### Everything You Ever Wanted to Know about Tapped Inductors but Were Too Afraid to Ask!

**About the writer:** Harvey Morehouse is a contractor/consultant
with many years of experience using circuit analysis programs. His primary
activities are in Reliability, Safety, Testability and Circuit Analysis.
He may be reached at harvey.annie@verizon.net. Simple questions for which
I know the answer are free. Complex questions, especially where I am ignorant
of the answers, are costly!!!

**Summary:** Tapped inductors are a useful circuit element. Typically
one specifies two separate inductors, and uses the inductive coupling
element with them. One problem that occurs is that the coefficient of
coupling cannot be a user defined value. Two solutions for this problem
are provided.

*Two devices are prepared. One is for use in normal autotransformer
applications, and the second is for use in tapped inductor applications
with SMPS power supplies.*

##### Background:

Before preparation of the models some introductory material is given to prepare the way.

##### ‘K’ inductive coupling element:

The ‘K’ inductive coupling element is shown in Figure 1 following:

Figure 1

K element symbol

TInterestingly enough, there are NO pins on this device model. The designations of the inductors are passed to the device as parameters, as is the coupling coefficient between the two inductors.

This device is a carry-over from SPICE2. Some SPICE variants have made
modifications to this device to add some capabilities. However, this is
what is present in B2SPICE^{TM}.

The parameters window for this device is shown in Figure 2 following:

Figure 2

‘K’ device model parameter entry window

One enters the **name** of the inductors to be coupled, and the **units**
of the inductors entered are in Henries.

A partial schematic showing the use of this device in a circuit is shown in Figure 3.

Figure 3

Coupled Inductor Model Usage

Now, the symbology shown in Figure 1 shows two inductors with polarity
marking, which is shown by selecting the display of the part pin names,
something not usually done. **Devices such as resistors, inductors,
capacitors and transformers do have inherent polarities within SPICE.***By this it is meant that IF one were to examine the current through
a device, or the voltage across a device, such as i(L1), or v(R2), the
sense of the reported currents through, and the voltages across many devices
is dependent on their orientation within a circuit.*

Note that the arrangement of the inductors is that they are additive. That is to say, the ‘dot’ ends of each portion of the inductor is to the left (or the right) of each portion of the inductor, They need not be directly connected as shown, however, that is how they will be utilized in an autotransformer application.

In an autotransformer circuit realization, where points (a), (b) and (c) of Figure 3 are all utilized and connected to other circuit elements, the realization will be different than where one connection is open for a portion of an cycle, as in a SMPS tapped inductor circuit.

##### Autotransformer model:

Refer to Figure 3. In an autotransformer model, as in a variac application, one might expect to know the total inductance from points (a) to (c) with the tap at one extreme or the other, or open.

Now inductance is proportional to the square of the turns. Defining the turns of Lp as np, and those of Ls as ns, the total inductance Lt is defined as being:

Lt = c*(ns + np)

^{2}Eq. 1

and

Ls = c*ns

^{2}Eq. 2

Lp = c*np^{2}Eq. 3

It should be evident that Lt does NOT equal the sum of Lp and Ls inductances save for the tap point being at zero or 100 percent values.

Lt/(ns + np)

^{2}= Lp/np^{2}= Ls/ns^{2}Eq. 4

Now some simplifications may be made. The exact number of turns is not important, as we are interested in their ratios, therefore we can assume that the total number of turns, ns + np, is unity. Now:

Lt/1 = Lp/np

^{2}Eq. 5

If we let np = kt*nt, then:

Lp = Lt* np

^{2}= kt^{2}*Lt Eq. 6

and

Ls = Lt* ns

^{2}= (1 –kt)^{2}*Lt Eq. 7

Let us define the turns ratio n as being:

n = ns/np Eq. 8

then

n = (Ls/Lp)

^{0.5}= (1 – kt)/kt Eq. 9

The range of suitable variables that could be used as inputs to a parameterized subcircuit model includes Lt, Lp, Ls, kt, n, np and ns in various combinations. We would also wish to be able parameterize kc, the coefficient of coupling. Now the inductive coupling device does not directly allow this, however, when used in a subcircuit and turned into a device, the resultant netlist can be edited to use such a parameter.

The choice was made to include Lt, kt and kc as parameters. A subcircuit was prepared similar to that of Figure 3 to create such a device, as shown in Figure 4.

Figure 4

AUTOXFMRft Coupled Inductor Model

The netlist for the created model is:

************************

* B2 Spice Subcircuit

************************

* Created by Harvey C. Morehouse

* and

* Thien Nguyen

*

* This model is copyright by B2SPICE and the creators

*

* Permission is given to all to freely use this model, however

* credits should be given to B2SPICE and the creators.* Pin # Pin Name

* vp vp

* vs vs

* Vtap Vtap

.Subckt AUTOXFMRft vp vs Vtap***** main circuit

Lp vp Vtap {Lt * kt ^2}

Ls Vtap vs {Lt*(1 - kt)^2}

K1 lp ls {kc}.ends

To test the device model a circuit was created as shown in Figure 5.

Figure 5

AUTOXFMRft Model Test Circuit

In Figure 5, V1 is 1V, 60 Hz generator. The first test was a transient sweep of the output voltage with kc varying from zero to unity in steps of 0.25. The output voltage is shown in Figure 6.

Figure 6

AUTOXFMRft Model kc Sweep

The smallest amplitude plot is with kc equal to zero. As kc increases, the output amplitude also increases as one might expect. Note that with kc between zero and unity leakage inductances will be present.

The next test will be with kc equal to unity, but kt varying from zero to unity in steps of 0.25. The results of this test are shown in Figure 7.

Figure 7

AUTOXFMRft Model kt Sweep

In Figure 7 the largest amplitude curve is where kt is zero. In this case Lp is zero and Ls is equal to Lt. Smaller and smaller output curves occur as Lp becomes progressively larger and Ls smaller.

Now another autotransformer device model was prepared as shown in Figure 8.

Figure 8

AUTOXFMRkc Model

In the AUTOXFMRkc device, the input parameters are kc, Lp and Ls. These inputs could be derived from other known characteristics of the autotransformer. For example, were one given the turns ratio n, one knows that this is, from Eq. 9:

n = (Ls/Lp)

^{0.5}= (1 – kt)/kt Eq. 10

then

kt = 1/(n + 1) Eq.11

The purpose of creating the autotransformer models was to be able to use them in a larger circuit with user defined variables. The user defined variables would then be transformed via expressions which were translated into the required inputs for the autotransformer/tapped inductor model devices. This was in fact what was done within the AUTOXFMRft device.

SMPS Autotransformer model

A primitive diode-tapped Buck converter is shown in Figure 9 following:

Figure 9

Primitive Diode-Tapped Inductor Buck circuit

Briefly, X1 represents a transistor switch while D1 represents a flywheel diode. When X1 is on, current flows through L1 and L2, charging the output capacitor and supporting the load. When the switch turns off, the voltage across L2 reverses. Current continues to flow through L2 and the output voltage decays. L1 is equivalent to Lp, and L2 to Ls with corresponding turns of np and ns..

Now what is interesting is that the discharge inductance is L2, whereas the charge inductance is the Lt equivalent of L1 and L2 – not their sum but somewhat larger. During the charge interval the voltage across Lt is Vin – Vout. The flux linkages are equal to (Vin – Vout)*D*(np + ns). During the discharge interval the voltage across L2 is Vout, and the flux linkages are Vout*(1-D)*ns.

Per the law of Conservation of Flux Linkages,

(Vin – Vout)*D*(np + ns) = Vout*(1-D)*ns Eq. 12

In this case, the turns ratio is defined as:

n = (ns + np)/ns Eq. 13

Solving for Vout/Vin – M(D) produces:

M(D) = D/(D + (1 – D)/n) Eq. 14

D is the normally commanded value for a plain buck converter. It is plain to see that if n = 1, that M(D) = 1. The circuit behaves as a plain Buck converter. Now with n = 5, \

M(D) = D/(D + 1/5- D/5) = D/(4*D/5 + 1/5) Eq. 15

M(D) = 5D/(4D + 1) Eq. 16

As n increases, the denominator decreases, increasing M(D This would have utility where one wished to use a Buck converter with an input voltage of 300V, and 5V output. The ‘normal’ Buck value of D would be 5/300, or 0.016.67. This small value could be troublesome given circuit turn on and turn off times. The diode tapped Buck inductor can increase this, as it can be seen that:

M(D) = (5*5/300)/(4*5/300 + 1) = (25/300)/(320/300) Eq. 17

M(D) = 25/320 = 0.078125 Eq. 18

At any rate, a circuit to test the use of a tapped inductor is shown in Figure 10.

Figure 10

Diode-Tapped Inductor Buck Test Circuit

In Figure 9 Vin = 300V, Vout = 5V, D = 5/300, Lp = 1200u, Ls = 25u, Lt
= 1875u -= (Lp^{0.5} + Ls^{0.5})^{2} , n = 4.

The U2 device is an Erickson CCM-DCM2d device, covered in my book “SMPS
Average Models for SPICE3”. The device is interesting in that the
flywheel diode is isolated, allowing easy modeling of devices where the
transistor switch is in the primary of a transformer and the flywheel
diode is in the secondary. The inductance parameter is the value of the
total inductance of a part with ns plus np turns, or Lt. It is also passed
a transformation ratio to relate the secondary flywheel diode current
to the primary side of the transformer (where the transistor switch is
usually located). In this case, because of the circuit topology, * for
the overall circuit*,

n = ns/(np + ns) = 1/(np/ns +1) = 1/5 Eq. 19

This value of n, one greater than the autotransformer value of n, is passed to the CCD-DCMd2 device as a value. In this case we are passing a numerical value for n, fs and L but one could also have used equations involving parameters to transform known quantities into those required by the subcircuits in this simulation. This is especially important if we wished to sweep those parameters for a simulation.

However the purpose of this example was to illustrate the working of the AUTOXMFkc device, consequently we will continue on

A single operating point analysis of the circuit fails . There is a bug
in the program that sometimes causes this to occur. (This has been reported
and is being studied by the makers of B2SPICE^{TM}.) Sometimes a sweep will
have this problem, but assuming it occurs for a OP sweep, one workaround
is to perform an operating point GRAPH as opposed to a table output. Then,
while in that graph window, click on the SPECIAL FUNCTIONS command and
in the drop down menu select CREATE TABLE FROM GRAPH. This will work for
an operating point sweep in most cases.

However, an operating point sweep does work, as shown in Figure 11.

Figure 11

Diode-Tapped Inductor Buck OP Sweep

The DC operating point analysis with the load resistance swept shows
that the idealized circuit value of M(D) is78.125 mv, as predicted over
most of the load range. At somevalue between 16 and 18 ohms, however,
the circuit enters into Discontinuous Conduction Mode (DCM). Then the
value of M(D) increases, and D decreases. As this is an * average*
simulation, the circuit still shows it regulates, but for these values
in a switching circuit simulation the ripple would increase.

Now the next simulation is a sweep of the AC response with the load resistance varied from two to 20ohms in steps of 2 ohms also. This is shown in Figure 12.

Figure 12

Diode-Tapped Inductor Buck Load AC Sweep

In Figure 11 are shown gain and phase plots of the circuit over the range of load resistance. Two main groups of curves for each are present. The bold curve is the phase plot. Most of these curves are close, however the two discrepant occur in DCM mode. The upper curves with the peak are the gain plots, with the two discrepant curves occurring when DCM mode is reached. The curves are typical and correct.

Next, disabling the resistive sweep parameter and sweeping the coefficient
of coupling from zero to unity in steps of 0.1 produces the family of
gain and phase curves shown in Figure 13.

Figure 13

Diode-Tapped Inductor Buck kc AC Sweep

In Figure 14 is seen the effects of varying kt (n , Lp and Ls) on the phase gain plots.

Figure 14

Diode-Tapped Inductor Buck kt AC Sweep

##### Summary:

Two device models have been prepared for an autotransfrmer/tapped-inductor device. In both models the library inductive coupling device was used, however the coefficient of coupling is now accessible for use as a parameterized device value – directly or indirectly.