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Capacitor discharge time. Transients in DC circuits with a capacitor

TRANSITION The process of transition from one regime established in the chain to another is called. An example of such a process is the charging and discharging of a capacitor. In some cases, laws direct current can also be applied to changing currents, when the change in current does not occur too quickly. In these cases, the instantaneous value of the current strength will be practically the same in all cross sections of the circuit. Such currents are called quasi-stationary

DISCHARGE OF THE CAPACITOR. If the plates of a charged capacitor capacitance WITH close through resistance R, then a current will flow through this resistance. According to Ohm's law for a homogeneous section of the chain

IR= U,

Where I And U- instantaneous values ​​of the current in the circuit and the voltage on the capacitor plates. Taking into account that and , we transform Ohm's law to the form

In that differential equation the variables are separated, and after integration we obtain the law of change in the charge of the capacitor with time

Where q 0 is the initial charge of the capacitor, e is the base of the natural logarithm. Work RC, which has the dimension of time, is called relaxation time t . Differentiating expression (2) with respect to time, we find the law of current change:

, (3)

Where I 0 - the current strength in the circuit at a time t= 0. Equation (3) shows that t is the time during which the current in the circuit decreases in e once.

Time dependence of the amount of heat released on the resistance R when a capacitor is discharged, it can be found from the Joule-Lenz law:

CHARGING THE CAPACITOR.

We assume that initially the capacitor is not charged. At the point in time t = 0 the key was closed, and a current went through the circuit, charging the capacitor. Increasing charges on the capacitor plates will increasingly impede the passage of current, gradually reducing it. Let's write Ohm's law for this closed circuit:

.

After separation variable equation will take the form:

After integrating this equation, taking into account the initial condition

q = 0 at t = 0 and taking into account the fact that when the time changes from 0 to t charge changes from 0 before q, we get

, or after potentiation

q = . (4)

An analysis of this expression shows that the charge approaches its maximum value equal to C asymptotically at t ® ?.

Substituting into formula (4) the function I(t) = dq/ dt, we get

. (5)

It follows from the law of conservation of energy that when the capacitor is charged for any moment of time, the operation of the current source dA ist wound sum of the amount of Joule heat dQ, released on the resistor R and a change in the energy of the capacitor dW:

dAist= dQ + dW,

Where dAist = IDt, dQ = I 2 Rdt, dW = d. Then for an arbitrary moment of time t we have:

A ist (t)= = =C . (6)

Q(t)= =C . (7)

W(t) = = . (8)

METHOD AND PROCEDURE OF MEASUREMENTS:

In real DC electrical circuits containing capacitors, the transient processes of discharging and charging capacitors take place in the order of 10 -6 - 10 -3 s. In order to make available for observation and measurement of electrical parameters during transients in this computer model this time is significantly increased by increasing the capacitance of the capacitor.

EXPERIMENT 1

Determination of the capacitance of a capacitor by the discharge method


1. Assemble a closed electrical circuit on the working part of the screen, shown below in Fig. 2. To do this, first click on the emf button located in the right part of the experiment window. Move the mouse marker to the working part of the screen where the dots are located, and click the mouse marker in the form of an extended index finger in the place where the current source should be located. Move the mouse marker to the slider of the emf controller that appears, press the left mouse button, keeping it pressed, change the emf value. and set to 10 V. Connect 4 other current sources in the same way. The total value of emf batteries must match the value shown in table 1 for your option.

In the same way, place further on the working part of the screen 7 lamps L1-L7 (button), Key K (button), voltmeter (button), ammeter (button), capacitor (button). All elements electrical circuit connect according to the scheme of Fig. 1 using mounting wires (button ).

2. Click on the "Start" button. The L7 lamp should light up, and the inscription on the button should change to “Stop”. Close the K key with the mouse cursor.

3. Once established in the circuit stationary current(lamps L5 and L6 should go out and lamps L1-L4 should light up) record the readings of electrical measuring instruments in table 2.

4. Click on the "Stop" button and open the K key with the mouse cursor.

5. With two short mouse clicks on the "Start" button, start and stop the process of discharging the capacitor. The ammeter reading will correspond to the initial discharge current of the capacitor I 0 . Record this value in Table 3.

6. Close the key again, charge the capacitor and repeat steps. 5, 6 4 more times.

7. For each experience, calculate It= I 0 / 2.7 - the current strength that should be in the capacitor discharge circuit after the relaxation time t and write these values ​​\u200b\u200bin table 3.

8. With the key open, by pressing the "Start" button, start the process of discharging the capacitor and at the same time turn on the stopwatch.

9. Carefully observe the ammeter reading as the capacitor discharges. Stop the stopwatch and simultaneously press the "Stop" button when the ammeter reading is equal to or close to I t . Record this value of time t 1 in table 3.

meaning

I 0 , A

It, A

t, With

Table 3. Results of measurements and calculations.

PROCESSING THE RESULTS:

1. According to Ohm's law for the circuit section L1-L4: and the measurement results given in Table 2, determine the resistance of one lamp.

2. According to the formula (when discharging a capacitor, a quasi-stationary current flows through 6 lamps connected in series), determine the capacitance of the capacitor and write these values ​​​​in table 3.

3. Calculate the measurement errors and formulate conclusions based on the results of the work done.

EXPERIMENT 2

Study of the time dependence of the amount of heat released on the load during the discharge of the capacitor

  1. Performing actions similar to those described in experiment 1, charge the capacitor to a voltage corresponding to the total emf value. for your option.
  2. Press the "Stop" button and turn off the K key.
  3. Run a 5 second process partial discharge capacitor through connected lamps. To do this, simultaneously press the "Start" button and the stopwatch start button, and after 5 seconds, by pressing the "Stop" button, stop the process of discharging the capacitor.
  4. Record the ammeter reading in Table 4 and charge the capacitor again to its original voltage.
  5. By successively increasing the duration of the process of discharging the capacitor by 5 s, perform these experiments until the discharge time corresponds to the complete disappearance of the charge on the capacitor. (The voltage on the capacitor and the discharge current through the lamps should be close to zero). Record the results of the discharge current measurements in the appropriate cells of Table 4.

Table 4. Results of measurements and calculations

PROCESSING THE RESULTS:

EXPERIMENT 3

Checking the law of conservation of energy in the process of charging a capacitor through resistance


Fig.3

  1. Assemble the circuit shown in Figure 3 in the working part of the experiment screen. A voltmeter connected in parallel with 5 lamps will show the voltage across the external resistance, and an ammeter will show the current through the load and current sources. The voltage on the capacitor is automatically determined by the program and indicated in volts on the monitor screen above the capacitor.
  2. Set the total emf. current sources corresponding to the value given in Table 1 for your option.
  3. With the key K open, press the "Start" button.
  4. By pressing the mouse button, close the key K and start the process of charging the capacitors. Simultaneously with the closing of the key, turn on the stopwatch.
  5. Through relaxation time t = RWITH by pressing the "Stop" button, stop the process and record the readings of electrical measuring instruments in table 5.
  6. Press the "Select" button and reset the voltage readings on all capacitors and electrical meters.
  7. Repeat these measurements 4 more times and fill in the top two rows of Table 5.

Table 5. Results of measurements and calculations

experience number

I, A

U R, B

A ist, J

DW, J

Q, J

PROCESSING THE RESULTS:

  1. According to formulas 6, 7, 8 and the measured values ​​​​of the voltage on the capacitor U c calculate the work of the current source A ist, the change in the energy of the capacitor DW and the amount of heat released on the load Q after a charge time equal to the relaxation time.
  2. Check the fulfillment of the law of conservation of energy in the process of charging the capacitor using the formula: A ist =DW + Q.
  3. Make conclusions based on the results of the work.

Questions and tasks for self-control

Questions and tasks for self-control

EXPERIMENT 14 RC Circuit Time Constant

Goals

After conducting this experiment, you will be able to demonstrate how the values ​​of capacitance and resistance control the time to charge and discharge a capacitor.

Required accessories

* Digital multimeter

* Dashboard

* Constant voltage source

* Stopwatch or watch with second hand

* Items:

one 22uF electrolytic capacitor, one 100uF electrolytic capacitor, one 33kΩ 1/4W resistor,

* one 100kΩ 1/4W resistor, one 220kΩ 1/4W resistor, one 1MΩ 1/4W resistor.

INTRODUCTION

A capacitor is an electrical element that stores electricity in the form of an electric field. When a constant voltage is applied to a capacitor, electrons leave one plate of the capacitor and accumulate on the other plate under the action of

external tension force. This causes the capacitor to charge up to a voltage equal to the applied voltage.

A positive charge on one side of the capacitor and a negative charge on the other side of the capacitor create a strong electric field between plates in a dielectric. This charge is retained even if the voltage source is disconnected. The capacitor can be discharged by connecting its terminals to each other to neutralize the charge on the plates.

Charging and discharging a capacitor to a certain voltage takes a finite period of time (called the time constant); this time depends mainly on the capacitance of the capacitor and the series resistance. The charge time constant is the time it takes the capacitor to charge to 63.2% of the applied voltage. This time (T) in seconds is expressed as follows:

The discharge time constant is the time it takes the capacitor to discharge to 36, 8% of the initial charge.

The time it takes for a capacitor to fully charge to an applied voltage or fully discharge to zero is approximately five times the time constant, i.e. 5T.

Summary

Many electronic circuits are based on the idea of ​​using the time constant for their work. Such circuits include, for example, time delay circuits, pulse and signal shaping circuits, and oscillator circuits. In this experiment, you will get acquainted with the charge and discharge time constant using three various groups resistors and capacitors.

PROCEDURE

Charging process

Resistor 100 kΩ; capacitor 100uF

1. Assemble the circuit shown in Figure 14-1. Observe polarity when connecting electrolytic capacitor.



Rice. 14-1.

2. Adjust the power supply to 12V.

3. Calculate the amount of voltage that will appear across the capacitor for one time constant.

Voltage (T) = ______ V

4. Calculate the time constant using the values ​​shown in Figure 14-1. Record your result in column 3 in Figure 14-2. Calculate also the value of the time it will take the capacitor to fully charge (5T). Record your result in column 4 in Figure 14-2.



Rice. 14-2.

5. Connect the test leads of your multimeter, observing polarity, to the capacitor leads. The multimeter should show 0 V. If this is not the case, there is some residual voltage on the capacitor plates. Remove it by briefly shorting the capacitor leads to each other for a few seconds. Measure the voltage again with your multimeter to verify that the capacitor voltage is zero.

6. Leave the test leads of the multimeter on the capacitor leads, connect the free end of the 100 kΩ resistor to the + 12 V terminal of the power supply. At the time of joining

start your stopwatch or start timing with the second hand of your watch. As the voltage across the capacitor begins to rise, note its magnitude. When the voltage across the capacitor reaches the value you calculated in step 2, note the time on the stopwatch or second hand. Record this value as the measured time constant in column 5 of Figure 14-2.

NOTE: Repeat this step a few times to make sure your timing is relatively accurate. After all, you are trying to watch both the voltmeter and the stopwatch to determine the time it takes to reach a particular voltage level. This is a rather tricky operation, so repeat it several times for more accurate measurements. ATTENTION:

if you need to repeat the experiment, remove the 10kΩ resistor and fully discharge the 100uF capacitor before each additional measurement. 7. Discharge the capacitor completely again and reconnect the test leads. Touch the free end of the 100 kΩ resistor to the +12 V terminal of the power supply. This time, measure the time it takes for the capacitor to fully charge to the applied voltage you measured in step 1. As before, start timing with the stopwatch or clock second hand the moment you apply voltage to the resistor. Record this measured time,

required for the capacitor to be fully charged in column 6 of Figure 14-2.

Resistor 11 k0m; capacitor 22uF

8. Repeat steps 4 through 7 using a 22µF capacitor and a 100k0m resistor. Complete the fields in the table in Figure 14-2 as you did before. Your calculated and measured values.

Resistor 220 k0m; capacitor 100uF

9. Repeat steps 4 through 7 again, but this time use a 100uF capacitor and a 220k0m resistor. Record your calculated and measured values ​​in the table in Figure 14-2.

Observation

10. Looking at the information in Figure 14-2 and noticing the different times obtained with different values resistance and capacitance, draw your own conclusions regarding the effect of resistance and capacitance values ​​on the time constant.

Discharging process

Resistor 100 k0m; capacitor 100uF

11. Rearrange the circuit to match the circuit shown in Figure 14-3. Observe the polarity when connecting the electrolytic capacitor. In this part of the experiment, you will demonstrate the process of discharging a capacitor. To do this, connect a resistor in parallel with the capacitor.



Rice. 14-3.

12. Calculate the time constant of the circuit and the time it takes for the capacitor to fully discharge, and record your data in column 3 of Figure 14-4.



the power source you measured in step 1. Calculate the amount of voltage that will be present on the Capacitor after it has been discharged for one time constant.

Voltage (t) = _______V

Resistor 100 kΩ; capacitor 22uF

14. Connect the test leads of your multimeter to a 22 microfarad capacitor. At this time, the voltage should be zero, since any charge on the capacitor plates has been eliminated by discharging the capacitor through the 1 MΩ resistor. Connect the circuit to the +12V terminal of the power supply. The capacitor charges immediately to the power supply voltage; there is no resistance connected in series with the capacitor.

15. Continue to fix the test leads of the multimeter in parallel with the capacitor leads. Remove the connecting wire from the + 12 V terminal of the power supply. Simultaneously with the removal of the wire, start counting the time on your stopwatch or on the second hand of the clock. Observe the voltage at the capacitor terminals. When the voltage reaches the desired value, note the time. Record the time constant in column 5 of the table in Figure 14-4. As before. You may wish to repeat steps 13 and 14 several times to improve measurement accuracy. After all, since you have to observe two values ​​at the same time, the measurement is quite tricky. By averaging several readings, you will get greater accuracy in the measurement.

Resistor 220 kOhm; capacitor 22uF

16. Repeat steps 12 through 15 again, but this time use a 22 microfarad capacitor and a 220 kΩ resistor. Calculate discharge times again for one time constant and for five time constants. Record all your data in the table in Figure 14-4.

Observation

17. Looking at the information in Figure 14-4 and noticing the different times obtained at different resistance and capacitance values, draw your conclusion regarding the relationship between the discharge time and the resistance and capacitance values.

18. Based on the comparison of your calculated and measured values, explain possible inconsistencies.

REVIEW QUESTIONS

1. It takes the same time to fully charge the capacitor as it takes to fully discharge it:

a) the statement is true

b) the statement is false.

2. To what voltage will a 5 microfarad capacitor be charged through a 10 kΩ resistor in one time constant when it is connected to a 6 V power source?

3. How long does it take for the capacitor in question 2 to fully discharge?

4. The capacitor takes 80 milliseconds to fully charge. So the time constant is:

5. For given values ​​of R (resistance) and C (capacitance), the capacitance is doubled, and the resistance is halved, while the time constant is:

a) remains the same

b) doubles

c) quadruples

d) doubled.

Using Peukert's law, you can find out the discharge efficiency of a battery. The German scientist Wilhelm Peikert (1855-1932) found that the available battery capacity decreases with increasing discharge rate and developed a formula to calculate the value of these losses. This formula is mainly applied to the lead-acid electrochemical system, helping to estimate the time battery life at various discharge loads.

Peikert's law takes into account internal resistance and battery recovery processes. The resulting value close to one (1) will indicate a good battery condition, with normal efficiency and minimal losses; received greater value reflects the reduced efficiency of the power supply under test. Peukert's law is exponential, standard values ​​for a lead-acid electrochemical system are between 1.3 and 1.5 and increase with age. The values ​​obtained are also influenced by temperature indicators. Figure 1 shows the available capacity depending on the strength of the discharge current of batteries with different values Peukert numbers.

For example, a 100Ah lead-acid battery discharged at 15A should theoretically deliver power for 6.6 hours (100Ah divided by 15A), but the actual time will be less. With a Peukert number of 1.3 digits, the time would be about 4.8 hours.

Figure 1: Available battery capacities with Peukert numbers from 1.08 to 1.50. A value close to 1 indicates the least internal loss, higher values ​​indicate a noticeable reduction in capacitance. The value of the Peukert number depends on the type and age of the battery, as well as on temperature. environment. Average values ​​of the Peukert number different types lead-acid batteries: AGM: 1.05 - 1.15; gel: 1.10 - 1.25; flooded: 1.20 - 1.60.

2. Ragon Plot

Nickel- and lithium-based batteries are usually rated using a Ragon chart. Named after David W. Ragon, this graph shows the relationship between battery capacity in watt-hours (Wh) and discharging power in watts (W). The big advantage of Ragon's chart over Peukert's law is the availability of battery life in minutes and hours; a specific diagonal line on the chart is responsible for each time value.


Figure 2: Ragon plot for 18650 Li-ion cells. Comparing discharge power and energy versus time. Not all curves are fully extended.

Key: A123 APR18650M1 is a 1,100mAh lithium iron phosphate (LiFePO4) battery rated for 30A continuous discharge current. Sony US18650VT and Sanyo UR18650W are 1,500mAh lithium manganese cells rated for 20A continuous discharge. Sanyo UR18650F is a capacity optimized cell (2,600mAh) with a moderate discharge current of 5A. This cell has the highest discharge energy but the lowest power.

Sanyo UR18650F has the highest energy density and can be used as a power source for a laptop or e-bike for several hours at moderate load. The Sanyo UR18650W, by comparison, has a lower power density but can deliver 20A. The A123 LFP technology has the lowest power density but delivers the highest power rating of 30A continuous current. Specific energy intensity implies the ratio of battery capacity to its weight (Wh / kg); the energy density is related to the volume (W * h / l).

The Ragon Plot can help in selecting the optimal Li-Ion system that meets the required discharge power requirements while maintaining the desired run time. If a high discharge current is needed, then the 3.3 minute diagonal line will point to A123 (battery 1). The A123 can provide up to 40W of power for 3.3 minutes. Sanyo F (battery 4) is somewhat weaker, and for the same time of 3.3 minutes it can already provide 36 watts. Focusing on battery life, let's analyze the 33-minute diagonal. A123 (battery 1) will provide 5.8 watts of power during this time before the energy is depleted. The Sanyo F (battery 4), which has a higher capacity, is capable of delivering approximately 17 watts in the same time.

But it should be taken into account that the Ragon plot shows the characteristics of new elements, a condition that, unfortunately, is temporary. When calculating the power and energy demand, degradation processes arising from cyclic operation and aging should be taken into account. Devices and systems that use batteries must be designed for some gradual degradation in the performance of their power supplies - down to about 70-80 percent of the original capacity. Another factor affecting battery performance is low temperature. In Ragon's schedule, this issue is not taken into account.

Structurally accumulator battery must be strong and resistant to regular use. Excessive range expansion allowable loads and the amount of capacity available leads to increased wear and eventually significantly reduces the life of the battery. If the requirements for regular high discharge currents are put forward, then the battery system must also be selected to meet these requirements. An analogy would be a comparison between a diesel truck and a supercharged sports car. With approximately the same power, these vehicles designed for absolutely different areas applications. This comparison applicable to batteries, the variety of characteristics of which determines the nuances of their operation.

The Ragon plot can also be used to calculate the power requirements of other power sources such as capacitors, flywheels, flow accumulators and fuel cells. But for internal combustion engines and fuel cells using fuel from a tank, this schedule is not applicable, since it does not take into account separately supplied fuel. Similar graphs are also used to find the optimal characteristics of renewable energy sources, such as solar panels and wind turbines.