Transformer Calculation, Losses and Leakage Inductance

How to calculate transformator, which losses and parasitic parameters are there in the transformer and how can they be measured and subsequently represented in a simulation model?

Key Takeaways

• Design procedure for real transformers: measure main inductances, leakage, winding capacitances and DCR, then map them into an equivalent circuit and LTspice model for accurate behavior prediction.​
• Main transformer losses are copper and core losses; leakage, proximity/skin and capacitive losses are usually secondary but become important at higher frequencies and in fast‑switching designs.​
• Leakage inductance and primary inductance are the dominant parameters for return loss and bandwidth, while copper and core loss primarily set efficiency and temperature rise.​
• Parasitic capacitances and inter‑winding capacitance have limited impact on return loss in most data transformers, but can matter when leakage is very low or operating frequency is high.​

Transformer Losses, Parasitic Parameters and Equivalent Circuit

“Ideal” transformer models are usually used to make it as easy as possible for the developer and to reduce the computation time in LTspice. Only the inductance values for the primary and secondary are required here, as well as the coupling factor K (here in statement K1 Lp LS set to 1 = ideal).

The simulation results are far closer to practice if the coupling factor is already taken into consideration [1], because transformers have stray inductance of 2% ~ 8%  depending on the construction.

We use the following equivalent circuit for further consideration and to determine the parasitic elements:

Measuring the Primary and Secondary Inductance

To measure the primary and secondary inductance, the respective winding not measured must remain open.

For Lp = 939 µH and Ls = 26.87 uH in Figure 1. example, the calculated turns ratio is 5.91.

Total Stray Inductance

Primary stray inductance and transferred secondary inductance can be measured by short-circuiting the secondary winding (pin 5/8) and measuring between pins 1/4.

Please note:
The stray inductance is as well in series with the transmission path . The stray inductance describes that part of the magnetic field, which is not enclosed from the respectively other winding and therefore contributes not to the coupling . The stray inductance results simply from the mechanical arrangement of the windings against each other . A decrease of the stray inductance comes along with the increase of the coupling capacitance . The total stray inductance (primary inductance + transferred secondary stray inductance) is measured by measuring at short circuited secondary winding (Please note: To not distort the measurement result a low impedance short circuit is necessary) .

Many applications demand as small a stray inductance as possible. It can be minimized using various winding techniques. The windings should be as wide as possible. A sandwich construction also helps, as in the case of the proximity effect. However, these techniques increase the coupling capacitance between the primary and secondary sides.

DC current winding resistances

RCu_prim and RCu_sec between pins 5/8 and 1/4 respectively can be measured with an ohmmeter. Example RCu_prim: 265 mΩ and RCu_sec: 858 mΩ

Coupling capacitance

Additional parasitic parameters include the coupling capacitance (capacitance between the primary and secondary sides) and the winding capacitance (capacitance between the turns of a winding). The influence of coupling capacitance on the circuit can be reduced by shielding windings between the primary and secondary sides. However, minimization of the coupling capacitance by winding in several sections or by inserting thick insulation between the primary and secondary side directly causes an increase in stray inductance. The coupling capacitance can be measured directly. The winding capacitance is measured indirectly via the resonance between the main inductance and the capacitance. An LCR bridge is used to measure from winding to winding, in this case between pins 1/5. For measurement reasons both windings should be separately short-circuited so the measurement result is not distorted.

Winding capacitances

The winding capacitances can only be determined indirectly from the resonances with the main inductance (L_prim/L_sec). The impedance with the secondary side “open” is measured with an impedance analyzer. The winding capacitance of the primary side is then calculated from the resonant frequency.

The example transformer resonant frequency is 875 KHz, the measurement resulted in L_prim with 939 µH. Rearranging the formula for C_wprim results in C_wprim 35 pF and for C_wsec 1,2 nF

The same approach is also taken on the secondary side. This produces the following simulation equivalent circuit shown in Figure 7.

The discrete equivalent circuit can presented in further simplified form, because LTspice offers the option of including the coupling factor, RCu_prim, RCu_sec; C_wsec and C_wprim in the components Lp and Ls, and of defining the stray inductance through the K statement.

Further calculation formulas for the equivalent parameters for the model with main inductance L_m:

The values for the resistances are determined by simple measurement with the ohmmeter. This model does not consider core losses, any capacitance or the frequency dependence of resistances due to the skin and proximity effects.

Transformer Loss Mechanisms in Practice

In real transformers, total losses are the sum of copper losses in the windings, core losses in the magnetic material, and additional stray and proximity‑related effects. For most low‑frequency and SMPS applications, copper and core losses dominate, while dielectric and stray losses are usually secondary but can become relevant at high frequencies.

Loss type	Dominant cause	Depends mainly on	Typical design lever
Copper (P_Cu)	I²R in windings	RMS current, DCR, temperature	Wire gauge, parallel strands, winding length
Core (P_core)	Hysteresis + eddy in core	B_pk, frequency, material	Core material, flux density, frequency
Leakage‑related	Energy in leakage inductance	I, L_leak, switching behavior	Winding geometry, interleaving, shunts
Proximity/skin	Non‑uniform current distribution	Frequency, conductor size	Litz wire, foil, layer structure
Dielectric/capacitive	Loss in insulation and C_ww, C_w	Frequency, voltage, C values	Insulation system, shielding, layout

In low‑frequency power transformers, copper and core losses are often comparable, whereas in high‑frequency data and telecom magnetics, copper, proximity and capacitive effects become increasingly important relative to core loss.

For practical design, a common target is that copper and core loss are of the same order of magnitude to avoid over‑stressing one part of the transformer. As a rule of thumb:

• Limit copper loss by selecting wire and window utilization such that winding temperature rise stays within the insulation class.
• Limit core loss by selecting a core material and flux density so that catalog $P\mathrm{\_}core$ at your frequency and B_pk stays below the thermal budget for the core area and cooling conditions. (Thermal Modeling of Magnetics)

Leakage Inductance

Leakage inductance is produced by the imperfect magnetic coupling between the transformer windings. The magnetic flux generated in the primary winding is never transferred 100% to the secondary winding. Its magnitude plays a key role in modern switched-mode power supplies. Whether it needs to be minimized or maximized, the use of complex prediction models, which are many times far from reality, is required.

In cases where higher leakage inductance is required, a magnetic shunt can be inserted between layers, while another possibility would be the use of fractional turns.

Leakage inductance depends on:

• Winding geometry.
• Core geometry.
• Number of turns.

Due to the fact that the winding geometry and arrangement have a high influence on the leakage inductance, it gets really complicated to calculate it analytically and get reliable results. However, if you know the effect these parameters have on the leakage, you will be able to control it.

From a loss perspective, energy stored in leakage inductance is usually dissipated in snubbers or clamp networks in SMPS, contributing indirectly to switching losses. In data transformers, leakage inductance mainly degrades bandwidth and return loss, while its direct power loss contribution is small; it is therefore treated as a performance, not a thermal, constraint.

further reference: How to Suppress Leakage Inductance in Transformer Design

Transformer Application Requirements

As illustrated with the transformer equivalent circuit, transformers have numerous parasitic properties, which can have a negative effect on the signal. In this chapter it is therefore explained why and where one applies transformers. An additional section deals with the requirements for signal transformers. To conclude the chapter, some standard commercially available transformers are described.

Function and Application Areas of Transformers

As a result of their functionality, transformers can be used for various tasks:

• Isolation: Transformers are constructed of several windings. Depend ing on the additional isolation, various potentials can be separated or isolated from one another
• Voltage transformation: Voltages are transformed proportionally to the turns ratio
• Current transformation: Currents are transformed inversely to the turns ratio (see chapter I/1.9)
• Impedance matching: Impedances are transformed as the square of the turns ratio

This gives rise to various applications for transformers:

• Voltage (power) supplies: Here the main functions of the trans former are voltage transformation and isolation
• Current converters: Here the main function is to convert high currents into small measurable currents
• Pulse transformers, e.g. drive transformers for transistors: The main function is isolation; sometimes higher voltages are also required to drive a transistor
• Data transformers: Here the main function is also isolation. In addition, sometimes different impedances have to be matched or voltages increased.

Requirements for Data and Signal Transformers

Transformers are used on data lines mainly for isolation and impedance matching. The signal should be largely unaffected in this case. The magnetizing current is not transferred to the secondary side. For this reason, the transformer should have the highest possible main inductance.

The signal profiles are usually rectangular pulses, i.e. they include a large number of harmonics. For the transformer, this means that its transformation properties should be as constant as possible up to high frequencies. Taking a look at the transformer equivalent circuit at the previous chapter, it is apparent that the leakage inductances contribute to addition frequency-dependent signal attenuation. The leakage inductances should therefore be kept as low as possible. Signal transformers therefore usually deploy ring cores with high permeability. The windings are at least bifilar; wound with twisted wires is better still. Because the power transfer is rather small, DCR is of minor importance.

The direct parameters, such as leakage inductance, interwinding capacitance etc. are usually not specified in the datasheets for signal transformers, but rather the associated parameters, such as insertion loss, return loss, etc.

The most important parameters are defined as follows:

• Common Mode Rejection: Measure of the suppression of DC interference

• Total harmonic distortion: The relationship between the total energy of the harmonics and the energy of the fundamental

• Bandwidth: Frequency range in which the insertion loss is lower than 3 dB

A Transformer’s Effect on Return Loss

Return loss is an expression in decibels (dB) of the power reflected on a transmission line from a mismatched load in relationship to the power of the transmitted incident signal. The reflected signal disrupts the desired signal and if severe enough will cause data transmission errors in data lines or degradation in sound quality on voice circuits.

The equation for calculating return loss in terms of characteristic complex line impedance, Z_O, and the actual complex load, Z_L, is shown below in eq. [3]. Expanding the return loss equation to terms of resistance and reactance we achieve formula as per eq. [4]:

Since return loss is a function of line and load impedance, the characteristic impedance of a transformer, inductor or choke will affect the return loss. A simple impedance sweep of a magnetic component reveals that the impedance varies over frequency, hence the return loss varies over frequency. We will discuss further the effects of a transformer on return loss later. Now let’s explore the relationship of return loss to other common reflection terms.

Reflection Coefficient

While return loss is generally used to denote line reflections in the magnetics industry; a more common term in the electronics industry for reflections is the complex reflection coefficient, gamma, which is symbolized either by the Roman character G or more commonly the equivalent Greek character Γ (gamma). The complex reflection coefficient Γ has a magnitude portion called ρ (rho) and a phase angle portion Φ (Phi). Those of you familiar with the Smith Chart know that the radius of the circle encompassing the Smith Chart is rho equal to one.

The reflection coefficient, gamma, is defined as the ratio of the reflected voltage signal in relationship to the incident voltage signal – see eq. 5 below.

Return loss expressed in terms of gamma is shown in the equation [7] below:

Standing Wave Ratio

The reflections on a transmission line caused by impedance mismatches reveal themselves in an envelope of the combined incident and reflected wave forms. The standing wave ratio, SWR, is the ratio of the maximum value of the resulting RF envelope E_MAX to the minimum value E_MIN.

Transmission Loss

The last signal reflection expression that we will discuss is the transmission loss. Transmission loss is simply the ratio of power transmitted to the load relative to the incident signal power. Transmission loss in a lossless network expressed in terms of the reflection coefficient is shown below:

Don’t forget that the magnitude of gamma (|Γ|) is rho (ρ) and either form can be found in publications and documents regarding reflections.

Related Terms

Reviewing the complex reflection coefficient formula we can see that the closer matched the load impedance Z_L is to the characteristic line impedance ZO the closer to zero the reflection coefficient is. As the mismatch between the two impedances increase the reflection coefficient increases to a maximum magnitude of one.

The table below shows how the varying complex reflection coefficient relates to SWR, return loss and transmitted loss. As can be seen a perfect match results in SWR equal to 1 and an infinite return loss. Similarly an open or short at the load will result in return an infinite SWR and 0 dB of return loss.

Maximum Power Transfer

Maximum power transfer is obtained from the source to the load when the source impedance is equal to the complex conjugate of the load impedance. This not only maximizes power but minimizes reflection energy back to the source.

Return Loss with Matched Load

Let’s take an example of a matched line and load. Let’s say that Z_O = 100 Ω in an ADSL application and that it is terminated with a purely resistive load of 100 Ω.

Since the load and source are purely resistive, the return loss will be the same at any frequency. Substituting and calculating shows that RL = ∞.

Return Loss with Unmatched load

Let’s take the same example of an ideal transformer, but with a slightly unmatched load. Let’s say that Z_O = 100+0j Ω as before, but now we will calculate return loss at a number of purely resistive load impedances to show how return loss if affected by mismatch. The resistive load is used again so that the return loss will be independent of frequency.

The results show that return loss is a function of mismatch and without regard to which direction the mismatch is in. If we look at the case of a slightly mismatched line versus load we see that it is independent of frequency if the line and load are purely resistive. Also note that if the match was perfect, the return loss would be infinite.

Return Loss with Nearly Ideal Transformer

Now let’s take the same example of a matched line and load, but add in a 1 : 1 transformer which is ideal except for having a primary inductance of L_P = 600 µH. Again we assume the line impedance is a purely resistive 100 Ω as well as the load impedance.

When we had an ideal transformer with both line and load impedances purely resistive, our return loss did not vary over frequency and was the same at any frequency. Now however, the inductance will vary over frequency thereby causing the effective load to vary over frequency. The return loss calculation also becomes more complex due to the load impedance now being complex.

Rather than go through all of the complex impedance calculations, I will show the steps required to calculate the return loss.

Step 1 : Using impedance transformation calculations, transform the impedance to the same side of the ideal transformer as the primary inductance. In this case the ideal transformer is a 1 : 1 transformer and the load does not change. See Figure 16.

Step 2: Combine the X_L the current Z_L = R_L+0j with a resultant Z_L’ which is complex.

Step 3: Calculate return loss using the resultant load and the original resistive line impedance as per eq. [4].

Results: Looking at the results over frequency we can see that the inductance at the lower end is mismatched due the inductance shorting out the load. The lower the primary inductance the more the load will be shunted. Looking at the graphed results we see that return loss due to the primary inductance will behave much like a filter in that it has a knee which will vary with inductance and the slope after the knee is 20 dB per decade.

Return Loss with Leakage Inductance Added

Consider return loss with leakage inductance as shown in Figure 19. Now let’s add leakage inductance of 1 µH to the same transformer under the same load conditions (Figure 20.). The effective load is calculated in the same manner with ZL’ the reactance of the primary in parallel with the load impedance after transformation. The ZL’’ is ZL’ with the series leakage inductance reactance added to it.

Using the same return loss formula we can then calculate our return loss at various frequencies. From the graphed results we see that the high frequency return loss is affected by the leakage inductance.

For most transformers the primary and leakage inductances will have the greatest effect on return loss, providing that the turns ratio chosen effectively matches the load resistance to the line impedance.

Return Loss with a Less-Than-Ideal Transformer

With the linear transformer model that is typically used in low frequency transformer design applications, we can calculate theoretical return loss based on lumped parameter analysis. With the exception of inter-winding capacitance, we can reduce the linear transformer model to a load impedance by either combining the elements in parallel or series. Keep in mind that the secondary DC resistance and the Z_L have to be transformed by dividing by n2 when brought to the line side of the model.

Inter-winding capacitance can not be so simply modeled because it resides on neither the line nor the load side of the model and can not be transformed into the equivalent load. At low frequencies the inter-winding capacitance acts as an open across the transformer and typically can be ignored. In fact most modeling programs for transformers do ignore inter-winding capacitance as leakage inductance and primary inductance are the dominant elements. However in certain designs where inter-winding capacitance is fairly large and the operating frequencies are high, it can become a very significant factor. Suffice it to say that if inter-winding capacitance needs to be included in the model it would be wise to use a more sophisticated analysis program such as LTspice.

Let’s take a look now at the linear transformer model for the theoretical ADSL transformer shown below with a load that is just off of the ideal 25 Ω for a perfect match. We will take this and model the effect of the various elements looking at it parameter by parameter.

Return Loss Effect of DCRs

The return loss effect of the DC resistance in the example on Figure 24. high lights two observations. First of all even though the secondary resistance is lower at 1.5 Ω in comparison to the primary resistance of 3.0 Ω, the effect on return loss is much greater. The reason for this is that the 1.5 Ω secondary when reflected to the primary side of the transformer is seen as 6.0 Ω.

Also note that a lower return loss number is only slightly affected by other elements that have significantly better return loss when standing alone. The return loss when due only to the secondary resistance is roughly 30 dB while the return loss due to the primary resistance is roughly 37 dB. When combined, the net effect is a return loss of 27 dB.

Return Loss Effect of Leakage Inductance and Distributed Capacitance

The effects on return loss by the leakage inductance and the distributive capacitance parameters of a transformer are interesting to compare as well. We see from the example on Figure 25. that the effects due solely to leakage inductance reveal a decaying return loss at the rate of 20 dB per decade. Now taking a look at the distributed capacitance we see that it causes a high-end decay at the same rate with the knee at a higher frequency.

The comparison gets interesting when we looked at the combined affect. When combined the net result is an improvement of return loss. Why is this? If you remember in our previous discussion, the return loss is a function of mismatch without regard to which direction the mismatch is in. With this example the mismatch is in opposite directions so the addition of the distributed capacitance effect actually improves the overall return loss.

Thinking about this in analytical terms, what is happening in the equivalent circuit? The reflected load is being increased by the reactance due to the leakage inductance thereby causing mismatch. However the reactance of the distributed capacitance is in parallel there by reducing the mismatch back toward the optimal 100 Ω reflected load.

Return Loss Effect of Inter-Winding Capacitance

As mentioned earlier, the effect of inter-winding capacitance is very difficult to calculate using simple equivalent impedance transformations. The problem is that the inter-winding capacitance is shared by both windings and is not clearly on one side of the ideal transformer or the other. The impact to the circuit model is therefore not so straight forward and requires more sophisticated modeling techniques. The example below was modeled with PSPICE rather than with simple calculations.

Typically however, the inter-winding capacitance has very little effect on the return loss in comparison to the leakage inductance and can be ignored. A word of warning is in order however since cases where leakage inductance is very low while at the same time the inter-winding capacitance is very high, inter-winding apacitance can become a factor to reckon with. See figure 26.

Return Loss Effect of Resistive Core Loss and Inductance

In this example we compare return loss due to the primary inductance as well as to the resistive core loss assuming that core loss factor, R_cAlpha, is equal to 0.44. As can be seen from the return loss due to the combined effect, the resistive core loss has very minimal impact.

For data and telecom transformers, core loss is usually dimensioned from thermal considerations rather than from its direct influence on return loss. In contrast, in offline SMPS and PoE power magnetics, core loss can be a major contributor to total losses and must be checked against the core manufacturer’s loss curves at the intended flux swing and frequency.

In very low frequency applications, such as audio, the resistive core loss can be a factor.

Return Loss Effect of All Parameters

Finally, looking at the effects of all parameters combined (Figure 28.) we can determine which are the significant factors in a typical transformer application. As can be seen from the results below, the leakage inductance and the primary inductance are the driving factors. While the other parasitic parameters do play a role in shaping the return loss response, they play a relatively minor role in a typical transformer design.

A Closer Look at Dominating Parameters

In closing we take a closer look at the dominating parameters of the transformer. The top graph on Figure 29. shows the return loss of various models in comparison to the ideal transformer with the slightly mismatched load. Then the lower graph on Figure 29. just zooms in on the non-ideal transformer cases.

Practical tip:

These graphs high-light the fact that primary and leakage inductance are the parameters that typically dictate return loss and that there is justification to ignoring inter-winding capacitance in most applications.

Read Also

• Winding Losses in Transformers and Inductors
• Transformer Testing, Parasitic Inductors/Capacitance & Lab traps

Conclusion

Practical transformer design always requires a compromise between ideal behavior, parasitic elements and acceptable loss levels. By identifying and measuring the key parameters (main inductances, leakage inductance, copper and core losses, winding and inter‑winding capacitances) and mapping them into a realistic SPICE model, the designer can predict bandwidth, return loss and thermal behavior with good accuracy. The examples for data, telecom and power transformers show that primary inductance and leakage inductance usually dominate return loss, while copper and core losses dictate efficiency and temperature rise. Using the presented methods and rules of thumb, engineers can systematically optimize transformer construction and select suitable catalog parts for their specific application.

FAQ