Difference between revisions of "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.
+
Controlling impedances in radio-frequency (RF) circuits is essential in order 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, <math>Q</math>, for a device or a circuit, and use this <math>Q</math> to build circuits that modify the impedance seen across a port.
  
 
== Device Quality Factor ==
 
== 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  
+
The quality factor, <math>Q</math>, is a measure of how good a device is at storing energy. Thus, the lower the losses, the higher the <math>Q</math>. In general, if we can express the impedance or admittance of a device as  
  
 
{{NumBlk|::|<math>Y\,\mathrm{or}\,Z=\frac{1}{A + jB}</math>|{{EquationRef|1}}}}
 
{{NumBlk|::|<math>Y\,\mathrm{or}\,Z=\frac{1}{A + jB}</math>|{{EquationRef|1}}}}
Line 13: Line 13:
  
 
=== Series RC Circuit ===
 
=== Series RC Circuit ===
A lossy capacitor can be modeled as a series RC circuit with the series resistance, <math>R_S</math>, could represent the series loss due to interconnect resistances. Thus, the admittance of the lossy capacitor can be expressed as  
+
[[File:RC Qseries.png|thumb|400px|Figure 1: A lossy capacitor.]]
 +
 
 +
A lossy capacitor, shown in Fig. 1, can be modeled as a series RC circuit with the series resistance, <math>R_S</math>, could represent the series loss due to interconnect resistances. Thus, the admittance of the lossy capacitor can be expressed as  
  
 
{{NumBlk|::|<math>Y_S =\frac{1}{R_S+\frac{1}{j\omega C}}</math>|{{EquationRef|3}}}}
 
{{NumBlk|::|<math>Y_S =\frac{1}{R_S+\frac{1}{j\omega C}}</math>|{{EquationRef|3}}}}
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We can therefore express the quality factor of the series circuit as:
 
We can therefore express the quality factor of the series circuit as:
  
{{NumBlk|::|<math>Q_S = \frac{\frac{1}{\omega C}}{R_S}=\frac{1}{\omega R_S C}</math>|{{EquationRef|4}}}}
+
{{NumBlk|::|<math>Q_S = \frac{\frac{1}{\omega C}}{R_S}=\frac{1}{\omega R_S C} = \frac{X_S}{R_S}</math>|{{EquationRef|4}}}}
  
Note that for the lossless case, <math>R_S=0</math>, leading to <math>Q_{S,\mathrm{ideal}} \rightarrow \infty</math>.
+
Note that for the lossless case, <math>R_S=0</math>, and consequently <math>Q_{S,\mathrm{ideal}} \rightarrow \infty</math>.
  
 
=== Parallel RC Circuit ===
 
=== Parallel RC Circuit ===
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Thus, the quality factor is:
 
Thus, the quality factor is:
  
{{NumBlk|::|<math>Q_P = \frac{\omega C}{\frac{1}{R_P}}=\omega R_P C</math>|{{EquationRef|6}}}}
+
{{NumBlk|::|<math>Q_P = \frac{\omega C}{\frac{1}{R_P}}=\omega R_P C=\frac{R_P}{X_P}</math>|{{EquationRef|6}}}}
  
We can see that in the lossless case, <math>R_P \rightarrow \infty</math>, resulting in <math>Q_{P,\mathrm{ideal}} \rightarrow \infty</math> respectively.
+
We can see that in the lossless case, <math>R_P \rightarrow \infty</math>, corresponding to <math>Q_{P,\mathrm{ideal}} \rightarrow \infty</math>.
  
 
=== RL Circuits ===
 
=== RL Circuits ===
Applying the ideas above to RL circuits, we can get the admittance of a series RL circuit as:
+
[[File:RL Qseries.png|thumb|400px|Figure 2: A lossy inductor.]]
 +
 
 +
Applying the ideas above to RL circuits in Fig. 2, we can get the admittance of a series RL circuit as:
  
 
{{NumBlk|::|<math>Y_S =\frac{1}{R_S+j\omega L}</math>|{{EquationRef|7}}}}
 
{{NumBlk|::|<math>Y_S =\frac{1}{R_S+j\omega L}</math>|{{EquationRef|7}}}}
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with quality factor:
 
with quality factor:
  
{{NumBlk|::|<math>Q_S = \frac{\omega L}{R_S}</math>|{{EquationRef|8}}}}
+
{{NumBlk|::|<math>Q_S = \frac{\omega L}{R_S}=\frac{X_S}{R_S}</math>|{{EquationRef|8}}}}
  
 
Similarly, for parallel RL circuits, the impedance and quality factor can be expressed as:
 
Similarly, for parallel RL circuits, the impedance and quality factor can be expressed as:
Line 47: Line 51:
 
{{NumBlk|::|<math>Z_P =\frac{1}{\frac{1}{R_P}+\frac{1}{j\omega L}}</math>|{{EquationRef|9}}}}
 
{{NumBlk|::|<math>Z_P =\frac{1}{\frac{1}{R_P}+\frac{1}{j\omega L}}</math>|{{EquationRef|9}}}}
  
{{NumBlk|::|<math>Q_P = \frac{\frac{1}{\omega L}}{\frac{1}{R_P}}=\frac{R_P}{\omega L}</math>|{{EquationRef|10}}}}
+
{{NumBlk|::|<math>Q_P = \frac{\frac{1}{\omega L}}{\frac{1}{R_P}}=\frac{R_P}{\omega L} = \frac{R_P}{X_P}</math>|{{EquationRef|10}}}}
 +
 
 +
Note that the quality factor is frequency dependent, and for <math>R_S=0</math> or <math>R_P \rightarrow \infty</math>, we get the ideal case where <math>Q_{S,\mathrm{ideal}} \rightarrow \infty</math> or <math>Q_{P,\mathrm{ideal}} \rightarrow \infty</math> respectively.
 +
 
 +
=== General Case ===
 +
[[File:Generic Qseries.png|thumb|400px|Figure 3: A lossy reactance.]]
 +
 
 +
In general, for a reactance <math>X_S</math> in series with a resistance <math>R_S</math>,
 +
 
 +
{{NumBlk|::|<math>Y_S =\frac{1}{R_S+jX_S}</math>|{{EquationRef|11}}}}
 +
 
 +
{{NumBlk|::|<math>Q_S = \frac{X_S}{R_S}</math>|{{EquationRef|12}}}}
 +
 
 +
and for a reactance <math>X_P = \tfrac{1}{B_P}</math> in parallel with a resistance <math>R_P = \tfrac{1}{G_P}</math>,
 +
 
 +
{{NumBlk|::|<math>Z_P =\frac{1}{G_P + jB_P}</math>|{{EquationRef|13}}}}
  
Note that the quality factor is frequency dependent, and in the ideal lossless case, either <math>R_S=0</math> or <math>R_P \rightarrow \infty</math>, leading to <math>Q_{S,\mathrm{ideal}} \rightarrow \infty</math> or <math>Q_{P,\mathrm{ideal}} \rightarrow \infty</math> respectively.
+
{{NumBlk|::|<math>Q_P = \frac{B_P}{G_P}= \frac{R_P}{X_P}</math>|{{EquationRef|14}}}}
  
 
== Series-Parallel Conversions ==
 
== 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.  
+
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, <math>\tfrac{1}{Y_S} = Z_S = Z_P</math>, we get
 +
 
 +
{{NumBlk|::|<math>R_S+ jX_S =\frac{1}{\frac{1}{R_P} + \frac{1}{jX_P}} = \frac{jR_P X_P}{R_P + jX_P}</math>|{{EquationRef|15}}}}
 +
 
 +
{{NumBlk|::|<math>\left(R_S+ jX_S\right)\left(R_P + jX_P \right) =jR_P X_P</math>|{{EquationRef|16}}}}
 +
 
 +
{{NumBlk|::|<math>R_S R_P + jR_P X_S+ jR_S X_P - X_S X_P = jR_P X_P</math>|{{EquationRef|17}}}}
 +
 
 +
We can look at the imaginary and real components separately. For the real components:
 +
 
 +
{{NumBlk|::|<math>R_S R_P - X_S X_P = 0</math>|{{EquationRef|18}}}}
 +
 
 +
{{NumBlk|::|<math>\frac{R_P}{X_P} = Q_P = \frac{X_S}{R_S} = Q_S = Q</math>|{{EquationRef|19}}}}
 +
 
 +
And for the imaginary components:
 +
 
 +
{{NumBlk|::|<math>R_P X_S + R_S X_P = R_P X_P</math>|{{EquationRef|20}}}}
 +
 
 +
Using eq. 18 and eq. 20, we get:
 +
 
 +
{{NumBlk|::|<math>\frac{X_S X_P}{R_S} X_S + R_S X_P = R_P X_P</math>|{{EquationRef|21}}}}
 +
 
 +
{{NumBlk|::|<math>\frac{X_S^2}{R_S^2} R_S + R_S = R_S\left(1 + Q^2\right)= R_P </math>|{{EquationRef|22}}}}
 +
 
 +
We can use eqs. 18 and 22 to get the relationship between the series and parallel reactances:
 +
 
 +
{{NumBlk|::|<math>R_S R_P - X_S X_P = R_S^2\left(1 + Q^2\right) - X_S X_P =0</math>|{{EquationRef|23}}}}
 +
 
 +
{{NumBlk|::|<math>R_S^2\left(1 + Q^2\right) - X_S X_P =R_S^2\left(1 + Q^2\right)\frac{X_S^2}{X_S^2} - X_S X_P = \frac{1 + Q^2}{Q^2} X_S - X_P = 0</math>|{{EquationRef|24}}}}
 +
 
 +
Rewriting eq. 24 and for <math>Q^2 \gg 1</math>:
 +
 
 +
{{NumBlk|::|<math>X_P = \frac{1 + Q^2}{Q^2} X_S = \left(1 + \frac{1}{Q^2}\right) X_S \approx X_S</math>|{{EquationRef|25}}}}
  
=== RC Circuits ===
+
{{Note|'''Key Results''':
For RC circuits, the series and parallel impedances are:
+
* The quality factor is the same for the parallel and series circuits when the impedances are the same.
  
{{NumBlk|::|<math>Z_S = R_S+\frac{1}{j\omega C_S}=\frac{j\omega R_S C_S + 1}{j\omega C_S}</math>|{{EquationRef|11}}}}
+
{{NumBlk|::|<math>\frac{R_P}{X_P} = Q_P = \frac{X_S}{R_S} = Q_S = Q</math>|{{EquationRef|26}}}}
  
{{NumBlk|::|<math>Z_P =\frac{1}{\frac{1}{R_P}+j\omega C_P}= \frac{R_P}{1 + j\omega R_P C_P}</math>|{{EquationRef|12}}}}
+
* The parallel resistance is larger than the series resistance.
 +
{{NumBlk|::|<math>R_P = R_S\left(1 + Q^2\right)</math>|{{EquationRef|27}}}}
  
Setting <math>Z_S = Z_P</math> and rewriting the espressions, we get:
+
* The parallel reactance is approximately equal to the series reactance for <math>Q^2 \gg 1</math>.
 +
{{NumBlk|::|<math>X_P = \left(1 + \frac{1}{Q^2}\right) X_S \approx X_S</math>|{{EquationRef|28}}}}
 +
}}
  
{{NumBlk|::|<math>\frac{j\omega R_S C_S + 1}{j\omega C_S} = \frac{R_P}{1 + j\omega R_P C_P}= \left(j\omega R_S C_S + 1\right)\left(1 + j\omega R_P C_P\right)=j\omega R_P C_S</math>|{{EquationRef|13}}}}
+
== Basic Matching Networks ==
 +
[[File:Matching block diagram.png|thumb|400px|Figure 4: An impedance matching network.]]
 +
We can use our series-parallel conversions to increase or decrease resistances based on the quality factor. Consider the circuit in Fig. 4, where we want to match the source resistance, <math>R_S</math> to a load resistance, <math>R_L</math> using a lossless matching network, i.e. a network make up of only reactances.
  
{{NumBlk|::|<math>j\omega R_S C_S + j\omega R_P C_P -\omega^2 R_S C_S R_P C_P + 1=j\omega R_P C_S</math>|{{EquationRef|14}}}}
+
=== L-Section Matching Circuits ===
 +
[[File:Matching L section.png|thumb|300px|Figure 5: An L-Section matching network and the network after the parallel-to-serial transformation of <math>R_L</math> and <math>jX_1</math>.]]
  
{{NumBlk|::|<math>j\omega \left(R_S C_S + R_P C_P - R_P C_S\right) -\omega^2 R_S C_S R_P C_P + 1 = 0</math>|{{EquationRef|15}}}}
+
Consider the circuit in Fig. 5, and assume <math>R_S<R_L</math>. We can use our parallel-series transformations to convert the parallel circuit into a series circuit so that the the series resistance <math>R_L^\prime</math> is equivalent to the source resistance, <math>R_S</math>
  
Equating the real part and the imaginary part to zero, we get:
+
{{NumBlk|::|<math>R_L^\prime = \frac{R_L}{1 + Q^2}</math>|{{EquationRef|29}}}}
  
{{NumBlk|::|<math>R_S C_S + R_P C_P - R_P C_S = 0</math>|{{EquationRef|16}}}}
+
Where <math>Q =\frac{R_L}{X_1}</math>. The reactance <math>X_1</math> also gets transformed into:
  
{{NumBlk|::|<math>\omega^2 R_S C_S R_P C_P = 1 = \frac{Q_P}{Q_S}</math>|{{EquationRef|17}}}}
+
{{NumBlk|::|<math>X_1^\prime = X_1 \frac{1}{\left(1 + \frac{1}{Q^2}\right)} \approx X_1</math>|{{EquationRef|30}}}}
  
Leading to the result that <math>Q_P=Q_S=Q</math>. Using eq. 17 to solve for <math>R_P C_P</math>, and plugging it into eq. 16, we get:
+
Thus, to keep the impedance seen by the source purely resistive, we can cancel out the reactance <math>X_1^\prime</math> by the reactance <math>X_2 = -X_1^\prime \approx -X_1</math>. Note that the negative sign relating <math>X_1</math> and <math>X_2</math> means that if <math>X_1</math> is a capacitance, then <math>X_2</math> should be an inductance, and vice-versa. Note that cancelling reactances at resonance could cause large voltages to appear across (or large currents through) the reactances. See this note on [[Resonance | resonance]] for more details. For the case when <math>R_S>R_L</math>, we just flip the circuit and the equations would still hold.
  
{{NumBlk|::|<math>R_S C_S + \frac{1}{\omega^2 R_S C_S} - R_P C_S = R_S + \frac{1}{\omega^2 R_S C_S^2} - R_P = R_S \left(1+ \frac{1}{\omega^2 R_S^2 C_S^2}\right) - R_P = R_S \left(1+ Q^2\right) - R_P = 0</math>|{{EquationRef|18}}}}
+
=== Design Example ===
 +
Let define the ''matching factor'' <math>m =\tfrac{R_{\mathrm{hi}}}{R_{\mathrm{lo}}}</math>, where <math>R_\mathrm{hi}=\max\left(R_S, R_L\right)</math> and <math>R_\mathrm{lo}=\min\left(R_S, R_L\right)</math>. Therefore for <math>R_L>R_S</math>, <math>R_{\mathrm{hi}}=R_L</math> and <math>R_{\mathrm{lo}}=R_S</math>. We can then use eq. 29 to solve for the value of <math>Q</math>, <math>X_1</math>, and <math>X_1^\prime</math>:
  
We can then express <math>R_P</math> as a function of <math>R_S</math> and <math>Q_S</math> as:
+
{{NumBlk|::|<math>Q=\sqrt{m - 1} = \frac{R_L}{X_1}=\frac{X_1^\prime}{R_S}</math>|{{EquationRef|31}}}}
  
{{NumBlk|::|<math>R_P = R_S \left(1+ Q^2\right)</math>|{{EquationRef|19}}}}
+
If we use a capacitor for <math>X_1</math>, then <math>C_1 = \tfrac{1}{\omega X_1}</math>. Thus, we need an inductance, <math>L_2=\tfrac{X_2}{\omega} =\tfrac{-X_1^\prime}{\omega}\approx \tfrac{-X_1}{\omega}</math>. Solving for <math>L_2</math>:
  
Substituting eq. 19 into eq. 17, we get:
+
{{NumBlk|::|<math>L_2 = \frac{1}{\omega^2 C_1^\prime} \approx \frac{1}{\omega^2 C_1}</math>|{{EquationRef|32}}}}
  
{{NumBlk|::|<math>\omega^2 R_S C_S R_P C_P = \omega^2 R_S^2 \frac{C_S^2}{C_S} \left(1+ Q^2\right) C_P =  \frac{1}{Q^2 C_S} \left(1+ Q^2\right) C_P =1</math>|{{EquationRef|20}}}}
+
Note that matching circuit is narrowband since the perfect cancellation occurs at only one frequency, <math>\omega</math>.
  
Thus, we can express <math>C_P</math> as:
+
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.
  
{{NumBlk|::|<math>C_P = \frac{Q^2}{1+ Q^2} C_S = \frac{1}{1 + \frac{1}{Q^2}} C_S</math>|{{EquationRef|21}}}}
+
== Loss in Matching Networks ==
 +
[[File:Lossy L match.png|thumb|400px|Figure 6: A lossy L-section.]]
 +
In typical real matching networks, the series resistance of the inductor is the dominant loss mechanism. We can model this loss using a resistor <math>R_2</math> in series with the inductor <math>L_2</math>, as shown in Fig. 6.
  
For <math>Q^2 \gg 1</math>, we get <math>R_P\approx Q^2 R_S</math> and <math>C_P\approx C_S</math>.
+
To determine the power loss due to <math>R_2</math>, we first calculate the power delivered by the input source:
  
=== RL Circuits ===
+
{{NumBlk|::|<math>P_{\mathrm{in}} = \frac{V_{\mathrm{in}}^2}{R_2 + R_L^\prime}</math>|{{EquationRef|33}}}}
In a similar way, we can convert the series RL circuit to its parallel equivalent, and vice versa. Thus, for <math>Z_P = Z_S</math> we get:
 
  
{{NumBlk|::|<math>\frac{1}{\frac{1}{R_P}+\frac{1}{j\omega L_P}} = \frac{j\omega R_P L_P}{R_P + j\omega L_P} = R_S+j\omega L_S</math>|{{EquationRef|22}}}}
+
Then we calculate the power delivered to the load, <math>R_L</math>:
  
{{NumBlk|::|<math>j\omega R_P L_P = \left(R_P + j\omega L_P\right) \left( R_S+j\omega L_S\right) = R_P R_S - \omega^2 L_P L_S + j\omega R_S L_P + j\omega R_P L_S</math>|{{EquationRef|23}}}}
+
{{NumBlk|::|<math>P_L = \left(V_{\mathrm{in}} \frac{R_L^\prime}{R_2 + R_L^\prime}\right)^2 \frac{1}{R_L^\prime}</math>|{{EquationRef|34}}}}
  
And once again, we can separate the real and imaginary components of eq. 23. Looking at the real components:
+
We can then define power loss as
  
{{NumBlk|::|<math>R_P R_S = \omega^2 L_P L_S </math>|{{EquationRef|24}}}}
+
{{NumBlk|::|<math>\mathrm{Loss} = \frac{P_{\mathrm{in}}}{P_L}=\frac{R_2 + R_L^\prime}{R_L^\prime}=1 + \frac{R_2}{R_L^\prime}=1 + \frac{R_2\left(1 + Q^2\right)}{R_L}</math>|{{EquationRef|35}}}}
  
{{NumBlk|::|<math>\frac{R_P}{\omega L_P} = \frac{ \omega L_S}{R_S } = Q_P = Q_S = Q</math>|{{EquationRef|25}}}}
+
Note that the loss is dependent on the ratio of <math>R_2</math> to <math>R_L^\prime</math>. Thus, even if <math>R_2</math> is small, <math>R_L^\prime</math> can be relatively small as well, depending on the value of <math>Q</math>. In some cases, it is more convenient to use the inverse of eq. 35, or commonly known as ''Insertion Loss'' (IL):
  
And for the imaginary components:
+
{{NumBlk|::|<math>\mathrm{IL}=\frac{P_L}{P_\mathrm{in}}=\frac{1}{\mathrm{Loss}}</math>|{{EquationRef|36}}}}
  
{{NumBlk|::|<math>R_P L_P = R_S L_P + R_P L_S</math>|{{EquationRef|26}}}}
+
If we define the inductor component quality factor as <math>Q_c = \tfrac{\omega L_2}{R_2}</math>, with <math>Q = \omega C_1 R_L =\tfrac{1}{\omega C_1^\prime R_L^\prime}</math> and <math>\omega^2 = \tfrac{1}{L_2 C_1^\prime}</math>, we can express the loss as:
  
From eq. 24, we get:
+
{{NumBlk|::|<math>\mathrm{Loss} = 1 + \frac{R_2}{R_L^\prime} = 1 + \frac{\frac{\omega L_2}{Q_c}}{\frac{1}{\omega C_1^\prime Q}}=1 + \frac{Q}{Q_c}</math>|{{EquationRef|37}}}}
  
{{NumBlk|::|<math>R_P = \frac{\omega^2 L_P L_S}{R_S} </math>|{{EquationRef|27}}}}
+
== T- and <math>\Pi</math>-Matching Networks ==
 +
[[File:Pi match.png|thumb|400px|Figure 7: A <math>\Pi</math>-matching network.]]
  
And plugging this into eq. 26 gives us:
+
Single L-section matching circuits are simple and elegant. However, we cannot choose the value of <math>Q</math> since it is determined by <math>m = \tfrac{R_\mathrm{hi}}{R_\mathrm{lo}}</math>. One way to increase the value of <math>Q</math> is to use two L-sections and an low intermediate resistance, <math>R_i</math>, where <math>R_i<R_S</math> and <math>R_i<R_L</math>, as shown in the <math>\Pi</math>-network in Fig. 7.  
  
{{NumBlk|::|<math>R_P L_P =R_S L_P + \frac{\omega^2 L_P L_S}{R_S} L_S</math>|{{EquationRef|28}}}}
+
The first L-section reduces the value of source resistance, <math>R_S</math>, to the intermediate resistance, <math>R_i</math>, and the second L-section brings resistance from <math>R_i</math> to <math>R_L</math>.
  
{{NumBlk|::|<math>R_P =R_S + \frac{\omega^2 L_S^2}{R_S^2} R_S = R_S\left(1 + Q^2\right)\approx R_S Q^2</math>|{{EquationRef|29}}}}
+
We can then compute the quality factor of the two L-sections as:
  
And substituting eq. 29 into eq. 27:
+
{{NumBlk|::|<math>Q_1 = \sqrt{\frac{R_S}{R_i} - 1} = \frac{X_1}{R_i}=\frac{R_S}{X_1^\prime}</math>|{{EquationRef|38}}}}
  
{{NumBlk|::|<math>R_P = \frac{\omega^2 L_P L_S}{R_S} = R_S\left(1 + Q^2\right)</math>|{{EquationRef|30}}}}
+
{{NumBlk|::|<math>Q_2 = \sqrt{\frac{R_L}{R_i} - 1} = \frac{X_2}{R_i}=\frac{R_L}{X_2^\prime}</math>|{{EquationRef|39}}}}
  
{{NumBlk|::|<math>\frac{\omega^2 L_S^2}{R_S^2} \frac{L_P}{L_S} = Q^2 \frac{L_P}{L_S}= \left(1 + Q^2\right)</math>|{{EquationRef|31}}}}
+
[[File:Pi match Q.png|thumb|400px|Figure 8: Calculating the overall <math>Q</math> of a <math>\Pi</math>-network.]]
  
{{NumBlk|::|<math>L_P = L_S \frac{1 + Q^2}{Q^2} = L_S \left(1 + \frac{1}{Q^2}\right)\approx L_S </math>|{{EquationRef|32}}}}
+
From Fig. 8, the overall <math>Q</math> can be calculated by converting the parallel circuits into their series equivalents. Therefore:
  
== Basic Matching Networks ==
+
{{NumBlk|::|<math>Q = \frac{X_1 + X_2}{2R_i} = \frac{Q_1 + Q_2}{2}</math>|{{EquationRef|40}}}}
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, <math>R_S</math> to a load resistance, <math>R_L</math> using a lossless matching network:
 
  
=== Case 1: <math>R_S<R_L</math> ===
+
[[File:T match.png|thumb|400px|Figure 9: A T-matching network.]]
Consider the circuit below. Since we know that the parallel resistance of the RC circuit gets reduced when converting it to a series RC circuit:
+
If we want an intermediate resistance, <math>R_i</math>, that is larger than both <math>R_S</math> and <math>R_L</math>, we can use a T-network instead of a <math>\Pi</math>-network, as shown in Fig. 9.
  
{{NumBlk|::|<math>R_L^\prime = \frac{R_L}{1 + Q^2}</math>|{{EquationRef|33}}}}
+
In this case,
  
Where <math>Q =\omega R_L C_1</math>. The capacitor <math>C_1</math> also gets transformed into:
+
{{NumBlk|::|<math>Q_1 = \sqrt{\frac{R_i}{R_S} - 1} = \frac{R_i}{X_1^\prime}=\frac{X_1}{R_S}</math>|{{EquationRef|41}}}}
  
{{NumBlk|::|<math>C_1^\prime = C_1 \left(1 +\frac{1}{Q^2}\right) \approx C_1</math>|{{EquationRef|34}}}}
+
{{NumBlk|::|<math>Q_2 = \sqrt{\frac{R_i}{R_L} - 1} = \frac{R_i}{X_2^\prime}=\frac{X_2}{R_L}</math>|{{EquationRef|42}}}}
  
Thus, to keep the impedance seen by the source purely resistive, we can cancel out the capacitance <math>C_1^\prime</math> by the inductance <math>L_1</math>:
+
Similar to the <math>\Pi</math>-network, the overall T-network quality factor is <math>Q=\tfrac{1}{2}\left(Q_1 + Q_2\right)</math>.
  
{{NumBlk|::|<math>L_1 = \frac{1}{\omega^2 C_1^\prime}</math>|{{EquationRef|35}}}}
+
== Multi-Section Low-<math>Q</math> Matching ==
 +
[[File:LowQ match.png|thumb|400px|Figure 10: A low-<math>Q</math> cascaded-L matching network.]]
  
If we set <math>R_S = R_L^\prime</math> and define <math>m = \tfrac{R_L}{R_S}</math> for <math>R_S<R_L</math>, we can write eq. 33 as
+
T- and <math>\Pi</math>-networks are used to increase the overall <math>Q</math>. However, in some applications, a lower <math>Q</math> is required, especially in broadband circuits. Low <math>Q</math> circuits are also more robust to process variations and have relatively lower losses. To lower the quality factor of a matching network, we can cascade L-sections with an intermediate resistance, <math>R_\mathrm{hi}>R_i>R_\mathrm{lo}</math>, as shown in Fig. 10 for <math>R_S<R_L</math>.
  
{{NumBlk|::|<math>Q=\sqrt{m - 1} = \omega R_L C_1</math>|{{EquationRef|36}}}}
+
The quality factor can be expressed as
  
We can then solve for the values of <math>C_1</math> and <math>L_1</math> for a certain frequency <math>\omega_0</math> as
+
{{NumBlk|::|<math>Q = \frac{1}{2}\left(Q_1 + Q_2\right) =\frac{1}{2}\left(\sqrt{\frac{R_i}{R_S} - 1} + \sqrt{\frac{R_L}{R_i} - 1}\right)</math>|{{EquationRef|43}}}}
  
{{NumBlk|::|<math>C_1=\frac{\sqrt{m - 1}}{\omega_0 R_L} </math>|{{EquationRef|37}}}}
+
To get the minimum possible <math>Q</math>, we can take the derivative of eq. 43 with respect to <math>R_i</math> and equate it to zero. This results in the optimum intermediate resistance:
  
{{NumBlk|::|<math>L_1=\frac{1}{\omega_0^2 C_1^\prime}</math>|{{EquationRef|38}}}}
+
{{NumBlk|::|<math>R_{i,\mathrm{opt}} = \sqrt{R_S R_L}</math>|{{EquationRef|44}}}}
  
=== Case 2: <math>R_S>R_L</math> ===
+
Which leads to the minimum quality factor for two cascaded L-sections:
  
== Loss in Matching Networks ==
+
{{NumBlk|::|<math>Q_\min = \sqrt{\sqrt{\frac{R_S}{R_L}} - 1}</math>|{{EquationRef|45}}}}
  
== T and Pi Matching Networks ==
+
{{Note|[[229-A1.2 | '''Activity A1.2''' Passive Matching Networks]] -- This activity walks you through the steps in designing and simulating passive matching networks.|reminder}}

Latest revision as of 14:36, 16 September 2020

Controlling impedances in radio-frequency (RF) circuits is essential in order 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, , for a device or a circuit, and use this to build circuits that modify the impedance seen across a port.

Device Quality Factor

The quality factor, , is a measure of how good a device is at storing energy. Thus, the lower the losses, the higher the . 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

Figure 1: A lossy capacitor.

A lossy capacitor, shown in Fig. 1, 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

Figure 2: A lossy inductor.

Applying the ideas above to RL circuits in Fig. 2, 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

Figure 3: A lossy reactance.

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

Figure 4: An impedance matching network.

We can use our series-parallel conversions to increase or decrease resistances based on the quality factor. Consider the circuit in Fig. 4, where we want to match the source resistance, to a load resistance, using a lossless matching network, i.e. a network make up of only reactances.

L-Section Matching Circuits

Figure 5: An L-Section matching network and the network after the parallel-to-serial transformation of and .

Consider the circuit in Fig. 5, 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. Note that cancelling reactances at resonance could cause large voltages to appear across (or large currents through) the reactances. See this note on resonance for more details. For the case when , we just flip the circuit and the equations would still hold.

Design Example

Let define the matching factor , where and . Therefore for , and . 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

Figure 6: A lossy L-section.

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 in Fig. 6.

To determine the power loss due to , we first calculate the power delivered by the input source:

 

 

 

 

(33)

Then we calculate the power delivered to the load, :

 

 

 

 

(34)

We can then define power loss as

 

 

 

 

(35)

Note that the loss is dependent on the ratio of to . Thus, even if is small, can be relatively small as well, depending on the value of . In some cases, it is more convenient to use the inverse of eq. 35, or commonly known as Insertion Loss (IL):

 

 

 

 

(36)

If we define the inductor component quality factor as , with and , we can express the loss as:

 

 

 

 

(37)

T- and -Matching Networks

Figure 7: A -matching network.

Single L-section matching circuits are simple and elegant. However, we cannot choose the value of since it is determined by . One way to increase the value of is to use two L-sections and an low intermediate resistance, , where and , as shown in the -network in Fig. 7.

The first L-section reduces the value of source resistance, , to the intermediate resistance, , and the second L-section brings resistance from to .

We can then compute the quality factor of the two L-sections as:

 

 

 

 

(38)

 

 

 

 

(39)

Figure 8: Calculating the overall of a -network.

From Fig. 8, the overall can be calculated by converting the parallel circuits into their series equivalents. Therefore:

 

 

 

 

(40)

Figure 9: A T-matching network.

If we want an intermediate resistance, , that is larger than both and , we can use a T-network instead of a -network, as shown in Fig. 9.

In this case,

 

 

 

 

(41)

 

 

 

 

(42)

Similar to the -network, the overall T-network quality factor is .

Multi-Section Low- Matching

Figure 10: A low- cascaded-L matching network.

T- and -networks are used to increase the overall . However, in some applications, a lower is required, especially in broadband circuits. Low circuits are also more robust to process variations and have relatively lower losses. To lower the quality factor of a matching network, we can cascade L-sections with an intermediate resistance, , as shown in Fig. 10 for .

The quality factor can be expressed as

 

 

 

 

(43)

To get the minimum possible , we can take the derivative of eq. 43 with respect to and equate it to zero. This results in the optimum intermediate resistance:

 

 

 

 

(44)

Which leads to the minimum quality factor for two cascaded L-sections:

 

 

 

 

(45)

Activity A1.2 Passive Matching Networks -- This activity walks you through the steps in designing and simulating passive matching networks.