Why Current lags voltage in an inductive circuit.

So you wanna play with the big boys?

If you want to get into AC theory there are a few key concepts that you will need to understand.  One of these key concepts is that in an inductive circuit current lags the voltage.

What does that mean?

In a purely resistive circuit, the current and voltage are in phase.  This means that they are starting and stopping at the same time.

In this drawing of the waveforms, we see that they are crossing the x axis at the same time.

In a purely inductive circuit, this isn’t the case.  In a purely inductive circuit, the current lags the voltage by 90°.  This means the voltage starts first and then the current starts a bit after.

pic 2

In this drawing, the voltage is the red waveform and the current in the purple waveform.  You notice that the red waveform crosses the x axis before the purple waveform does.

Why does current lag the voltage in a purely inductive circuit?

Well in a purely resistive circuit the voltage across a resistor is dependent on ohms law.  Remember that one?

E= IR

E is the voltage.

I is the current in the circuit.

R is the resistance in the circuit.

This allows current and voltage to be in phase with one another.

In a purely inductive circuit, we have a bit of an issue.  There is no resistance.  If there is no resistance we can’t use ohms law to determine the voltage at the inductor.

We determine the voltage at the inductor by using Faraday’s law of electromagnetic induction.

E= -L Δi/Δt

E is the voltage at the inductor.

L is the inductance of the inductor.

This is a physical property much like resistance and is constant.  The negative in front of the L is to show that the voltage that is induced at the inductor is in opposition to the applied voltage (this concept is covered in another lesson on electromagnetic induction)

Δi/Δt is what is known as rate of change. How fast the current is changing across the inductor.  It is this factor that is the key as to why current lags the voltage.

Why the rate of change will blow your effing mind.

 If we are trying to determine the voltage (which is determined for us) and our inductance is a given everything depends on the rate of change.

Lets look at a voltage waveform first:

pic 3

In this picture, we have two waveforms.  The red wave form is the applied voltage (the source).  The green waveform is the induced waveform (voltage at the inductor).

If we look at when the voltage is zero for both the applied and induced voltage and use the formula E= -L Δi/Δt, we see that in order for E to equal Zero, the rate of change must be zero (remember that L is a physical value that can’t change).

Now comes the time we need to talk about the rate of change.

 Before we continue we need to understand the concept of rate of change when it comes to current.

Let’s look at the sign wave below:

pic 4

When determining the minimum rate of change, where would you guess that occurs?  If you said when current is at zero you would be wrong.  That is a natural assumption.  When current is at zero it actually has the most to gain.

It is when current is at its peak that the rate of change is zero (watch the video for a more in depth analysis).  When Current is at zero its rate of change is at maximum.

Remember this:

The rate of change is at a maximum when current is at a minimum.

The rate of change is at a minimum when current is at maximum.

Now back to our regularly scheduled program.

 It is this relationship with the rate of change that causes the current to lag the voltage.  When the voltage is at zero our rate of change must be at zero.  This means that current is at maximum.

When the voltage is at its maximum, the rate of change must be at its maximum.  This occurs when current is at zero.  As we chart it out you start to see that the current waveform and the voltage waveform are no longer in sync.

pic 5

That leaves us with this one truth:

In a purely inductive circuit current lags the voltage by 90°.  Make sure you check out the video for a more in depth look.

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Any questions or issue?  Email me.  I respond to each and every one.  I can be reached at chad@theelectricacademy.com

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