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 circuits

### RL circuit

Consider 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 90o.

·         In an inductor circuit, the current lags the inductor voltage by 90o.

·         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 VR is in phase with I, we can draw a phasor diagram to show the phase relationship.  We will show IR leading, and IL lagging. We can show the resultant current, I. We can easily see that I is the vector sum of IL and IR.  So we write: where: and: We can write an expression for the phase angle: We also know that: Question 1 What is Z? 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: Question 2 Why can we not write: ? Question 3 A parallel circuit consists of a perfect inductor of 400 mH and a 200 ohm resistor.  The components are connected to a supply of 6 V at a frequency of 300 Hz. a.       What is the reactance of the inductance? b.      What is the impedance of the circuit? c.       What is the resultant current? 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. Question 4 Use the data from Question 3.  The frequency is now 50 Hz. What is the impedance of the series part of the circuit, if the resistance of the inductor is 15 ohms? Question 5 Work out the phase angle between the voltage and the resultant current. 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 152 to 7542, 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 circuit

Consider 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 90o.

·         In an inductor circuit, the current lags the inductor voltage by 90o.

·         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 VR is in phase with I, we can draw a phasor diagram to show the phase relationship.  We will show IR leading, and IL lagging. We can show the resultant current, I. We can easily see that I is the vector sum of IL and IR.  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.

 Question 6 A parallel circuit consists of a capacitor of 400 mF and a 2 ohm resistor.  The components are connected to a supply of 6 V at a frequency of 300 Hz. a.      What is the reactance of the capacitor? b.      What is the impedance of the circuit? c.      What is the resultant current? Question 7 What is a perfect capacitor?  Is it possible to have a perfect capacitor? 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 micro-amps.

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. Question 8 Assuming that the capacitor was working correctly, what can we say about the value of resistance in the 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. Question 9 Use the data from Question 6.  The frequency is now 50 Hz. What is the value of r if the leakage current is 0.8 mA? Question 10 What is the reactance of the capacitor? Question 11 Work out the resulting impedance.  Is the difference significant? 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.