Tutorial 12 A  Parallel RL and RC Circuits 

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

Key Questions 
What is a parallel RL circuit? What is a parallel RC circuit? What do the phasor diagrams look like for the parallel RL circuit? What do the phasor diagrams look like for the parallel RC circuit? How do we work out the impedance? How can we deal with resistance in the inductor? 

Parallel RL circuitsRL circuitConsider this circuit that consists of an inductor of inductance L, and resistor of resistance R. It is connected to an alternating voltage V that has a frequency of f.
We know that: · In a parallel circuit, the voltage is the same across each branch. · In a capacitor circuit, the current leads the capacitor voltage by 90^{o}. · In an inductor circuit, the current lags the inductor voltage by 90^{o}. · In a parallel reactive circuit, the currents add up as a vector sum. · The current is always in phase with the voltage across the resistor.
Since V_{R} is in phase with I, we can draw a phasor diagram to show the phase relationship. We will show I_{R} leading, and I_{L} lagging.
We can show the resultant current, I.
We can easily see that I is the vector sum of I_{L} and I_{R}. So we write:
where:
and:
We can write an expression for the phase angle:
We also know that:
So we can get an expression for Z by simple substitution into the current equation:
Since the voltage across a parallel circuit is the same, we can rewrite this as:
This reminds us of the equation for parallel resistors:
We work out the reactance of the inductor simply by using:
Now we know that no inductor is perfect; it has a definite value for the resistance. Let’s look at this further. It does complicate things, but it’s not impossible. We model the real inductor as a perfect inductor in series with a resistor, r.
The first thing we need to do is to work out the series impedance between L and r. We will call this z.
As the frequency goes up with an inductor, the reactance also increases. Therefore the ohmic resistance of the inductor becomes much less significant. Using the numbers that we did in Question 3, we found that the reactance was 754 W. When you add 15^{2} to 754^{2}, the square root hardly changes from 754. (Try it for yourselves.)
Even at a lower frequency, there was not much difference. However at low frequencies the change becomes significant.


RC circuitConsider this circuit that consists of a capacitor of capacitance C, and resistor of resistance R. It is connected to an alternating voltage V that has a frequency of f.
We know that: · In a parallel circuit, the voltage is the same across each branch. · In a capacitor circuit, the current leads the capacitor voltage by 90^{o}. · In an inductor circuit, the current lags the inductor voltage by 90^{o}. · In a parallel reactive circuit, the currents add up as a vector sum. · The current is always in phase with the voltage across the resistor.
Since V_{R} is in phase with I, we can draw a phasor diagram to show the phase relationship. We will show I_{R} leading, and I_{L} lagging.
We can show the resultant current, I.
We can easily see that I is the vector sum of I_{L} and I_{R}. So we write:
where:
and:
We can write an expression for the phase angle:
We also know that:
So we can get an expression for Z by simple substitution into the first equation:
Since the voltage across a parallel circuit is the same, we can rewrite this as:
This reminds us of the equation for parallel resistors:
We work out the reactance of the capacitor simply by using:
In this discussion above, we have assumed that the capacitor is perfect.




Capacitors with a Leakage Current Some capacitors are NOT perfect. Electrolytic capacitors have a measurable leakage current. This arises due to imperfections in the dielectric, making it conduct slightly. The leakage current is small; if it were large, the capacitor would be defective, and useless. Generally the leakage current is a few microamps.
Leakage current can also occur with electronic components that are connected to the capacitor, due to the electrical properties of diodes and transistor, even when turned off.
So how do we model this? As there is a current going through the capacitor, we can model it as a capacitor of infinite resistance in parallel with a resistor, r.
The first thing we need to do is to work out the parallel impedance between C and r. We will call this z.
In the last example we saw that there was no difference at all between the overall impedance and the reactance of the capacitor. We would need to go to many more significant figures to observe and difference. For most practical purposes, an answer to 2 to 3 significant figures is quite sufficient for the electrical engineer.
As the frequency goes up with a capacitor, the reactance decreases. Therefore any leakage current in a capacitor becomes even less significant. There is no need to go into the effect of leakage current.


