Understanding the Non-Idealities of Magnetically Coupled RF Transformers

Transformers perform a number of useful functions in RF designs, including:

• Impedance matching.
• Isolation between two parts of the circuit.
• Transformation between balanced and unbalanced signal environments.

There are multiple ways to implement an RF transformer. The simplest method is to use a pair of magnetically coupled coils. To help increase the coupling between the coils, transformers of this kind usually also include a magnetic core.

Figure 1 shows a push-pull RF power amplifier with transformers at the input and output.

 

Figure 1. A transformer-coupled push-pull amplifier uses two transformers to produce the input and output signals. Image used courtesy of Steve Arar

 

To understand the limitations of the above circuit, we need to understand the non-idealities of practical transformers. This article explores the major non-idealities exhibited by magnetically coupled, magnetic-core transformers at high frequencies. We’ll start with an overview of this transformer’s ideal version.

 

The Ideal Magnetically Coupled Transformer

An ideal transformer provides perfect magnetic coupling between the primary and secondary coils, and exhibits no loss of energy. Figure 2 shows the schematic symbol of an ideal 1:n transformer. Note the polarity dots—these identify which terminals are in phase.

 

Figure 2. Transformer schematic symbol. Image used courtesy of Steve Arar

 

The transformer dot convention specifies the direction in which each coil is wrapped around the core with respect to the other. In Figure 2, according to the dot convention, current that flows into the primary coil's dotted end will come out of the secondary coil's dotted end.

With the voltage polarities and current directions shown in the figure, the two defining equations for the ideal transformer are:

$$v_{2}~=~nv_{1}$$

Equation 1.

 

$$i_{2}~=~-\frac{i_{1}}{n}$$

Equation 2.

 

where:

i₁ is the primary current.

i₂ is the secondary current.

v₁ is the primary voltage.

v₂ is the secondary voltage.

The negative sign in Equation 2 results from our drawing i₂ as an input to the transformer’s dotted secondary terminal in Figure 2. If we had drawn i₂ as exiting the dot terminal, the sign would be positive.

 

Imperfect Magnetic Coupling

In practice, only a portion of the magnetic flux produced by one coil couples to the other. The degree of coupling between the primary and secondary windings is characterized by the mutual inductance (M), as shown in Figure 3.

 

Figure 3. An ideal transformer made up of two inductors with a mutual inductance. Image used courtesy of Steve Arar

 

The magnetic coupling between the coils depends on the following factors:

• The separation between the coils.
• The coils’ orientation.
• The number of turns in each coil.
• The magnetic properties of the core.

For the above circuit, the primary and secondary voltages can be expressed by the following phasor equations:

$$V_{1}~=~j \omega L_{1}I_{1}~+~j \omega MI_{2}$$

Equation 3.

 

$$V_{2}~=~j \omega L_{2}I_{2}~+~j \omega MI_{1}$$

Equation 4.

 

where:

L₁ is the inductance of the primary coil alone, with the secondary open-circuited.

L₂ is the self-inductance of the secondary coil.

M is the mutual inductance, and must be a positive value.

According to the dot convention, if currents are sent into the dotted terminals of the coupled inductors, the magnetic flux linking the coils reinforces the self-flux of the coils. This is why the mutual induction terms are added to the self-inductance terms in Equations 3 and 4. If, instead, one coil current enters the dotted end of a coil while the other coil current enters the undotted end of the coil, we subtract the mutual inductance term from the self-inductance term.

The degree of magnetic coupling can also be specified through the coupling coefficient parameter (k), defined as:

$$k~=~ \frac{M}{\sqrt{L_{1} L_{2}}}$$

Equation 5.

 

The maximum value of M is \(\sqrt{L_{1} L_{2}} \). Since M must have a positive value, this means that k has a minimum value of 0 and a maximum value of 1. When k = 0, no coupling exists.

Most power system transformers have a k that approaches 1. Because no magnetic materials are available in most integrated circuit processes, RF integrated circuit inductors typically have an overall coupling coefficient of 0.8 to 0.9.

 

Modeling the Transformer’s Magnetic Flux Leakage

Figure 4 uses the ideal transformer from Figure 3 to create a useful model of flux leakage. The inductors L_pl and L_pm account for imperfect coupling between the coils.

 

Figure 4. Modeling flux leakage in a transformer. Image used courtesy of Steve Arar

 

The voltage and current quantities of this circuit are governed by Equation 2, and the two inductors divide the total primary self-inductance into two portions. The leakage inductance (L_pl) is the portion that doesn’t contribute to the magnetic coupling between the primary and secondary windings. The magnetizing inductance (L_pm) is the portion that does contribute to the magnetic coupling between the coils.

The leakage inductance in the above model is given by:

$$L_{pl}~=~(1~-~k^2) L_1$$

Equation 6.

 

and the magnetizing inductance, by:

$$L_{pm}~=~k^2 L_1$$

Equation 7.

Finally, the turns ratio is defined as:

$$n~=~\frac{L_2}{M}$$

Equation 8.

 

The Lower Bound of the Transformer Frequency Range

The magnetizing inductance reminds us that a real-world transformer can’t operate at DC—even though the leakage inductance has a negligible effect at low frequencies, the magnetizing inductance tends to short out the signal path. This inductance, along with the source resistance driving the primary winding (R_S), forms a high-pass filter with cutoff frequency \(\frac{R_S}{L_{pm}} \).

The impedance of the magnetizing inductance must reach a minimum value before the transformer will operate properly. The impedance only reaches this level, however, when the input frequency is a decade or so above the cutoff frequency. To reduce the cutoff frequency, we have to raise the inductance of the windings. This increases the parasitic capacitances of the windings, eventually limiting the high-frequency response of the transformer.

 

Major Core Loss Mechanisms

As we mentioned earlier in the article, magnetically coupled transformers usually include magnetic cores. In addition to the imperfect coupling between the coils, we need to account for the two major loss mechanisms that affect these cores—hysteresis loss and eddy current loss. These can make conventional magnetic cores extremely lossy at high frequencies.

The application of an AC signal to the transformer causes the core material’s magnetic domains to vibrate. Because the core particles exhibit inertia and friction, the motion of the magnetic domains results in the loss of energy we know as hysteresis loss. At higher frequencies, the magnetic domains of the core switch faster, which is why the hysteresis loss increases as the signal frequency rises. Increasing the current also increases hysteresis loss.

Some magnetic core materials are conductors. When the magnetic flux through an electrically conductive core changes, it creates small current loops. These current loops, known as eddy currents, produce a load-independent power loss. The eddy current loss is proportional to the square of frequency.

 

Modeling the Transformer’s Losses and Non-Idealities

Figure 5 shows a more elaborate model of the transformer that includes several non-idealities. The core losses are modeled by a frequency-dependent resistance (R_c) in parallel with the primary winding.

 

Figure 5. Equivalent circuit model of a transformer with non-idealities. Image used courtesy of Mini-Circuits

 

Above, R₁ and R₂ model the resistive losses of the primary and secondary windings. Due to the skin effect, these loss terms increase with frequency. They also increase with temperature, producing higher losses in higher-power applications. Separately, note that a series inductance (L₂) is present in the model to account for the leakage inductance of the secondary winding.

Energy stored in the electric fields between the windings also adversely affect the transformer’s performance at high frequencies. The windings need to be placed close to one another to maximize the coupling factor, and this proximity can produce significant parasitic capacitances. Most magnetic core materials have a relative dielectric constant greater than one, further increasing these capacitances.

Figure 5 models the following parasitic capacitances:

• C₁: The parasitic capacitance of the primary winding, which might also include other parasitic capacitances at the input.
• C₂: The parasitic capacitances associated with the secondary winding, known as either the intra-winding capacitances or the self-capacitances of the windings.
• C/2: The interwinding capacitances, which account for the capacitive coupling between the two windings.

Though the parasitic capacitances are modeled as lumped components in Figure 5, it’s important to remember that they’re actually distributed components. Figure 6 illustrates the distributed nature of the interwinding capacitance.

 

Figure 6. The interwinding capacitance is actually a distributed component. Image used courtesy of Steve Arar

 

The Upper Bound of the Transformer Frequency Range

The leakage inductance and the parasitic capacitances determine the upper limit of the transformer’s frequency range. As we increase the frequency, the reactance presented by the leakage inductance also increases, eventually blocking the signal. The parasitic capacitances present a decreasing reactance at high frequencies. C₁ and C₂ short out the signal path, and the interwinding capacitance bypasses the transformer.

 

A Better RF Transformer

Magnetically coupled transformers are best suited for low-frequency applications. To extend the transformer’s frequency range upward, we would need to decrease the leakage inductance and the parasitic capacitances. Doing so requires balancing contradictory requirements.

For example, we can reduce the interwinding capacitance by increasing the physical distance between the windings—but this lowers the coupling factor, leading to a higher leakage inductance. We could increase the coupling factor by using a magnetic core, but hysteresis loss and eddy current loss make magnetic cores unsuitable for high frequencies. Additionally, some RF applications need a relatively large impedance transformation ratio, which requires a transformer with extremely low leakage inductance.

Fortunately, there’s an elegant solution to this problem—we can use a transmission line transformer instead. With this type of transformer, the interwinding capacitance and the leakage inductance are assumed to be the distributed elements of a transmission line. In the next article, we’ll see how treating the coils of a transformer as a transmission line enables us to build transformers that successfully operate at RF and microwave frequencies.

As a final aside, proofs for some of the math in this article can be found in the following texts:

• “Introduction to Electric Circuits” by James A. Svoboda and Richard C. Dorf.
• “Electromagnetics for High-Speed Analog and Digital Communication Circuits” by Ali M. Niknejad.

 

Featured image used courtesy of Adobe Stock