Tutorial 11 A  Series RLC Circuits 

Learning Objective 
To understand phase relationships in series RLC circuits; To draw phasor diagrams; To calculate impedance. To measure impedance 

Key Question 
What is a series RLC circuit? What do the phasor diagrams look like for the RLC circuit? How do we work out the impedance? How can we deal with multiple components? 

Series RLC circuits Consider this circuit that consists of an inductor of inductance L (H), resistor of resistance R (W), and capacitor of capacitance C (F). It is connected to an alternating voltage V that has a frequency of f.
We know that:
Let’s look at the voltages in the circuit:
Since V_{R} is in phase with I, we can draw a phasor diagram to show the phase relationship. We will show V_{L} leading, and V_{C} lagging. Since V_{L} is greater than V_{C} the circuit is considered to be inductive.
We can see easily that V_{L} and V_{C} are 180^{o} out of phase. This means that the vector sum of the two vectors is V_{L} – V_{C}. We can call this the reactive voltage:
Reactive Voltage = V_{L} – V_{C}
So our vector diagram looks like this:
We can now do the vector sum of the voltages:
The impedance is calculated with the equation:
We use the equation below to work out the phase angle:
In these examples below we will use calculations. You can, of course, use accurate drawing of the phase vectors to calculate the resultant voltage.
Look at the proof HERE on the extension page.




Capacitative CircuitsIf V_{C} were greater than V_{L}, our phasor diagram would look like this:
We can now do the vector sum of the voltages:
Using a similar argument to that above we can write:
And:
This circuit is capacitative. 

ReactanceFrom previous topics we know that reactance is defined as the ratio of the voltage across a reactive component, and the current. For a capacitor:
For an inductor:
Real inductors have an ohmic resistance. We treat a real inductor as: a perfect inductor in series with a resistor.
Often we will need to work out the reactance of these components before we are able to work out the total impedance and other numerical parameters in the series circuit. We will look at this in the worked example.
We can draw the phasor diagram for this circuit:
This phasor diagram shows what we would expect if we had a perfect inductor of 120 mH in series with a 5 ohm resistor and a 100 mF capacitor. 

Multiple Components in SeriesSo far we have looked at a circuit that is simply a series capacitor, inductor, and resistor. But what happens if we have more than one resistor, or more than one inductor? It’s actually quite simple. We find the single equivalent resistor, or single equivalent inductor. In series circuits they simply add up.
With series capacitors, it’s a little more involved, but not too difficult:
Now we will look at adding an extra component. We will use the same example that we looked at on page 8, but this time, we will add in an extra 10 ohm resistor.
In real inductive components, the V_{L} vector is never quite 90^{o} to the V_{R} resistor. This is because the coil has a definite resistance, so there is a phase difference that is less than 90^{o}.
This is less of a problem with a nonelectrolytic capacitor, as a capacitor has an insulating layer. If it stops insulating, it’s not much good as a capacitor.
An electrolytic capacitor has a certain leakage current, which can be modelled as a resistor.




