Topic 10  RC Networks with DC

RC Network

At its simplest the RC network is a series circuit consisting of a capacitor and a resistor connected to a source. 

If we discharge a capacitor, we find that the charge decreases by the same fraction for each time interval.  If it takes time t for the charge to decay to 50 % of its original level, we find that the charge after another t seconds is 25 % of the original (50 % of 50 %).  This time interval is called the half-life of the decay.  The decay curve against time is called an exponential decay.

Discharging a Capacitor

The voltage, current, and charge all decay exponentially during the capacitor discharge.

  We can note the voltage and current at time intervals and plot the data, which gives us the exponential graph, using a circuit like this.  

The graph is like this:

We should note the following about the graph:

• Its shape is unaffected by the voltage.

• The half life of the decay is independent of the voltage.

The product RC (capacitance × resistance) is called the time constant.  The units for the time constant are seconds.  We can go back to base units to show that ohms × farads are seconds. 

  After RC seconds the voltage is 37 % of the original.    To increase the time taken for a discharge we can:

• Increase the resistance.

• Increase the capacitance.

  The half-life is 69 % of the time constant. 

Electronic engineers use the time constant in preference to the half-life.  In theory the exponential decay should never allow a capacitor to discharge completely, but in practice, a rule of thumb is that the capacitor is discharged completely after 5 RC seconds. 

Click HERE for a worked example

Charging a Capacitor

When we charge up a capacitor, we get an exponential rise in charge and voltage.  We get an exponential fall in the current.  This is because when we start to charge up the capacitor, the current is a maximum and the voltage is zero.  When the voltage is at a maximum, the current is zero, because no charge can flow on.

The graphs are like this:

After RC seconds the capacitor has charged up to 63 % of its final voltage.

  As in a discharge there is a half life in the charging of a capacitor; we can relate it to the time constant by the relationship:

                        t_1/2 = 0.69 RC

After 5 RC time constants, the capacitor is almost completely charged up, so the voltage is almost Vo. 

Useful Websites

http://www.s-cool.co.uk/topic_index.asp?subject_id=2 

http://www.phy.ntnu.edu.tw/java/rc/rc.html