Mutual Coupling & Dot Convention

[Marcin H]

Homework Statement

Homework Equations

KVL and KCL

The Attempt at a Solution

I am very confused by the dot convention and how it works when writing your loop equations... I can't find any good videos or explanations online and I am completely lost with this dot convention. So I understand where every component comes from in the loop equations, but I have no clue why the signs on the mutually coupled components (M) are what they are.

The way I think it should be based on what I learned in class is that once you reach that mutually coupled component in your loop you look at the direction of the current in the opposite dot and if the current in positive going into the dot (on the opposite side) then you will have +M. If it's going away from the dot you will have -M. The problem is this all changes based on the way you choose your current... How do you choose your current? How do you know if it's right or wrong? I thought choosing current directions never mattered, but here it seems to change the problem completely... Can anyone clarify this whole process or dumb it down for me? I feel lost atm...

Also according to the solutions the sign on both M's is positive like I have in my solutions above. I just don't understand why... especially for the second loop...

 

[gneill]

Hi Marcin,

The first thing to do is establish your assumed loop current directions. It doesn't matter which directions you choose as long as you then remain consistent with them when writing your equations. The loop currents then define the "positive" directions for those currents.

When writing your equations, note whether a given loop current is flowing into or out of the dotted end of the inductor that the current is passing through. If the local loop current is entering the dot then the current that will be induced in the coupled inductor will be flowing out of its dot. Similarly, if the current is flowing out of the dotted end of the inductor then the induced current in the other inductor will be flowing into its dot.

One way to keep this straight and get the signs correct is to re-draw your circuit inserting a voltage supply symbol in series with each coupled inductor. The polarity of each of these voltage supplies is then easily determined by inspection using the above "rule"; the supply polarities are chosen so that they will tend to push current in the desired direction in their loops, causing the induced currents to enter or leave the dot according to the rule. Once that's done you can go ahead and write the loop equations from the diagram.

 

[Marcin H]

When writing your equations, note whether a given loop current is flowing into or out of the dotted end of the inductor that the current is passing through. If the local loop current is entering the dot then the current that will be induced in the coupled inductor will be flowing out of its dot. Similarly, if the current is flowing out of the dotted end of the inductor then the induced current in the other inductor will be flowing into its dot.

So just to make sure. You are saying that you have to check if the current (that you arbitrarily assumes at the beginning) enters or exits on the dot on the left side and then that current will exit on the dot on the right side. If the current that exits on the right is in the same direction as the current you chose at the beginning, then you will have a positive current and therefore a positive sign on your mutual inductance term. Here is a picture of what I am trying to get at...

One way to keep this straight and get the signs correct is to re-draw your circuit inserting a voltage supply symbol in series with each coupled inductor. The polarity of each of these voltage supplies is then easily determined by inspection using the above "rule"; the supply polarities are chosen so that they will tend to push current in the desired direction in their loops, causing the induced currents to enter or leave the dot according to the rule. Once that's done you can go ahead and write the loop equations from the diagram.

I have never heard about this method and I don't think our book mentions anything about a method like this. Can you explain further? Or do you have any good examples of that method? I am not too sure what you would do with them once you place them in there or how to determine their polarity based off of the current rule you mentioned for the dot convention.EDIT***
Also, is this correct for finding the correct sign for the second loop?

 

[gneill]

You've got the current direction correct for the current induced in the second loop due to the first loop's current.

Personally I find trying to sort out the various current directions and dot locations in order to assign the correct sign to a term in an equation to be a bit too abstract to do confidently by inspection; I seem to get it wrong about half the time! Also, the sign will depend on whether you write your loop equations as a sum of potential drops or a sum of potential rises. Too many things to juggle. That's why I use the "insert voltage supply" technique, which always works and be done practically without thinking. Then I can tackle writing the loop equations in the same way I always do, making far fewer errors because it's a "by rote" process.

Let's take a look at your circuit.

Okay, in your diagram you've chosen the first loop's current to be clockwise and the second loop's current to be counterclockwise. Call them $I_1$ and $I_2$ respectively. In the first loop you can see that $I_1$ flows out of the dot on that loop's inductor. That means the induced current in the second loop will flow into the dot in the second loop, thus a counterclockwise induced current. So you can orient the voltage supply in the second loop accordingly so that it drives an induced current counterclockwise. You can also see that $I_2$ flows into the dot in its loop. That means that the induced current in the first loop will flow out of the dot on its inductor. Again the added voltage supply should be oriented to induce the correct current direction.

Now you can write the KVL loop equations from the new circuit diagram using whatever convention (drops or rises) that you prefer, and the signs will take care of themselves.

 

[Marcin H]

I am getting confused by what is considered into the dot and what is considered out of the dot exactly. How do you know what is in or out? Which direction is in or out? Looking at some other circuits I am struggling to understand this...

Here is another example from class:

for the first loop equation (v1) when we get to the mutal inductance term M21 (the dots not the squares), we chose current goes into the dot. Therefore the current will come out of the dot in the second circuit (L2) but why is the current to the left? Why is that considered out? It goes against the current we chose in that circuit (L2) so that is why we have a negative, but I don't see why that direction is out and not into the inductor...

 

[gneill]

I am getting confused by what is considered into the dot and what is considered out of the dot exactly. How do you know what is in or out? Which direction is in or out?

Current entering a dot is current that passes through the dot and then enters the inductor. Current exiting a dot is current that first passes through the inductor and then leaves it via the dot.

 

[Marcin H]

Current entering a dot is current that passes through the dot and then enters the inductor. Current exiting a dot is current that first passes through the inductor and then leaves it via the dot.

Ok so when the dot is on the bottom of the inductor then the current has to pass through the inductor and out the dot. That means it's exiting the dot. So that means that current has to enter the dot on the other side, but that is where I get confused again. How do we know what is entering and exiting on the other side? It's not based off of i2 because we use i2 to determine whether or not M will be negative or positive...

Here is another example...

For the left circuit we know current goes into the dot because that is what we chose... But then how do we know what is considered out when looking at the second dot?

 

[Marcin H]

Current entering a dot is current that passes through the dot and then enters the inductor. Current exiting a dot is current that first passes through the inductor and then leaves it via the dot.

Also, I think I am noticing a pattern... It looks like when the dots are on the same side (for example the top of both inductors or bottom of both inductors) then the M term is positive. If they are opposite sides the M terms are negative. Is this always true?

 

[gneill]

Ok so when the dot is on the bottom of the inductor then the current has to pass through the inductor and out the dot.

That depends upon your choice of direction for the loop current. Remember, you get to choose your assumed direction for loop currents before your analysis begins. Choose the loop currents first, then determine whether those chosen currents are entering or exiting the inductors via their dotted ends. This is important because eventually you will come across a circuit where a dotted inductor is shared by two loops and the currents may be passing in opposite directions in the inductor branch.

That means it's exiting the dot. So that means that current has to enter the dot on the other side, but that is where I get confused again. How do we know what is entering and exiting on the other side? It's not based off of i2 because we use i2 to determine whether or not M will be negative or positive...

The induced currents are separate from the loop currents. They are an "extra" current flowing in the loop that is being driven by the current of another loop (hence we talk about "induced current"). You write separate terms for the loop currents and the induced currents.

Here is another example...

View attachment 211901
For the left circuit we know current goes into the dot because that is what we chose... But then how do we know what is considered out when looking at the second dot?

"Out" is always away from the inductor. "In" is into the inductor.

The loop current $i$ enters the dot of the top inductor, so it will cause an induced current to flow out of the dot in the second inductor. That means that this induced current will flow upwards out of the bottom inductor. Similarly, the loop current $i$ enters the dot of the second inductor, too, so it too will cause an induced current to flow out of the dot of the top conductor.

 

[gneill]

Also, I think I am noticing a pattern... It looks like when the dots are on the same side (for example the top of both inductors or bottom of both inductors) then the M term is positive. If they are opposite sides the M terms are negative. Is this always true?

Again, it depends upon your choice of loop current directions and how you choose to sum the potential changes when you write your loop equations.

 

[Marcin H]

Again, it depends upon your choice of loop current directions and how you choose to sum the potential changes when you write your loop equations.

Thanks! It finally clicked.