Capacitors in D.C.

Capacitors - National 5 Recap

A Capacitor is a device for storing Electrical Charge. A Capacitor consists of two parallel plates separated by an insulator. The symbol for a Capacitor is shown below :-

Once connected within the circuit, negative charge builds up on one of the electrodes, causing a positive charge to build up on the other electrode. This causes Energy to be stored between the plates. 

Capacitors and Thunderstorms 

A natural example of a Capacitor is a thunderstorm. During a thunderstorm, supercooled water droplets and ice crystals rub together generating a massive negative charge within the cloud. When this charge reaches a large enough size, there is an electrical discharge between the cloud and the ground which is called lightning

The thunderstorm and the ground act as the two plates in a Capacitor, with the air acting as the insulator between.

The amount of charge stored by a thunderstorm is massive, so when this is discharged over a very short time (~50 microseconds on average), this generates a very high Current (1000s of Amperes) which is partly why lightning is so dangerous. 

The video below shows the effects of being struck by lightning ( on a gherkin ) :-

Capacitors in Circuits

The diagram below shows a simple Capacitor circuit :-

At the first instant that the switch is closed and Current starts to flow, there is no Voltage across the Capacitor (VC reads 0 V). This means that the Resistor has the full Voltage share of the Battery (VR reads 12 V). 

As the Capacitor begins to store Charge, the Voltage across the plates increases (VC reading increases). This means that the Voltage share across the Resistor decreases (VR reading decreases). 

The Capacitor will continue to store Charge until the Voltage across it is equal to the supply Voltage (VC reads 12 V), at which point no more Current will flow. 

Capacitor Graphs

As the Voltage across the Capacitor increases, the Potential Difference between the Battery and Capacitor gets smaller, so the Current decreases. This causes the curved shape of the following Voltage and Current graphs of a charging and discharging Capacitor :-

Note - This shape is called an exponential curve. This type of curve will tend towards a set value, but will only reach it at infinity. In a real-life situation, however, the value will be reached as all meters that scientists use will round the number to a certain degree. 

'Filling' a Capacitor

To describe how a capacitor stores charge, it can be useful to use the analogy of a bucket being filled using a hose. 

The time taken to fully charge a Capacitor depends on two quantities :-

1. The Capacitance of the Capacitor (The size of the bucket). 

2. The Resistance in the circuit (The amount of water flowing into the bucket). 

The larger the Capacitance of the Capacitor, the longer it takes to charge up. Similarly, the larger the Resistance of the circuit, the longer it takes to charge up.

The graphs below show the effect on the Current flow due to Capacitance and Resistance :- 

Note - As in V-T Graphs within the Higher course, the area under the Current-Time graph is important as it allows the calculation of the total Charge stored. As the total Charge stored for a set Capacitance is a particular value, the area under each curve in the effect of Resistance graph must be equal.  

The embedded website below allows a simulation of charging and discharging a capacitor :-

Capacitor Case Study : Traffic Lights

One function of a Capacitor in a circuit is to act as a time-delay system. An example of this is the button pushed at a Pelican Crossing. The button is pushed and after a certain length of time, the traffic lights change. 

The diagram below shows the Transistor circuit that would allow this system to work :-

When the switch is closed, the Capacitor is initially uncharged. This means that the Voltage across the Capacitor is equal to zero and the Transistor is off. 

As the Capacitor charges, the Voltage across it increases. After a short time, when the Voltage across the Capacitor reaches 0.7 V, the Transistor 'turns on' and the LED lights.

Time Constant

In the above Case Study, the LED turned on after a short time. The exact length of time it takes to charge the Capacitor depends upon the Resistance within the circuit and the Capacitance of the Capacitor. 

However, because the Current and Voltage in a Capacitor do not follow linear relationships, but instead follow exponential relationships which only reach full charge at infinity, we cannot talk about full charge time. Instead, the term Time Constant is used. 

The Time Constant is defined as: 

1. Time taken by the Capacitor for the Charge stored to reduce to 37 % of its full value when discharging.  

2. Time taken by the Capacitor for the Charge stored to increase to 63 % of its full value when charging.  

Note - The values of the above percentages are based on the Time Constant being equal to 1/e where "e" is Euler's Number ( 2.7182818459045235... ) . The derivation as to why this is the case is beyond the scope of this course. 

The graph below shows how to graphically identify the Time Constant value :-

The process is identical to the method for graphically measuring Radioactive Half-Life, except in this case we a looking for the time taken to reach 0.37 of full value, instead of half of full value.

Calculation of Time Constant Numerically

The Time Constant can also be calculated numerically from the Resistance and Capacitance of the circuit :-

t   =   R   x   C 

Where :-

t  =  Time Constant  (S) 

R  =  Resistance of Resistor (Ω) 

C  =  Capacitance of Capacitor (F)