current changes discontinuously from increasing (the inductor is 'charging') to decreasing (the inductor is 'discharging') The voltage on a charging and discharging capacitor through a reverse-biased diode is calculated from basic equations and is found to be in good agreement with experimental measurements.


Current is negative because it flows opposite to reference direction Discharge transients last five time constants All voltages and currents are at zero when capacitor has fully discharged /W 0 t v C V e W W 0 / / 0 t C t R e R V i v V e C-C Tsai 22 Charging and discharging of a capacitor 67 o ) the capacitor gets discharged through the load. As soon as the capacitor starts discharging, the time becomes over. With the switch closed, the capacitor now starts to discharge as shown, with the decay in the RC discharging curve steeper at the beginning because the discharging rate is fastest at the start and then tapers off as the capacitor looses charge at a slower rate. The current through the resistor as a function of time is i(t)=−qmax τ(e− τ).


When the key K is switched on, the capacitor starts discharging through inductor. As out falls, at somepoint , driving M 1 into the triode region. (Fig 2. = (1400 )(4 F) = 5.


60 ms = 5. Van Bramer 1/15/98 Constants and Definitions: k 1000 . 2 Answers. it falls to 37% of its maximum value .


Thus the value of RLoad at the discharge time will also be high and have just a little less value that the output of RLoad. 63Vo, i. i. A foretaste .


Instead of using an actual step function, I'm going to use a DC input and assume the capacitor starts out discharged. max. µF Capacitance of Circuit Equations for Charging and Discharging RC Circuits t 0. 2 Voltage across capacitor as a function of time A decreasing capacitor voltage requires that the charge differential between the capacitor’s plates be reduced, and the only way that can happen is if the direction of current flow is reversed, with the capacitor discharging rather than charging.


It discharges through the resistor when the switch is at B. We note that for V out 2 (DD TH), the transistor can be viewed as a resistor equal to R on =[ n C ox (W=L)(V DD TH)] 1. 99% of the transition at t = 5RC . At Charging and discharging in RC Circuits - Example 1 (rising exponential) continued - Input node Output node ground R V in C + V out-time Vin 0 0 10 Vout 1nsec 6.


Introduction: A capacitor is a passive two-terminal electrical component used to store energy electrostatically in an electric field. During charging of condenser, current decreases exponentially and during an interval equal to time 0 constant the value of current falls to 1/e of its maximum value I , i. Capacitance values reaching up to 800 Farads in a single standard case size are available. Summary: Solving the Charging Differential equation for a Capacitor The charging capacitor satisfies a first order differential equation that relates the rate of change of charge to the charge on the capacitor: dQ Q1 dt R C =− ε This equation can be solved by the method of separation of variables.


That shows the charging time of the capacitor increase with the increase in the time constant RC. The voltage on each of the capacitors is the same, V B The charge on the capacitor C 1, Q 1 = C 1×VB The charge on the capacitor C 2, Q 2 = C 2×VB Total charge stored in both capacitors: QTot = Q The charge and discharge of a capacitor. In one time constant (5. This reverse biases the diode D and the capacitor C starts discharging through R L till the voltage across (which is the voltage at the cathode) it becomes less than Vs which is rising sinusoid.


Inductors also get charged and discharged, but it is current that is increased and decreased rather than charge. It will be seen, therefore, that the rate of energy transfer will depend on RC where C is the capacitance and R some e ective resistance in the circuit. Capacitance, Chapter Notes, Class 12, Physics (IIT-JEE & AIPMT) notes for Class 12 is made by best teachers who have written some of the best books of Class 12. Determine the DC state (current, voltages, etc.


•Capacitors store energy in their electric fields •The force on a charge in E field E is F=qE •The work done moving a charge q across a potential V is qV •Lets treat a capacitor as a storage device we are charging •We start from the initial state with no charge and start adding charge until we reach the final state with charge Q −t [NOTE: - This is the charging equation only, for discharging equation proceed the same way but only remove E from Kirchhoff law’s equation] Thus equation for discharging, (e ) −t RC V=V0 When RC=t, then equation becomes, 12 V=Vo ( 1−e ) −1 Which on solving gives V=0. 7V 10F capacitors versus one string of eight (in series) of the same capacitor. 6 rating. While the same total charge and energy can be stored, Supercapacitors also called ultracapacitors and electric double layer capacitors (EDLC) are capacitors with capacitance values greater than any other capacitor type available today.


Cell A cell is a charge giving energy source which consist of a positive and a negative terminal and there exists a potential difference between two terminals. 5. The electrochemical capacitor is a capacitor that employs the electrochemical reaction (oxidation-reduction or redox) to improve the energy density, and is categorized as “capacitor” in a wide sense, assuming it possesses rapid charge/discharge property. A MATHEMATICAL MODELING OF GROWTH & DECAY OF ELECTRIC CURRENT ATOKOLO .


If a resistor is also connected in series, it will resist the flow of the electrons through the circuit, and delay the charge's building up on the plates of the capacitor. File Capacitor Square Wave Charge Discharge Svg Wikimedia Commons. 2 Capacitors It’s the capacitor discharging – this current is equal to the rate of discharge of the capacitor, so: So substituting the above expression for I Ooh look, a differential equation. on the plates of the capacitor: the excess charges on one plate °ow ofi and neutralize the deflcitofchargesontheotherplate.


3 Answers. 1d) the circuit is complete Discharging of capacitor through inductor Let us consider a charged capacitor of capacitance C is connected to a pure inductor of inductance L through a key K in series. When the switch is closed, and charging starts, the rate of flow of charge is large (i. The term RC is the resistance of the resistor multiplied by the capacitance of the capacitor, and known as the time constant, which is a unit of time.


1a Chapter 14 CAPACITORS IN AC AND DC CIRCUITS So far, all we have discussed have been electrical elements in which the desired voltage at the top of the string. q is magnitude of charge on plates V= q/C V across plates dU= V dq increase in potential energy two ways to express the result Parallel connection of capacitors Capacitors C 1 and C 2 are connected in parallel: both terminal of each capacitor are connected to the same wires. My attempt was that the voltage gain across the capacitor should equal the voltage drop across the resistor, therefore q(t)/C = i(t)*R, or q(t) - RC*q'(t) = 0. C Range of time for graphing, from 0 to 5*RC stored in a capacitor is given by U = ½ CΔV2 = ½ Q2/C = ½ QΔV (all equivalent).


37% of its maximum value. Capacitor Wikipedia. The proposed SSC For the discharging process, τis equal to the time for V(t) to fall to 63% from its initial value. Can use a battery to do this.


63% of its maximum value (CV). Thepotentialdifierencewhichdrivescurrentsthusbecomessmaller,andsothecurrent°ow should reduce. Hz MHz 10 6. For instance, consider the choice of using two strings of four 2.


The propagation delay – that is the time it takes to remove the charge CLVDD/2 by PDF | The voltage on a capacitor discharging through a forward biased diode is calculated from basic equations and is found to be in good agreement with experimental measurements. Capacitor Charging From an Initial Voltage: E o is the initial capacitor voltage at t = 0 and t = 0 + From the derivation of notes #18a we found i c = i s. It is charged by connecting the switch to A. W, OMALE.


This will enable you to explore the effects of three quantities: applied potential difference, capacitor Before discussing series and parallel combination, it is important to know the charging and discharging of a capacitor. Symbols Appealing Capacitor Charging Equation Slide A Very Slowly. Consider the circuit which shows a capacitor connected to a d. Ripple factor: Ripple factor is a measure of effectiveness of a rectifier circuit Action 1 causes electric charge to flow onto one capacitor plate and off the other.


The charging and discharging of the capacitor causes ripple voltage in the output. Battery does work that increases potential energy of capacitor. The charge stored on the capacitor as a function of time is q(t)=qmaxe− τ. Circuits having capacitors: • At DC – capacitor is an open circuit, like it’s not there.


1) for a constant-current source is of course very simple since the voltage will change linearly with time. These instructions will use the notation τ=RC for the time constant of either a charging or discharging RC circuit. charge in a capacitor – higher capacitance means more charge can be stored using the same voltage V. One terminal of each capacitor should be marked with a definite polarity sign.


b. Usually, capacitors of the size specified to have a negative (-) marking or series of negative markings pointing toward the negative terminal. I. k Resistance of Circuit C 1.


We can apply Kirchhoff's second rule (conservation of energy) to get: V R + V C = V o or, using Ohm’s Law (V Discharging case. Chapter 14--Capacitors 521 FIGURE 14. Capacitors (1 of 11) in Series, Calculating Voltage, Charge and Total Capacitance - Duration: 8:37. Q.


Reference Designer Calculators RC Time Constant Derivation The circuit shows a resistor of value $R$ connected with a Capacitor of value $C$. Charging and Discharging of Capacitors. • Long term behavior of Capacitor: Current through aLong term behavior of Capacitor: Current through a Capacitor is eventually zero. When the power is negative, the capacitor is discharging, supplying energy to the circuit (the resistor receives the energy initially stored in the capacitor).


Fig. 1b standard symbol for a capacitor + - alternate symbol--a DC capacitor FIGURE 14. Objective: To investigate the charge and the discharge of a capacitor. Charge Capacitor 0.


As the value of time ‘t’ increases, the term reduces and the voltage across the capacitor reaches its saturation value. Solving a circuit with transient changes. the voltage on capacitor at time t=RC becomes 63% of the max voltage, which means 63% of total charge has been stored in the capacitor. Charging of an RC Circuit S.


Charging and discharging a load capacitor through MOSFET constant current sources. Now the charge stored on the capacitor is free to leave the plates and will cause a current to flow. Capacitor Charging Figure 4. The voltage on a charging and discharging capacitor through a reverse-biased diode is calculated from basic Charging and discharging in RC Circuits (Example 1 of the EE42 Easy Method) 1) Simplify the circuit : Find Vc (t) for the following circuit: (input switches from 2V to -1V at t = 0) 2) The time constant of the transient is τ = RC = 20nsec 1K Vin 2pF Vc +-2-1 9K 10K Vin 2pF Vc +-3) Before the transient Vin = 2V so Vc =2V 10K 2V Vc +- behave in circuits, by charging up and discharging a capacitor.


Charging the capacitor and timing how long it took to reach our target value of volts yielded the experimental value of RC. The general circuit equation is, 𝑉 + 𝑉𝐶 = 𝐸 1 𝐶 𝑉𝐶 + 𝑉𝐶 = 𝐸 (2) The general solution for a capacitor with an initial voltage on it is, 𝑉𝐶 = 𝐾 1 + 𝐾 2 When a capacitor is connected to a battery, charge is immediately stored on the plates of the capacitor. C 500 5. THE EXPONENTIAL The exponential voltage function, which is derived from equation (1), V(t) V (2) o e t-is shown in Figure 3.


2409 Discharging a Capacitor - 3 Saved: 1/5/16, printed: 1/5/16 As you see, the discharging of a capacitor is an exponentially decreasing process. It has a slope (rate of change) which is proportional to the value of the function (V) no matter where you are on the curve. 37 I. Derivation Of Charging And Discharging Equations For Rc Circuit Rc Circuits Charging Capacitors Discharing Capactiors And Derive Equation For Charge On A Capacitor Wmv You Charging Discharging Curves Equations For Inductor You Symbols Wonderful Capacitor And Inductor Video Charging Symbols Easy The Eye Capacitor And Inductor Video Time.


What I don't understand is why we are again taking ds = - dr as we already did that while modifying the below equation. R . Charging a lead acid battery typically have two tasks to accomplish which are to restore the capacity as quickly as possible and to maintain the capacity compensating or self discharge. Pdf Theory And Experiment On Charging Discharging A Capacitor.


Very large capacitors are often polarity-labeled by a positive (+) marking next to one terminal. For this, the circuit that we used included the resistor and the capacitor with the power supply. Capacitor Discharging Equations If a capacitor is charged to voltage V 0 and then discharged. Chapter 12.


To extend the charging process, the resistors were used. During charging electrons flow from the negative terminal of the power supply to one plate of the capacitor and from the other plate to the positive terminal of the power supply. 4. Derivation Of Charging And Discharging Equations For Rc Circuit.


This process however can take seconds, hours, days, weeks or months, depending on individual circumstances. TAP 125- 3: Charging and discharging capacitors . • Every wire in a circuit has resistance. For the discharging process, ˝ is equal to the time for V(t) to fall 63% from its initial value.


Hz µF 10 6. Let's go through this. Solving the differential equation in (4. Discharging the capacitor Suppose that with the capacitor fully charged, the switch is now closed in position B.


The device nevertheless continues to discharge C H until V out approaches zero. Complete the circuit by connecting the lead to the battery/power supply and observe the charging of the capacitor via the oscilloscope or meter. Charge and Discharge of a Capacitor INTRODUCTION Capacitors1 are devices that can store electric charge and energy. charging and discharging of capacitor derivation pdf

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