ClassX Video Lessons Youtube Channel The Engineering Mindset Capacitor charge time calculation – time constants
Imagine you have a simple electrical circuit consisting of a 9-volt battery, a 100-microfarad capacitor, a 10-kilohm resistor, and a switch, all connected in series. Initially, the capacitor is fully discharged, meaning it has 0 volts across its leads. When you close the switch, the capacitor begins to charge, and its voltage gradually increases until it matches the battery’s voltage. However, this increase doesn’t happen instantly; it follows an exponential curve. At first, the voltage rises quickly, then slows down as it approaches the battery’s voltage.
The charging process can be divided into segments known as time constants. In this context, we focus on the first five time constants because, by the fifth, the capacitor is nearly fully charged. It takes five time constants for the capacitor to charge from zero to almost 100% of the battery voltage.
The time constant is calculated using the formula: Time Constant (seconds) = Resistance (ohms) × Capacitance (farads). For our circuit, the resistor is 10,000 ohms, and the capacitor is 0.0001 farads (100 microfarads converted to farads). Multiplying these gives a time constant of 1 second. Therefore, it takes 5 seconds for the capacitor to charge to 9 volts.
If the resistor’s value were reduced to 1,000 ohms, the time constant would decrease to 0.1 seconds, meaning the capacitor would charge in just 0.5 seconds. Conversely, if the capacitor’s capacitance were increased to 1,000 microfarads, the total charge time would extend to 50 seconds. Thus, larger capacitors or resistors result in longer charge times.
Let’s examine the voltage across the capacitor at each time constant:
Although the voltage never quite reaches 100%, these points are close enough for practical purposes. For example, after 1 second, the capacitor voltage is approximately 5.68 volts; after 2 seconds, it’s 7.78 volts; after 3 seconds, it’s 8.55 volts; after 4 seconds, it’s 8.83 volts; and after 5 seconds, it’s 8.94 volts.
When the capacitor discharges, the voltage decreases following a similar exponential curve. If you disconnect the battery and provide a path for discharge, the voltage will drop over time. For instance, with a 9-volt battery, a 500-ohm resistor, and a 2,000-microfarad capacitor, the time constant is 1 second. Initially, the capacitor holds 9 volts, but after 1 second, the voltage drops to 36.8% (about 3.12 volts). After 2 seconds, it’s 1.215 volts; after 3 seconds, it’s 0.45 volts; after 4 seconds, it’s 0.162 volts; and after 5 seconds, it’s 0.063 volts.
In this scenario, if the resistor were a lamp, it would shine brightly at first and then gradually dim as the capacitor discharges.
Understanding time constants and how they affect capacitor charge and discharge times is crucial in electronics engineering. For more insights into electronics, explore additional resources and videos available online.
Use a circuit simulation software like LTSpice or Tinkercad to model the given circuit with a 9-volt battery, a 100-microfarad capacitor, and a 10-kilohm resistor. Observe the charging curve and verify the time constant calculation by measuring the voltage across the capacitor at each second. This hands-on experience will help you visualize the exponential charging process.
Calculate the time constants for different resistor and capacitor values. Create a table comparing the time constants and the corresponding charge times. Discuss how these changes affect the charging process and the practical implications in real-world circuits.
In small groups, discuss how changing the resistor or capacitor values impacts the circuit’s behavior. Consider scenarios where quick charging is essential versus situations where a slower charge is beneficial. Share your findings with the class to deepen your understanding of component selection in circuit design.
Plot a graph of the capacitor voltage over time for the first five time constants. Use software like Excel or Google Sheets to create the graph. Analyze the curve and annotate key points, such as the voltage levels at each time constant, to reinforce the concept of exponential growth in charging.
Set up a simple experiment to discharge a capacitor through a resistor. Measure the voltage drop over time and compare it to the theoretical values. Document your observations and reflect on how the discharge process mirrors the charging process, reinforcing the concept of time constants in both scenarios.
Here’s a sanitized version of the provided YouTube transcript:
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Let’s say we have a 9-volt battery, a 100-microfarad capacitor, a 10-kohm resistor, and a switch, all in series. The capacitor is fully discharged, and we read 0 volts across the two leads. When we close the switch, the capacitor will charge, and the voltage will increase until it matches the battery voltage. The voltage increase is not instant; it follows an exponential curve. Initially, the voltage increases rapidly and then slows down until it reaches the same voltage level as the battery.
We can split this curve into six segments, but we’re only interested in the first five because at the fifth marker, we are essentially at full voltage. Each segment represents something called a time constant. Therefore, with five segments, we have five time constants. It will take five time constants to charge the capacitor from zero to just under 100%.
To calculate the time constant, we use the formula: time constant in seconds equals the resistance in ohms multiplied by the capacitance in farads. We convert our resistor to ohms and our capacitor value to farads. In this case, 10,000 ohms multiplied by 0.01 farads equals 1, so the time constant is 1 second. Therefore, five of these is 5 seconds, meaning it takes 5 seconds for the capacitor to fully charge to 9 volts.
If the resistor were just 1,000 ohms, the time constant would be 0.1 seconds, so it would take 0.5 seconds to reach 9 volts. If the capacitor were 1,000 microfarads, it would take 50 seconds in total. As the capacitor size increases, the time taken will also increase. Similarly, if the resistor value increases, the time taken also increases.
Returning to our original circuit, we can calculate the voltage level at each time constant:
– At 0 seconds, the voltage is always 63.2%
– At 1 second, it is 86.5%
– At 2 seconds, it is 95%
– At 3 seconds, it is 98.2%
– At 4 seconds, it is 99.3%
So the voltage will never actually reach 100%. That’s why we stop at just five points. In this example, after 1 second, the capacitor voltage is 5.68 volts; after 2 seconds, it’s 7.78 volts; after 3 seconds, it’s 8.55 volts; after 4 seconds, it’s 8.83 volts; and after 5 seconds, it’s 8.94 volts.
If you need a more precise answer, we could calculate each point like this. Remember, because this is in series, the current of the circuit decreases while the voltage of the capacitor increases. Once at full voltage, no current will flow in the circuit. If the resistor were a lamp, it would instantly reach full brightness when the switch was closed, but then become dimmer as the capacitor reaches full voltage.
When we provide a path for the capacitor to discharge, the electrons will leave the capacitor, and the voltage will reduce. It doesn’t discharge instantly but follows an exponential curve. We can split this curve into six segments, but again, we’re only interested in the first five.
For example, if we had a 9-volt battery, a lamp with a resistance of 500 ohms, and a 2,000 microfarad capacitor, our time constant would be 500 ohms multiplied by 0.2 farads, which equals 1 second. At the moment the battery is disconnected, the capacitor will be at 9 volts, and as it powers the circuit, the lamp will also experience 9 volts.
After one time constant (1 second), the voltage will be 36.8%, which is approximately 3.12 volts. At 2 seconds, it will be 1.215 volts; at 3 seconds, it will be 0.45 volts; at 4 seconds, it will be 0.162 volts; and at 5 seconds, it will be 0.063 volts. So the lamp will be illuminated for just under 3 seconds, obviously becoming dimmer towards the end of that time.
Thank you for watching! Check out one of the videos on screen now to continue learning about electronics engineering. Don’t forget to follow us on social media and visit the engineering mindset website for more resources.
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This version removes any informal language and maintains a professional tone while conveying the same information.
Capacitor – A device used in electrical circuits to store charge temporarily, consisting of one or more pairs of conductors separated by an insulator. – The capacitor in the circuit was used to smooth out fluctuations in the power supply.
Charge – A fundamental property of matter that causes it to experience a force when placed in an electromagnetic field, measured in coulombs. – The charge on the electron is negative, which affects how it interacts with electric fields.
Time – A scalar quantity representing the progression of events from the past to the future, often used in physics to describe the duration of events or the intervals between them. – The time constant of the RC circuit determines how quickly the capacitor charges and discharges.
Constant – A quantity that remains unchanged under specified conditions, often used to describe fixed values in equations or systems. – Planck’s constant is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency.
Resistance – A measure of the opposition to the flow of electric current in a conductor, typically measured in ohms. – The resistance of the wire increased as its temperature rose, affecting the overall current in the circuit.
Voltage – The electric potential difference between two points, which causes current to flow in a circuit, measured in volts. – The voltage across the resistor was measured to ensure it did not exceed the component’s rating.
Discharge – The process by which a capacitor releases its stored charge, often resulting in a flow of current. – The rapid discharge of the capacitor provided the necessary power to start the motor.
Circuit – A closed loop or pathway that allows electric current to flow, typically consisting of various electrical components. – The engineer designed a complex circuit to control the robotic arm’s movements precisely.
Engineering – The application of scientific and mathematical principles to design, build, and analyze structures, machines, and systems. – Engineering students often work on projects that require them to apply their theoretical knowledge to practical problems.
Electronics – The branch of physics and engineering that deals with the study and application of electronic devices and circuits. – The course on electronics covered topics such as semiconductor devices and digital circuit design.
