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Mutual Inductance

When one particular loop of a circuit affects another loop by means of current conduction, we have what is called a "conductively-coupled circuit." However, when two loops with no physical connections between them are able to affect one another by means of a magnetic field generated by one of the loops, we have what is called a "magnetically-coupled circuit."

Self Inductance:

Consider a circuit having a single inductor with a coil of "N" number of turns as well as the indicated magnetic field:

Self inducting coil

By Faraday's Law of Induction, the voltage across the coil is defined as: vL=NdΦBdt,(Eqn1) ...where: ΦB=magneticflux While we won't address the physics associated with electromagnetism, know that the magnetic flux will change if the current changes and if this is an AC circuit, the current will obviously be time varying. This knowledge allows us to rewrite equation #1 as: vL=NdΦBdididt,(Eqn2) Since it is also known that the definition of inductance is: L=NdΦBdi ...we can rewrite equation #2 as: vL=Ldidt,(Eqn3) You may recognize equation #3 as being the voltage-current relationship for an inductor.

Mutual Inductance:

Now consider what happens when we have two coils in close physical proximity to one another, each having a self-inductance value of L1 and L2:

Two coils displaying mutual inductance

We will assume that coil 1 has "N1" number of turns and coil two has "N2" number of turns. Additionally we will assume that no current flows through the second conductor. Additionally, let: ΦB1=totalmagneticfluxgeneratedfromcoil1 ΦB1=ΦB11+ΦB12 The magnetic flux labeled "B11" is the component of the total flux which passes solely through coil #1. The magnetic flux labeled "B12" is the component of the total flux which passes through both coil #1 and coil#2. The two coils are said to be coupled due to the fact that the total magnetic flux (B1) links both coils.

Induced Voltage in Coil #1:

By Faraday's Law we have: V1=N1dΦB1dt We see that all of the magnetic flux "B1" passes through coil #1 and this flux is caused by the current flowing through coil #1 (i1). V1=N1dΦB1di1di1dt V1=L1di1dt ...where: L1=selfinductanceofcoil1

Induced Voltage in Coil #2:

Once again, by Faraday's Law we have: V2=N2dΦB12dt We see that only magnetic flux "B12" passes through coil #2. However, this flux is also caused by the current flowing in coil #1 (i1). V2=N2dΦB12di1di1dt

V2=M21di1dt,(Eqn4) ...where: M21=N2dΦB12di1 =mutualinductanceofcoil2withrespect tocoil1

Above, we see that the voltage induced in coil #2 is related to the current in coil #1.

Current flows through coil #2 but not coil #1

We now consider a situation that is opposite of the one above. Now a current (i2) is allowed to flow through coil #2 but not coil #1:

Two coils displaying mutual inductance

ΦB2=totalmagneticfluxgeneratedfromcoil2 ΦB2=ΦB21+ΦB22 The magnetic flux labeled "B22" is the component of the total flux which passes solely through coil #2. The magnetic flux labeled "B21" is the component of the total flux which passes through both coil #1 and coil#2. Once again, by Faraday's Law we have the following expression for the voltage across coil #2: V2=N2dΦB2dt Since all of the magnetic flue ("B2") flows through coil #2 and is caused by the current i2, we have: V2=N2dΦB2di2di2dt V2=L2di2dt L2=selfinductanceofcoil2 For the voltage across coil #1 we have: V1=N1dΦB21dt Since only the magnetic flue ("B21") flows through coil #1 and is caused by the current i2, we have: V1=N1dΦB21di2di2dt

V1=M12di2dt,(Eqn5) ...where: M12=N1dΦB21di2 =mutualinductanceofcoil1withrespect tocoil2

While the following is not proven here, know that: M12=M21=M "M" is the mutual inductance between the two coils and is measured in units of Henrys. This means we now have the following expressions for the mutually induced voltage across the two coils: V1=M(di2dt) V2=M(di1dt) However, if you take a look at the section on Phasor Calculus you may realize that we can make the following substitution: didt=jωI Therefore, the mutually induced voltages across the coils can also be expressed in the frequency domain as: V1=jωMI2 V2=jωMI1

Determining polarity of mutually induced voltages (Dot-Convention):

In regards to the self-induced voltage (L di/dt) the polarity is determined by the reference direction of the current flow in relation to the passive sign convention. The polarity of the mutually induced voltage (M di\dt) is somewhat more difficult as it relies on Lenz's Law and the Right Hand Rule as well as the knowledge of how the coils are physically wound. Since it is impractical to annotate such information in an electrical schematic something called the "Dot Convention" is often used. In the dot convention, a dot is placed at one end of each magnetically coupled coil. The dot indicates the direction of magnetic flux if the current was to enter that particular terminal of the coil. The following statement explains this convention in relation to polarity

Conversely:

The following schematics will helpfully illustrate the above statements:

Dot convention for polarity of mutually induced voltages.

click here for PDF version of above illustration

Dot convention for polarity of mutually induced voltages.

Steps for solving mutually coupled circuits:

  1. Determine the sign for the induced voltages.
  2. Determine the value of the induced voltages.
  3. V2=jωMI1 V1=jωMI2
  4. Analyze this circuit containing two dependent sources.

For purposes of our analysis here, the value of "M" and the location of the dots are taken as given.

Recap

For two coils in close proximity to one another, the mutually induced voltage across them is defined as: V1=M(di2dt),V1=jωMI2 V2=M(di1dt),V2=jωMI1

In the next page we will look at an example problem involving mutual inductance

Continue on to Mutual Inductance example problem #1...