Passive Matching Networks

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Controlling impedances in RF circuits are essential to maximize power transfer between blocks and to reduce reflections caused by impedance discontinuities along a signal path. In this module, we will define the quality factor, Q, for a device or a circuit, and use this Q to build circuits that modify the impedance seen across a port.

Device Quality Factor

The quality factor, Q, is a measure of how good a device is at storing energy. Thus, the lower the losses, the higher the Q. In general, if we can express the impedance or admittance of a device as

 

 

 

 

(1)

The quality factor can then be expressed as

 

 

 

 

(2)

The imaginary component, represents the energy storage element, and the real component, , represents the loss (resistive) component.

Series RC Circuit

A lossy capacitor can be modeled as a series RC circuit with the series resistance, , could represent the series loss due to interconnect resistances. Thus, the admittance of the lossy capacitor can be expressed as

 

 

 

 

(3)

We can therefore express the quality factor of the series circuit as:

 

 

 

 

(4)

Note that for the lossless case, , and consequently .

Parallel RC Circuit

A lossy capacitor can also be modeled as a parallel RC circuit, with the parallel resistance, could represent the energy loss due to the dielectric leakage of the capacitor. In this case, we can write the impedance of the circuit as:

 

 

 

 

(5)

Thus, the quality factor is:

 

 

 

 

(6)

We can see that in the lossless case, , corresponding to .

RL Circuits

Applying the ideas above to RL circuits, we can get the admittance of a series RL circuit as:

 

 

 

 

(7)

with quality factor:

 

 

 

 

(8)

Similarly, for parallel RL circuits, the impedance and quality factor can be expressed as:

 

 

 

 

(9)

 

 

 

 

(10)

Note that the quality factor is frequency dependent, and for or , we get the ideal case where or respectively.

General Case

In general, for a reactance in series with a resistance ,

 

 

 

 

(11)

 

 

 

 

(12)

and for a reactance in parallel with a resistance ,

 

 

 

 

(13)

 

 

 

 

(14)

Series-Parallel Conversions

It is very convenient to be able to convert the series RC or RL circuit to its parallel equivalent or vice-versa, especially in the context of matching circuits. If we equate the parallel and series impedances, , we get

 

 

 

 

(15)

 

 

 

 

(16)

 

 

 

 

(17)

We can look at the imaginary and real components separately. For the real components:

 

 

 

 

(18)

 

 

 

 

(19)

And for the imaginary components:

 

 

 

 

(20)

Using eq. 18 and eq. 20, we get:

 

 

 

 

(21)

 

 

 

 

(22)

We can use eqs. 18 and 22 to get the relationship between the series and parallel reactances:

 

 

 

 

(23)

 

 

 

 

(24)

Rewriting eq. 24 and for :

 

 

 

 

(25)

Key Results:
  • The quality factor is the same for the parallel and series circuits when the impedances are the same.

 

 

 

 

(26)

  • The parallel resistance is larger than the series resistance.

 

 

 

 

(27)

  • The parallel reactance is approximately equal to the series reactance for .

 

 

 

 

(28)

Basic Matching Networks

We can use our series-parallel conversions to increase or decrease resistances based on the quality factor. Consider the circuit below, where we want to match the source resistance, to a load resistance, using a lossless matching network:

L-Section Matching Circuits

Consider the circuit below, and assume . We can use our parallel-series transformations to convert the parallel circuit into a series circuit so that the the series resistance is equivalent to the source resistance,

 

 

 

 

(29)

Where . The reactance also gets transformed into:

 

 

 

 

(30)

Thus, to keep the impedance seen by the source purely resistive, we can cancel out the reactance by the reactance . Note that the negative sign relating and means that if is a capacitance, then should be an inductance, and vice-versa. For the case when , we just flip the circuit and the equations would still hold.

Design Example

Let . We can then use eq. 29 to solve for the value of and :

 

 

 

 

(31)

If we use a capacitor for , then . Thus, we need an inductance, . Solving for :

 

 

 

 

(32)

Note that matching circuit is narrowband since the perfect cancellation occurs at only one frequency, .

Instead of a capacitor, we can place an inductor in parallel with the larger resistance, and use a series capacitor to cancel out the series equivalent inductance. However, in this case, the capacitor will block any DC and low-frequency signals.

Loss in Matching Networks

In typical real matching networks, the series resistance of the inductor is the dominant loss mechanism. We can model this loss using a resistor in series with the inductor , as shown below.

The power delivered by the input source can be expressed as:

 

 

 

 

(33)

T and Pi Matching Networks