# Back to Basics: Impedance Matching (Part 1)

## Rationale And Concept

The maximum power-transfer theorem says that to transfer the maximum amount of power from a source to a load, the load impedance should match the source impedance. In the basic circuit, a source may be dc or ac, and its internal resistance (Ri) or generator output impedance (Zg) drives a load resistance (RL) or impedance (ZL) (Fig. 1):
RL = Ri or ZL = Zg

A plot of load power versus load resistance reveals that matching load and source impedances will achieve maximum power (Fig. 2).

A key factor of this theorem is that when the load matches the source, the amount of power delivered to the load is the same as the power dissipated in the source. Therefore, transfer of maximum power is only 50% efficient.

The source must be able to dissipate this power. To deliver maximum power to the load, the generator has to develop twice the desired output power.

## Applications

Delivery of maximum power from a source to a load occurs frequently in electronic design. One example is when the speaker in an audio system receives a signal from a power amplifier (Fig. 3).

Maximum power is delivered when the speaker impedance matches the output impedance of the power amplifier.

Another example involves power transfer from one stage to another in a transmitter (Fig. 4).

The complex (R ± jX) input impedance of amplifier B should be matched to the complex output impedance of amplifier A. It’s crucial that the reactive components cancel each other. One other example is the delivery of maximum power to an antenna (Fig. 5). Here, the antenna impedance matches the transmitter output impedance.

## Transmission-Line Matching

This last example emphasizes another reason why impedance matching is essential. The transmitter output is usually connected to the antenna via a transmission line, which is typically coax cable. In other applications, the transmission line may be a twisted pair or some other medium.

A cable becomes a transmission line when it has a length greater than λ/8 at the operating frequency where:

λ = 300/fMHz

For example, the wavelength of a 433-MHz frequency is:

λ = 300/fMHz = 300/433 = 0.7 meters or 27.5 inches

A connecting cable is a transmission line if it’s longer than 0.7/8 = 0.0875 meters or 3.44 inches. All transmission lines have a characteristic impedance (ZO) that’s a function of the line’s inductance and capacitance:

ZO = √(L/C)

To achieve maximum power transfer over a transmission line, the line impedance must also match the source and load impedances (Fig. 6).

If the impedances aren’t matched, maximum power will not be delivered. In addition, standing waves will develop along the line. This means the load doesn’t absorb all of the power sent down the line.

Consequently, some of that power is reflected back toward the source and is effectively lost. The reflected power could even damage the source. Standing waves are the distributed patterns of voltage and current along the line. Voltage and current are constant for a matched line, but vary considerably if impedances do not match.

The amount of power lost due to reflection is a function of the reflection coefficient (Γ) and the standing wave ratio (SWR). These are determined by the amount of mismatch between the source and load impedances.

The SWR is a function of the load (ZL) and line (ZO) impedances:

SWR = ZL/ZO (for ZL > ZO)

SWR = ZO/ZL (for ZO > ZL)

For a perfect match, SWR = 1. Assume ZL = 75 Ω and ZO = 50 Ω:

SWR = ZL/ZO = 75/50 = 1.5

The reflection coefficient is another measure of the proper match:

Γ = (ZL – ZO)/(ZL + ZO)

For a perfect match, Γ will be 0. You can also compute Γ from the SWR value:

Γ = (SWR – 1)/(SWR + 1)

Calculating the above example:

Γ = (SWR – 1)/(SWR + 1) = (1.5 – 1)/(1.5 + 1) = 0.5/2.5 = 0.2

Looking at amount of power reflected for given values of SWR (Fig. 7), it should be noted that an SWR of 2 or less is adequate for many applications. An SWR of 2 means that reflected power is 10%. Therefore, 90% of the power will reach the load.

Keep in mind that all transmission lines like coax cable do introduce a loss of decibels per foot. That loss must be factored into any calculation of power reaching the load. Coax datasheets provide those values for various frequencies.

Another important point to remember is that if the line impedance and load are matched, line length doesn’t matter. However, if the line impedance and load don’t match, the generator will see a complex impedance that’s a function of the line length.

Reflected power is commonly expressed as return loss (RL). It’s calculated with the expression:

RL (in dB) = 10log (PIN/PREF)

PIN represents the input power to the line and PREF is the reflected power. The greater the dB value, the smaller the reflected power and the greater the amount of power delivered to the load.

## Impedance Matching

The common problem of mismatched load and source impedances can be corrected by connecting an impedance-matching device between source and load (Fig. 8). The impedance (Z) matching device may be a component, circuit, or piece of equipment.

A wide range of solutions is possible in this scenario. Two of the simplest involve the transformer and the λ/4 matching section. A transformer makes one impedance look like another by using the turns ratio (Fig. 9):

N = Ns/Np = turns ratio

N is the turns ratio, Ns is the number of turns on the transformer’s secondary winding, and Np is the number of turns on the transformer’s primary winding. N is often written as the turns ratio Ns:Ns.

The relationship to the impedances can be calculated as:

Zs/Zp = (Ns/Np)2

or:

Ns/Np = √(Zs/Zp)

Zp represents the primary impedance, which is the output impedance of the driving source (Zg). Zs represents the secondary, or load, impedance (ZL).

For example, a driving source’s 300-Ω output impedance is transformed into 75 Ω by a transformer to match the 75-Ω load with a turns ratio of 2:1:

Ns/Np = √(Zs/Zp) = √(300/75) = √4 = 2

The highly efficient transformer essentially features a wide bandwidth. With modern ferrite cores, this method is useful up to about several hundred megahertz.

An autotransformer with only a single winding and a tap can also be used for impedance matching. Depending on the connections, impedances can be either stepped down (Fig. 10a) or up (Fig. 10b).

The same formulas used for standard transformers apply. The transformer winding is in an inductor and may even be part of a resonant circuit with a capacitor.

A transmission-line impedance-matching solution uses a λ/4 section of transmission line (called a Q-section) of a specific impedance to match a load to source (Fig. 11):

ZQ = √(ZOZL)

where ZQ = the characteristic impedance of the Q-section line; ZO = the characteristic impedance of the input transmission line from the driving source; and ZL = the load impedance.

Here, the 36-Ω impedance of a λ/4 vertical ground-plane antenna is matched to a 75-Ω transmitter output impedance with a 52-Ω coax cable. It’s calculated as:

ZQ = √(75)(36) = √2700 = 52 Ω

Assuming an operating frequency of 50 MHz, one wavelength is:

λ = 300/fMHz = 300/50 = 6 meters or about 20 feet

λ/4 = 20/4 = 5 feet

Assuming the use of 52-Ω RG-8/U coax transmission line with a velocity factor of 0.66:

λ/4 = 5 feet (0.66) = 3.3 feet

Several important limitations should be considered when using this approach. First, a cable must be available with the desired characteristic impedance. This isn’t always the case, though, because most cable comes in just a few basic impedances (50, 75, 93,125 Ω). Second, the cable length must factor in the operating frequency to compute wavelength and velocity factor.

In particular, these limitations affect this technique when used at lower frequencies. However, the technique can be more easily applied at UHF and microwave frequencies when using microstrip or stripline on a printed-circuit board (PCB). In this case, almost any desired characteristic impedance may be employed.

The next chapter of this book will explore more popular impedance-matching techniques.

# Back to Basics: Impedance Matching (Part 2)

## L-Network Applications And Configurations

The primary applications of L-networks involve impedance matching in RF circuits, transmitters, and receivers. L-networks are useful in matching one amplifier output to the input of a following stage. Another use is matching an antenna impedance to a transmitter output or a receiver input. Any RF circuit application covering a narrow frequency range is a candidate for an L-network.

There are four basic versions of the L-network, with two low-pass versions and two high-pass versions (Fig. 1). The low-pass versions are probably the most widely used since they attenuate harmonics, noise, and other undesired signals, as is usually necessary in RF designs. The key design criteria are the magnitudes and relative sizes of the driving generator output impedance and load impedance.

1. There are four basic L-network configurations. The network to be used depends on the relationship of the generator and load impedance values. Those in (a) and (b) are low-pass circuits, and those in (c) and (d) are high-pass versions.

The impedances that are being matched determine the Q of the circuit, which cannot be specified or controlled. If it is essential to control Q and bandwidth, a T or π-network is a better choice. These choices will be covered in a subsequent article.

While the L-network is very versatile, it may not fit every need. There are limits to the range of impedances that it can match. In some instances, the calculated values of inductance or capacitance may be too large or small to be practical for a given frequency range. This problem can sometimes be overcome by switching from a low-pass version to a high-pass version or vice versa.

## Design Example #1

The goal is to match the output impedance of a low-power RF transistor amplifier to a 50-output load, and 50 Ω is a universal standard for most receiver, transmitter, and RF circuits. Most power amplifiers have a low output impedance, typically less than 50 Ω.

2. The RF source is a transistor amplifier with an output impedance of 10 Ω that is to be matched to 50-Ω output impedance load. The L-network with a parallel output capacitor is used.

Figure 2 shows the desired circuit. Assume an amplifier output (generator) impedance of 10 Ω at a frequency of 76 MHz. Calculate the needed inductor and capacitor values using the formulas given in Figure 1a:

Q = √[(RL/Rg) – 1]

Q = √[(50/10) – 1] = √[(5) – 1] = √4 = 2

XL = QRg = 2(10) = 20 Ω

L = XL/2πf

L = 20/[2(3.14)(76 x 106)]

L = 42 nH

XC = RL/Q

XC = 50/2 = 25 Ω

C = 1/2πfXC

C = 1/[2(3.14)(76 x 106)(25)]

C = 83.8 pF

This solution omits any output impedance reactance such as transistor amplifier output capacitance or inductance and any load reactance that could be shunt capacitance or series inductance. When these factors are known, the computed values can be compensated.

The bandwidth (BW) of the circuit is relatively wide given the low Q of 2:

BW = f/Q = 76 x 106/2 = 38 x 106 = 38 MHz

You can see how this matching network functions by converting the parallel combination of the 50-Ω resistive load and the 25-Ω capacitive reactance into its series equivalent (Fig. 3):

Rs = Rp/(Q2 + 1)

Rs = 50/(22 + 1) = 10 Ω

Xs = Xp/[(Q2 + 1)Q2]

Xs = 25/(5/4) = 25/1.25 = 20 Ω

Note how the series equivalent capacitive reactance equals and cancels the series inductive reactance. Also the series equivalent load of 10 Ω matches the generator resistance for maximum power transfer.

## Parallel And Series Circuit Equivalents

Sometimes it’s necessary to convert a series RC or RL circuit into an equivalent parallel RC or RL circuit or vice versa. Such conversions are useful in RLC circuit analysis and design (Fig. 4).

4. These are all the possible practical series and parallel RC and RL circuit equivalents. The text provides the calculations for RS, RP, XS, and XP.

These equivalents also can help explain how the L-networks and other impedance-matching circuits work. The designations are:

Rs = series resistance

Rp = parallel resistance

Xs = series reactance

Xp = parallel reactance

The conversion formulas are:

Rs = Rp/(Q2 + 1)

Xs = Xp/[Q2 + 1)Q2]

Rp = Rs (Q2 + 1)

Xp = Xs [(Q2 +1)/Q2]

Q = √[Rp/(Rs – 1)]

Q = XL/Rs

Q = Rp/XC

If the Q is greater than 5, you can use the simplified approximations:

Rp =Q2Rs

Xp = Xs

## Design Example #2

Match the output impedance of 50 Ω from a 433-MHz industrial-scientific-medical (ISM) band transmitter to a 5-Ω loop antenna impedance (Fig. 5).

5. The RF source is a transmitter at 433 MHz with an output impedance of 50 Ω. The load is a loop antenna with an impedance of 5 Ω.

Q = √[(Rg/RL) – 1]

Q = √[(50/5) – 1] = √[(10) – 1] = √9 = 3

XL = QRL = 3(5) = 15 Ω

L = XL/2πf

L = 15/2(3.14)(433 x 106)

L = 5.52 nH

XC = Rg/Q

XC = 50/3 = 16.17 Ω

C = 1/2πfXC

C = 1/2(3.14)(433 x 106)(16.67)

C = 22 pF

In this example, the capacitor, inductor, and load resistance form a parallel resonant circuit (Fig. 6).

6. The equivalent circuit of the L-network and load is a parallel resonant circuit. At resonance, the parallel circuit has an equivalent resistance equal to the generator resistance of 50 Ω for a match.

Recall that a parallel resonant circuit acts like an equivalent resistance. That resonant equivalent resistance (RR) of a parallel RLC circuit can be calculated by:

RR = L/CR

or:

RR = R(Q2 +1)

RR = L/CR = 5.52 x 10–9/(22 x 10–12)(5) = 50.18 Ω

RR = R(Q2 +1) = 5(32 + 1) = 50 Ω

In both cases the parallel resonant load equivalent resistance is 50 Ω and equal to the generator resistance allowing maximum power transfer. Again, adjustments in these values should be made to include any load reactive component. The equivalent high-pass networks could also be used. One benefit is that the series capacitor can block dc if required.

## A Modern Application

In radio communications, a common problem is matching a transmitter, receiver, or transceiver to a given antenna. Most transceivers are designed with a standard 50-Ω input or output impedance. Antenna impedances can vary widely from a few ohms to over a thousand ohms.

To meet the need to match a transceiver to an antenna, the modern antenna tuner has been developed. Manual versions with tunable capacitors and switched tapped inductors have been available for years. Today, modern antenna tuners are automated. When the transceiver is in the transmit mode, the tuner automatically adjusts to ensure the best impedance match possible for maximum power transfer.

7. The basic impedance-matching circuit in the MFJ Enterprises MFJ-928 automatic antenna tuner is an L-network with a switched tapped inductor and switched capacitors. SWR, impedance, and frequency sensors provide inputs to the microcontroller whose algorithm switches the network seeking an impedance match.

Figure 7 shows a representative tuner. It is essentially an L-network that is adjusted automatically by switching different values of capacitance in or out and/or switching different taps on the inductor to vary the inductance. A microcontroller performs the switching according to some algorithm for impedance matching.

The criterion for determining a correct match is measuring the standing wave ratio (SWR) on the transmission line. The SWR is a measure of the forward and reflected power on a transmission line. If impedances are properly matched, there will be no reflected power and all generated power will be sent to the antenna. The most desirable SWR is 1:1 or 1. Anything higher indicates reflected power and a mismatch. For example, an SWR value of 2 indicates a reflected power of approximately 11%.

In Figure 7, a special SWR sensor circuit measures forward and reflected power and provides proportional dc values to the microcontroller. The microcontroller has internal analog-to-digital converters (ADCs) to provide binary values to the impedance-matching algorithm. Other inputs to the microcontroller are the frequency from a frequency counter circuit and the actual complex load impedance as measured by an impedance-measuring circuit.

One typical commercial automated antenna tuner, the MFJ Enterprises MFJ-928, has an operating frequency range of 1.8 to 30 MHz and can handle RF power up to 200 W. It has an SWR matching range of 8:1 for impedances less than 50 Ω and up to 32:1 for impedances greater than 50 Ω.

The total impedance-matching range is for loads in the 6- to 1600-Ω range. The range of capacitance is 0 to 3900 pF in 256 steps, and the range inductance is 0 to 24 µH in 256 steps. Note that the capacitance may be switched in before or after the inductor. This provides a total of 131,072 different L/C matching combinations.

Such automatic tuners are widely used in amateur radio and the military where multiple antennas on different frequencies may be used. The approach is also used in high-power industrial applications, such as matching the complex impedance of a semiconductor etch chamber in an RF plasma etcher to the 50-Ω kilowatt power amplifier output. Motors are often used to vary the capacitors and inductors in a closed loop servo feedback system.

## References

Automatic L-network calculator: http://bwrc.eecs.berkeley.edu/research/rf/projects/60ghz/matching/impmatch.html
Frenzel, Louis E. Jr., RF Power for Industrial Applications. Prentice Hall/Pearson, 2004, www.ece.msstate.edu/~donohoe/ece4333notes5.pdf

# Back to Basics: Impedance Matching (Part 3)

## Rationale For Use

The main reason to employ a T-network or π-network is to get control of the circuit Q. In designing an L-network, the Q is a function of the input and output impedances. You end up with a fixed Q that may or may not meet your design specs. In most cases the Q is very low (<10). This may be too low for applications where you need to limit the bandwidth to reduce harmonics or help filter out adjacent signals without the use of additional filters. Remember the relationship for determining Q:

Q = f/BW

where f is the operating frequency and BW is the bandwidth.

The T-networks and π-networks provide enough variety to fit almost any situation.

## π-Networks

The basic π-network’s primary application is to match a high impedance source to lower value to load impedance. It can also be used in reverse to match a low impedance to a higher impedance. The low pass version in Figure 1a is the most common configuration, though the high pass version in Figure 1b also can be used.

1. The π-network matching circuit is used mostly in high- to low-impedance transformations. The basic circuit (a) is a low pass circuit. A high pass version (b) can also be used. The π-network also can be considered two back-to-back L-networks with a virtual impedance between them (c).

You may design a π-network using L-network procedures. The π-network can be considered as two L-networks back to back. To use the L-network procedures, you need to assume an intermediate virtual load/source resistance RV as shown in Figure 1c. You can estimate RV from:

RV = RH/(Q2 + 1)

RH is the higher of the two design impedances Rg and RL. The resulting RV will be lower than either Rg or RL depending on the desired Q. Typical Q values are usually in the 5 to 20 range. An example will illustrate the process.

## π-Network Design Example

Assume you want to match a 1000-Ω source to a 100-Ω load at frequency (f) of 50 MHz. You desire a bandwidth (BW) of 6 MHz. The Q must be:

Q = f/BW = 50/6 = 8.33

RV = RH/(Q2 + 1) = 1000/[(8.33) 2 + 1] = 1000/70.4 = 14.2 Ω

The design procedure for the first L-section uses the formulas from “Back to Basics: Impedance Matching (Part 2).” Use the desired Q of 8.33 with an RL value equal to RV.

The inductor L1 value is:

XL = QRL = 8.333(14.2) = 118.3 Ω

L = XL/2πf

L1 = 118.3/2(3.14)(50 x 106)

L1 = 376.7 nH

The capacitor C1 value is:

XC1 = Rg/Q

XC1 = 1000/8.33 = 120 Ω

C1 = 1/2πfXC

C1 = 1/2(3.14)(50 x 106)(120)

C1 = 26.54 pF

Now calculate the second section with L2 and C2 using an Rg value of RV or 14.2 Ω with the load RL of 100 Ω. The Q is now defined by the L-network relationship:

Q = √[(RL/Rg) – 1]

Rg in this case is RV or 14.2 Ω.

Q = √[(100/14.2) – 1] = √[(7) – 1] = √6 = 2.46

The inductance L2, then, is:

XL2 = QRg = 2.46(14.2) = 35 Ω

L2 = XL/2πf

L2 = 35/[2(3.14)(50 x 106)]

L2 = 111.25 nH

The capacitance C2 is:

XC2 = RL/Q

XC = 100/2.46 = 40.65 Ω

C2 = 1/2πfXC

C2 = 1/[2(3.14)(50 x 106)(40.65)]

C2 = 78.34 pF

Note that the two inductances are in series so the total is just the sum of the two or:

L1 + L2 = 376.7 + 111.25 = 487.97 nH

Figure 2 shows the final circuit.

2. The π-network resulting from the example problem matches a 1000-Ω generator to a 100-Ω load at a frequency of 50 MHz with a bandwidth of 6 MHz and a Q of 8.33.

Two Web tools provide the same results:

http://bwrc.eecs.berkeley.edu/research/rf/projects/60ghz/matching/impmatch.html%20

http://www.raltron.com/cust/tools/network_impedance_matching.asp%20

## T-Networks And LCC Design Example

Figure 3 illustrates the basic T-networks. The basic T shown in Figure 3a is not widely used, but its variation in Figure 3b is. The second network is called an LCC network.

3. There are two versions of the T-network, an alternate matching network: the low pass version (a) and the more popular LCC network (b).

To design these networks, you can also consider them as two cascaded L-networks. However, since the version in Figure 3b is so common, you can also use some shortcut formulas. Here is the procedure:

1. Select the desired bandwidth and calculate Q.
2. Calculate XL = QRg
3. Calculate XC2 = RL√[Rg (Q2 + 1)/RL – 1]
4. Calculate XC1 = Rg (Q2 + 1)/Q[QRL/(QRL – XC2)]
5. Calculate the inductance L= XL/2πf
6. Calculate the capacitances C = 1/2πfXC

Assume a source or generator resistance of 10 Ω and a load resistance of 50 Ω. Let Q be 10 and the operating frequency be 315 MHz.

XL = QRg = 10(10) = 100 Ω

L = XL/2πf = 100/2(3.14)(315 x 106) = 50 nH

XC2 = RL√[Rg (Q2 + 1)/RL – 1] = 50√{[10(101)/50] – 1} = 219 Ω

XC1 = Rg (Q2 + 1)/Q [QRL/(QRL – XC2)] = 10(101)/10[500/(500 – 219)] = 179 Ω

C2 = 1/2πfXC = 1/2(3.14)(315 x 106)(219) = 2.31 pF

C1 = 1/2πfXC = 1/2(3.14)(315 x 106)(179) = 2.82 pF

## Applications With Tunable Networks

Impedance-matching networks must be tunable before they can be used over a wider frequency range or match a wider range of impedances for lower standing wave ratio (SWR) values. One or more of the components must be variable to make such networks. While manually variable capacitors and inductors are available, they are too large for practical modern circuits and their variability cannot controlled electronically. Electronic control permits automatic tuning and matching circuits to be built.

Different types of electronically variable capacitors are now available to implement such automatic tuners and matching circuits. The varactor diode or voltage variable capacitor has been available for years, and its continuously variable nature over a wide capacitance range is desirable. However, it is nonlinear and requires a significantly high bias voltage for control. Two other available options are the digitally tunable capacitor and the microelectromechanical-systems (MEMS) switched capacitor. Both are available in IC form and are ideal for making high-frequency variable matching networks.

4. Peregrine Semiconductor’s solid-state UltraCMOS DTC for wireless applications uses five MOSFET switched capacitors.

Peregrine Semiconductor’s PE64904 and PE64905 Digitally Tunable Capacitors (DTCs) consist of five fixed capacitors switched by an array of MOSFET switches (Fig. 4). These devices are made using a unique silicon-on-sapphire process. The capacitance is changed by a serial 5-bit code word using either a serial peripheral interface (SPI) or an I2C interface. The capacitor may be used in a series or parallel format. Maximum shunt capacitance is 5.1 pF with a minimum of 1.1 pF. Maximum series capacitance is 4.6 pF with a minimum of 0.6 pF. The capacitors can be put in series or parallel for larger or smaller values.

Figure 5 shows a circuit using several of these capacitors. It’s both a tunable filter and impedance-matching circuit. It’s also a reconfigurable coupled resonator topology that’s widely used in band-pass filters and impedance-matching networks. Using external inductors, this network provides wide impedance coverage over the cellular phone bands of 698 to 960 MHz and 1710 to 2170 MHz. Such tunable networks can provide automatic adjustment when cell sites are changed or can correct for the antenna detuning when the phone is held.

5. Tunable matching networks can use UltraCMOS DTCs to target multi-band LTE/WCDMA/GSM applications on UMTS-FDD Bands I, II, III, IV, V, VIII, and XII.

Wispry’s MEMS devices also are commercially available tunable capacitors. The basic element is a MEMS capacitor that can be switched to produce an incremental change of 0.125 pF per bit. Each capacitor comprises a fixed plate with a dielectric on its surface. Above it is a mechanical flap forming the other plate. When an electrostatic charge is applied to the plates, they’re attracted to one another. The upper plate is pulled downward, significantly decreasing plate spacing and increasing capacitance. By forming an array of these tiny capacitors, larger values can be created.

The company makes several versions of this device including custom circuits. Capacitance values to a maximum of 10, 20, or 30 pF can be formed with a tuning range of 10:1. The capacitance is controlled digitally with a serial word using either an SPI or the MIPI radio-frequency front-end (RFFE) interface. Capacitive devices use external inductors.

Also, Wispry’s WS2017 standard impedance matching network is a variable π-network matching device featuring on-chip inductors. Again, the primary application is automatic tuning and impedance matching in cell phones and other small RF equipment. It operates over the 824- to 2170-MHz frequency range. An on-chip dc-dc converter provides the high voltage to operate the switchable capacitors. The device uses an SPI for control. A similar product, the WS2018, has similar specifications as well as the MIPI RFFE interface.

## References

Bostic, Chris, RF Circuit Design, Howard W. Sams, 1982.
Frenzel, Louis E., Principles of Electronic Communication Systems, McGraw Hill, 2008.