Series Connection of Mutually Coupled Coils

In Chapter 12, we found that the total inductance of series isolated coils was determined simply by the sum of the inductances. For two coils that are connected in series but also share the same flux linkages, such as those in Fig. 1(a), a mutual term is introduced that will alter the total inductance of the series combination. The physical picture of how the coils are connected is indicated in Fig. 1(b).
Mutually coupled coils connected in series.
Fig. 1: Mutually coupled coils connected in series.
An iron core is included, although the equations to be developed are for any two mutually coupled coils with any value of coefficient of coupling k. When referring to the voltage induced across the inductance $L_1$ (or $L_2$) due to the change in flux linkages of the inductance $L_2$ (or $L_1$, respectively), the mutual inductance is represented by $M_{12}$. This type of subscript notation is particularly important when there are two or more mutual terms.
Due to the presence of the mutual term, the induced voltage $e_1$ is composed of that due to the self-inductance $L_1$ and that due to the mutual inductance $M_{12}$. That is,
$$e_{1}=L_{1} \frac{d i_{1}}{d t}+M_{12} \frac{d i_{2}}{d t}$$
However, since $i_{1}=i_{2}=i$,
$$e_{1}=L_{1} \frac{d i}{d t}+M_{12} \frac{d i}{d t}$$
$$\bbox[10px,border:1px solid grey]{e_{1}=\left(L_{1}+M_{12}\right) \frac{d i}{d t}} (volts, V) \tag{1}$$
and, similarly,
$$e_{2}=\left(L_{2}+M_{12}\right) \frac{d i}{d t} \quad \text { (volts, V) } \tag{2}$$
For the series connection, the total induced voltage across the series coils, represented by $e_{T}$, is
$$e_{T}=e_{1}+e_{2} =\left(L_{1}+M_{12}\right) \frac{d i}{d t} + \left(L_{2}+M_{12}\right) \frac{d i}{d t}$$
$$e_{T}=\left(L_{1}+L_{2}+M_{12}+M_{12}\right) \frac{d i}{d t}$$
and the total effective inductance is
$$\bbox[10px,border:1px solid grey]{L_{T(+)}=L_{1}+L_{2}+2 M_{12}} \text{(henries, H)} \tag{3}$$
The subscript (+) was included to indicate that the mutual terms have a positive sign and are added to the self-inductance values to determine the total inductance.
Mutually coupled coils connected in series
with negative mutual inductance.
Fig. 2: Mutually coupled coils connected in series with negative mutual inductance.
If the coils were wound such as shown in Fig. 2, where $\phi_1$ and $\phi_1$ are in opposition, the induced voltages due to the mutual terms would oppose that due to the self-inductance, and the total inductance would be determined by
$$\bbox[10px,border:1px solid grey]{L_{T(-)}=L_{1}+L_{2}-2 M_{12}} \text{(henries, H)} \tag{4}$$
Through Eqs. (1) and (2), the mutual inductance can be determined by
$$ \bbox[10px,border:1px solid grey]{M_{12} = { 1\over 4}( L_{T(+)} - L_{T(1)})} \tag{5}$$
Equation (5) is very effective in determining the mutual inductance between two coils. It states that the mutual inductance is equal to one-quarter the difference between the total inductance with a positive and negative mutual effect.
From the preceding, it should be clear that the mutual inductance will directly affect the magnitude of the voltage induced across a coil since it will determine the net inductance of the coil. Additional examination reveals that the sign of the mutual term for each coil of a coupled pair is the same. For $L_{T(+)}$ they were both positive, and for $L_{T(-)}$ they were both negative.
On a network schematic where it is inconvenient to indicate the windings and the flux path, a system of dots is employed that will determine whether the mutual terms are to be positive or negative. The dot convention is shown in Fig. 21.27 for the series coils of Figs. 1 and 2.
Dot convention for the series coils.
Fig. 3: Dot convention for the series coils of (a) Fig. 1 and (b) Fig. 2.
If the current through each of the mutually coupled coils is going away from (or toward) the dot as it passes through the coil, the mutual term will be positive, as shown for the case in Fig. 3(a). If the arrow indicating current direction through the coil is leaving the dot for one coil and entering the dot for the other, the mutual term is negative.
A few possibilities for mutually coupled transformer coils are indicated in Fig. 4(a). The sign of M is indicated for each. When determining the sign, be sure to examine the current direction within the coil itself.
In Fig. 4(b), one direction was indicated outside for one coil and through for the other. It initially might appear that the sign should be positive since both currents enter the dot, but the current through coil 1 is leaving the dot; hence a negative sign is in order.
Defining the sign of M for mutually coupled transformer coils
Fig. 4: Defining the sign of M for mutually coupled transformer coils
The dot convention also reveals the polarity of the induced voltage across the mutually coupled coil. If the reference direction for the current in a coil leaves the dot, the polarity at the dot for the induced voltage of the mutually coupled coil is positive. In the first two figures of Fig. 4(a), the polarity at the dots of the induced voltages is positive.
In the third figure of Fig. 4(a), the polarity at the dot of the righthand coil is negative, while the polarity at the dot of the left-hand coil is positive, since the current enters the dot (within the coil) of the righthand coil. The comments for the third figure of Fig. 48(a) can also be applied to the last figure of Fig. 4(a).
Example 1: Find the total inductance of the series coils of Fig. 5.
Fig. 5: Example 1.
Coil 1: $L_{1}+M_{12}-M_{13} $
Coil 2: $L_{2}+M_{12}-M_{23} $
Coil 3: $L_{3}-M_{23}-M_{13} $
$$\begin{aligned} L_{T} &=\left(L_{1}+M_{12}-M_{13}\right)+\left(L_{2}+M_{12}-M_{23}\right)+\left(L_{3}-M_{23}-M_{13}\right)\\ &=L_{1}+L_{2}+L_{3}+2 M_{12}-2M_{23}-2M_{13} \end{aligned}$$
Substituting values, we find
$$\begin{aligned} L_{T} &=5 H+10 H+15 H+2(2 H)-2(3 H)-2(1 H) \\ &=34 H-8 H=26H \end{aligned}$$

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