Capacitance is the storing ability of a capacitor, which is measured in Farad. Capacitors are frequently used to store electrical energy and release it when needed. In combination with other circuit components, capacitors are employed to create a filter that allows some electrical impulses to flow while blocking others. Many modern devices, such as pacemakers, mobile phones, or computers, use capacitors as key components of electrical circuits.
How does a capacitor charge?
In conductive materials, there are many negatively charged electrons that create the electrical current. These electrons can easily move around in the electric field and break away from the atom. In insulator materials, however, electrons occur only in very small numbers, and as they are strongly bonded to the atomic nucleus, they can’t break away from the atom easily.
This is important to know in order to understand how a capacitor charges, as the capacitor’s charging ability comes from the electric field that is pushing or pulling the electrons. The capacitor becomes charged when positive and negative charges merge on the opposite capacitor plates.
Figure 1. Diagram of a charged capacitor.
The positive and negative charges on the plates attract but never reach each other, so these opposite charges are constantly pushing and pulling each other in an electric field between two conductive plates, allowing a capacitor to maintain its charge.
The insulator between the two plates holds this ‘charge’. In practice, there are small leakage currents going through insulators. So, how long a capacitor can hold a charge depends on the quality of the insulator. This ‘charge’ is actually the potential energy difference between the two plates, which comes from the voltage difference between the two ends.
How does a capacitor’s charge behave in AC and DC circuits?
Let us now explore the differences in how a capacitor charges in DC circuits compared to its charging behaviour in AC circuits.
A capacitor’s charging behaviour in DC circuits
To understand how a capacitor works and how its charge behaves in DC circuits, take a look at the basic circuit below.
Figure 2. A simple capacitor charge circuit.
When the switch is in position 2, there is no voltage being applied to the capacitor and thus no electric field. The electrons in the conductive plates are stationary, and the plates don’t charge with a positive or negative charge. The potential difference between them, therefore, is zero, and the voltmeter reads the value 0.
When you move the switch to position 1, you will see that the ammeter’s pointer moves up before quickly going back down. This is because there is an electron movement when the switch is moved to position 1.
By means of the force of the electric field, the DC supply’s positive pole pulls the electrons in the upper conductive plate while the negative pole pushes the electrons to the bottom conductive plate. The upper plate charges positively, having lost electrons, while the bottom plate charges negatively, having gained electrons.
Figure 3. A capacitor’s two opposite plates charged with the opposite charges.
There is now a potential difference between the two plates of the capacitor, which is in the opposite direction of the DC potential. So, how do the values read by the ammeter and voltmeter change? Take a look at the scatter charts below.
Figure 4. Scatter charge of the voltage value of the capacitor during the time period.
The period during which a capacitor is charging is called the temporary state. It is during this period that the ammeter’s pointer moves up and then back down again. When the capacitor is fully charged, it has reached the steady state. At this point, the voltmeter reads V, which is the value of the DC supply’s voltage.
Figure 5. Scatter charge of the current value of the capacitor during the time period.
The reading of the ammeter value is the opposite of the voltage value. The reason for this is that the capacitor is charging in the temporary state, so the current continues to go through it.
As it charges, the potential difference between the capacitor plates rises, approaching the DC supply’s potential difference. As it gets closer, the current begins to decrease because the potential difference between the DC supply and the capacitor is decreasing. When the capacitor is fully charged, it enters the steady state, and the potential differences of the DC supply and the capacitor are the same.
The electrical load a capacitor can store in a DC circuit is:
\[Q = CV\]
Here:
- V = the voltage applied to the capacitor.
- C = the capacitance of the capacitor.
- Q = the electrical load of the capacitor.
A capacitor’s charging behaviour in AC circuits
A capacitor in an AC circuit behaves differently. While in DC circuits, the capacitor’s plates charge positively and negatively only once, in AC circuits, its value changes continuously, depending on the AC supply. The current flow also differs. In DC circuits, the current flows in one direction until the capacitor is charged when the current stops its flow. In AC circuits, the current flow is continuous, and it flows in both directions.
To understand the concept of a capacitor charging in an AC circuit, we need to look at the process in different parts of a charging period.
Figure 6. A capacitor’s current and voltage have a 90-degree phase difference in AC circuits.
We are going to look at the behaviour of the circuit in 4 different parts of a charging period. These parts are for an angle named a between 0 - π/2, π/2 - π, π - 3π/2, and 3π/2 - 2π.
0 < a < π/2
Figure 7. A capacitor’s charge in AC current (Diagram 1).
When you close the switch at the time t = 0, the capacitor begins to charge. Because the voltage is changing at a high rate, there is a high electron flow, which means that the current is at its maximum level. As we get closer to π/2, the capacitor’s voltage is getting closer to Um (the AC source’s peak value), the electron flow is decreasing, and the current is also decreasing.
π/2 < a < π
At a = π/2, the value of the current is zero, and the voltage of the capacitor is at its maximum value (V = Vm). The capacitor’s load, therefore, is at its maximum level as well: q = Qm = Vm ⋅ C, where q is the load, Qm is the maximum load, Vm is the AC source’s peak value, and C is the capacitance.
Figure 8. A capacitor’s charge in AC current (Diagram 2).
After the a = π/2 point, because the AC source’s voltage value is decreasing, the capacitor’s voltage is also decreasing. This also means that the capacitor’s load is going to decrease as well, which means that the extra electrons in the bottom plate are going to move to the upper plate. This is the reason behind the change in the current’s direction. As we move towards the a = π point, the AC source’s voltage begins to change rapidly, causing the value of the current to increase.
π < a < 3π/2
At the a = π point, because the voltage changes direction, the pace of the change (dV/dt) and the value of the current will be at their maximum levels. But what is the connection between these two? Take a look at the equation below for the current going through the capacitor.
\[i(t) = C \cdot \frac{dV}{dt}\]
Although it includes differentiation, the explanation is pretty simple. The current going through the capacitor is directly proportional to its capacitance value and how fast the voltage changes in time.
Figure 9. A capacitor’s charge in AC current (Diagram 3).
After the a = π point, the capacitor’s voltage begins to increase as the AC source voltage increases. The electrons in the bottom plate are being pulled by the source, while extra electrons are moving to the upper plate. As we move towards the a = 3π/2 point, because the pace of the change of voltage decreases and the voltage of capacitor approach -Vm, the value of the current decreases.
3π/2 < a < 2π
At the a = 3π/2 point, since the voltage of the capacitor is at its maximum level, the load is at its maximum value as well: q = Qm = Vm ⋅ C, where q is the load, Qm is the maximum load, Vm is the AC source’s peak value, and C is the capacitance. The capacitor is fully charged, so there will be no current going through it at this exact point. The current, therefore, is i = 0.
Figure 10. A capacitor’s charge in AC current (Diagram 4).
After the a = 3π/2 point, the voltage of the source decreases, which means that the voltage of the capacitor is going to decrease as well. As we move towards the 2π point, the pace of the change of the voltage (dV/dt) increases, and so does the current. At the 2π point, the value of the current is at its maximum, and the value of the AC source’s voltage is 0. The load of the capacitor (q) is also zero because it has discharged at this point.
Lastly, let’s explore the connection between the capacitor’s current, capacitance, maximum voltage (Vm), and maximum current (Im).
We know that V = Vm ⋅ sin (wt + θ) and θ is the phase difference (if any) of the AC source’s wave, w is the angular velocity, Vm is the peak value of the voltage, and t is time in seconds. If you combine this with the equation for finding the current going through the capacitor, you get:
\[i(t) = C \cdot \frac{dV}{dt} = C \cdot \frac{d(V_m \sin(wt))}{dt}\]
This gives you:
\[i(t) = C \cdot w \cdot V_m \cos(wt) = \frac{V_m}{\frac{1}{w \cdot C}} \cos(wt)\]
And from the previous equation, we get:
\[C \cdot w \cdot V_m = I_m\]
Putting Im into the last equation, we then get:
\[i(t) = I_m \cos(wt)\]
Capacitor Charge - Key takeaways
- The positive and negative charges on each of the conductive plates in a capacitor attract but never reach each other, so these opposite charges are constantly pushing and pulling each other in an electric field between two conductive plates, allowing a capacitor to maintain its charge.
- There is a difference in how a capacitor charges in various circuits since the voltage levels are steady in DC but constantly changing in AC.
- In DC circuits, the period of time during which a capacitor is charging is called the temporary state. When the capacitor is fully charged, it has reached the steady state.
- In DC circuits, the capacitor gets charged once and then doesn’t allow any further current to go through it. In AC circuits, the capacitor’s voltage, current, and load are constantly changing.
- In AC circuits, the current going through a capacitor is directly proportional to its capacitance value and how fast the voltage changes in time.
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