Impedance Matching Networks | T and π | Transformers and Baluns

Impedance Matching Networks

Impedance Matching Networks: Matching networks that connect one stage to another are very important parts of any transmitter. In a typical transmitter, the oscillator generates the basic carrier signal, which is then amplified, usually by multiple stages, before it reaches the antenna.

Since the idea is to increase the power of the signal, the inter stage coupling circuits must permit an efficient transfer of power from one stage to the next. Finally, some means must be provided to connect the final amplifier stage to the antenna, again for the purpose of transferring the maximum possible amount of power. The circuits used to connect one stage to another are known as impedance-matching networks. In most cases, they are LC circuits, transformers, or some combination.

The basic function of a matching network is to provide for an optimum transfer of power through impedance-matching techniques. Matching networks also provide filtering and selectivity. Transmitters are designed to operate on a single frequency or selectable narrow ranges of frequencies.

The various amplifier stages in the transmitter must confine the RF generated to these frequencies. In class C, D, and E amplifiers, a considerable number of high-amplitude harmonics are generated. These must be eliminated to prevent spurious radiation from the transmitter. The impedance-matching networks used for inter stage coupling accomplish this.

Impedance Matching Networks | T and π | Transformers and Baluns

The basic problem of coupling is illustrated in Fig. 8-37(a). The driving stage appears as a signal source with an internal impedance of Zi. The stage being driven represents a load to the generator with its internal resistance of Zl. Ideally, Zi and Zl are resistive. Recall that maximum power transfer in dc circuits takes place when Zi equals Zl.

This basic relationship is essentially true also in RF circuits, but it is a much more complex relationship. In RF circuits, Zi and Zl are seldom purely resistive and, in fact, usually include a reactive component of some type. Further, it is not always necessary to transfer maximum power from one stage to the next. The goal is to transfer a sufficient amount of power to the next stage so that it can provide the maximum output of which it is capable.

In most cases, the two impedances involved are considerably different from each other, and therefore a very inefficient transfer of power takes place. To overcome this problem, an impedance-matching network is introduced between the two, as illustrated in Fig. 8-37(b). There are three basic types of LC impedance-matching networks: the L network, the T network, and the π network.

Networks

L networks consist of an inductor and a capacitor connected in various L-shaped configurations, as shown in Fig. 8-38. The circuits in Fig. 8-38(a) and (b) are low-pass filters; those in Fig. 8-38(c) and (d) are high-pass filters. Typically, low-pass networks are preferred so that harmonic frequencies are filtered out.

The L-matching network is designed so that the load impedance is matched to the source impedance. For example, the network in Fig. 8-38(a) causes the load resistance to appear larger than it actually is. The load resistance ZL appears in series with the inductor of the L network. The inductor and the capacitor are chosen to resonate at the transmitter frequency. When the circuit is at resonance, XL equals XC.

To the generator impedance Zi, the complete circuit appears as a parallel resonant circuit. At resonance, the impedance represented by the circuit is very high. The actual value of the impedance depends upon the L and C values and the Q of the circuit. The higher the Q, the higher the impedance. The Q in this circuit is basically determined by the value of the load impedance. By proper selection of the circuit values, the load impedance can be made to appear as any desired value to the source impedance as long as Zi is greater than ZL.

By using the L network shown in Fig. 8-39(b), the impedance can be stepped down or made to appear much smaller than it actually is. With this arrangement, the capacitor is connected in parallel to the load impedance. The parallel combination of C and ZL has an equivalent series RC combination. Both C and ZL appear as equivalent series values Ceq and Zeq. The result is that the overall network appears as a series resonant circuit, with Ceq and L resonant. Recall that a series resonant circuit has a very low impedance at resonance. The impedance is, in fact, the equivalent load impedance Zeq, which is resistive.

Impedance Matching Networks | T and π | Transformers and Baluns
Impedance Matching Networks | T and π | Transformers and Baluns
Impedance Matching Networks | T and π | Transformers and Baluns

The design equations for L networks are given in Fig. 8-39. Assuming that the internal source and load impedances are resistive, Zi = Ri and ZL = RL. The network in Fig. 8-39(a) assumes RL , Ri, and the network in Fig. 8-39(b) assumes Ri , RL.

Suppose we wish to match a 6-V transistor amplifier impedance to a 50-V antenna load at 155 MHz. In this case, Ri , RL, so we use the formulas in Fig. 8-38(b).

To find the values of L and C at 155 MHz, we rearrange the basic reactance formulas as follows:

In most cases, internal and stray reactances make the internal impedance and load impedances complex, rather than purely resistive. Fig. 8-40 shows an example using the figures given above. Here the internal resistance is 6 V, but it includes an internal inductance Li of 8 nH. There is a stray capacitance CL of 8.65 pF across the load.

The way to deal with these reactance’s is simply to combine them with the L network values. In the example above, the calculation calls for an inductance of 16.7 nH. Since the stray inductance is in series with the L network inductance in Fig. 8-40, the values will add. As a result, the L network inductance must be less than the computed value by an amount equal to the stray inductance of 8 nH, or L=16.7 – 8 = 8.7 nH. If the L network inductance is made to be 8.7 nH, the total circuit inductance will be correct when it adds to the stray inductance.

A similar thing occurs with capacitance. The circuit calculations above call for a total of 55.65 pF. The L network capacitance and the stray capacitance add for they are in parallel. Therefore, the L network capacitance can be less than the calculated value by the amount of the stray capacitance, or C= 55.65-8.65 = 47 pF. Making the L network capacitance 47 pF gives the total correct capacitance when it adds to the stray capacitance.

Impedance Matching Networks | T and π | Transformers and Baluns

T and π Networks

which is determined by the values of the internal and load impedances and may not always be what is needed to achieve the desired selectivity. To overcome this problem, matching networks using three reactive elements can be used. The three most widely used impedance matching networks containing three reactive components are illustrated in Fig. 8-41. The network in Fig. 8-41(a) is known as a π network because its configuration resembles the Greek letter π. The circuit in Fig. 8-41(b) is known as a T network because the circuit elements resemble the letter T.

Impedance Matching Networks | T and π | Transformers and Baluns
Impedance Matching Networks | T and π | Transformers and Baluns

Design Procedure:

  1. Select a desired circuit Q
  2. Calculate XL QRi
  3. Calculate XC1:

4. Calculate XC2:

5. Compute final L and C values:

The circuit in Fig. 8-41(c) is also a T network, but it uses two capacitors. Note that all are low-pass filters that provide maximum harmonic attenuation. The π and T networks can be designed to either step up or step down the impedance as required by the circuit. The capacitors are usually made variable so that the circuit can be tuned to resonance and adjusted for maximum power output.

tuned to resonance and adjusted for maximum power output. The most widely used of these circuits is the T network of Fig. 8-41(c). Often called an LCC network, it is used to match the low output impedance of a transistor power amplifier to the higher impedance of another amplifier or an antenna.

The design procedure and formulas are given in Fig. 8-42. Suppose once again that a 6-V source Ri is to be matched to a 50-V load RL at 155 MHz. Assume a Q of 10. (For class C operation, where many harmonics must be attenuated, it has been determined in practice that a Q of 10 is the absolute minimum needed for satisfactory suppression of the harmonics.) To configure the LCC network, the inductance is calculated first

Transformers and Baluns

One of the best impedance-matching components is the transformer. Recall that iron-core transformers are widely used at lower frequencies to match one impedance to another. Any load impedance can be made to look like a desired load impedance by selecting the correct value of the transformer turns ratio. In addition, transformers can be connected in unique combinations called baluns to match impedances.

Impedance Matching Networks | T and π | Transformers and Baluns

Transformer Impedance Matching

Refer to Fig. 8-43. The relationship between the turns ratio and the input and output impedances is

Impedance Matching Networks | T and π | Transformers and Baluns

That is, the ratio of the input impedance Zi to the load impedance ZL is equal to the square of the ratio of the number of turns on the primary NP to the number of turns on the secondary NS. For example, to match a generator impedance of 6 V to a 50-V load impedance, the turns ratio should be as follows:

This means that there are 2.89 times as many turns on the secondary as on the primary.

The relationship given above holds true only on iron-core transformers. When air-core transformers are used, the coupling between primary and secondary windings is not complete and therefore the impedance ratio is not as indicated. Although air-core transformers are widely used at Fs and can be used for impedance matching, they are less efficient than iron-core transformers.

Ferrite (magnetic ceramic) and powdered iron can be used as core materials to provide close coupling at very high frequencies. Both the primary and the secondary windings are wound on a core of the chosen material.

The most widely used type of core for RF transformers is the toroid. A toroid is a circular, doughnut-shaped core, usually made of a special type of powdered iron. Copper wire is wound on the toroid to create the primary and secondary windings. A typical arrangement is shown in Fig. 8-44. Single-winding tapped coils called autotransformers are also used for impedance matching between RF stages. Fig. 8-45 shows impedance step-down and step-up arrangements. Toroids are commonly used in autotransformers.

Unlike air-core transformers, toroid transformers cause the magnetic field produced by the primary to be completely contained within the core itself. This has two important advantages. First, a toroid does not radiate RF energy. Air-core coils radiate because the magnetic field produced around the primary is not contained. Transmitter and receiver circuits using air-core coils are usually contained with magnetic shields to prevent them from interfering with other circuits. The toroid, on the other hand, confines all the magnetic fields and does not require shields. Second, most of the magnetic field produced by the primary cuts the turns of the secondary winding. Thus the basic turns ratio, input-output voltage, and impedance formulas for standard low-frequency transformers apply to high-frequency toroid transformers.

Transformer Impedance Matching
Transformer Impedance Matching

In most new RF designs, toroid transformers are used for RF impedance matching between stages. Further, the primary and secondary windings are sometimes used as inductors in tuned circuits. Alternatively, toroid inductors can be built. Powdered iron core toroid inductors have an advantage over air-core inductors for RF applications because the high permeability of the core causes the inductance to be high. Recall that whenever an iron core is inserted into a coil of wire, the inductance increases dramatically. For RF applications, this means that the desired values of inductance can be created by using fewer turns of wire, and thus the inductor itself can be smaller. Further, fewer turns have less resistance, giving the coil a higher Q than that obtainable with air-core coils.

Powdered iron toroids are so effective that they have virtually replaced air-core coils in most modern transmitter designs. They are available in sizes from a fraction of an inch to several inches in diameter. In most applications, a minimum number of turns are required to create the desired inductance.

Transmission Line Transformers and Baluns

A transmission line or broadband transformer is a unique type of transformer that is widely used in power amplifiers for coupling between stages and impedance matching. Such a transformer is usually constructed by winding two parallel wires (or a twisted pair) on a toroid, as shown in Fig. 8-46. The length of the winding is typically less than one-eighth wavelength at the lowest operating frequency. This type of transformer acts as a 1:1 transformer at the lower frequencies but more as a transmission line at the highest operating frequency.

Transformer Impedance Matching
Transformer Impedance Matching

Transformers can be connected in unique ways to provide fixed impedance-matching characteristics over a wide range of frequencies. One of the most widely used configurations is shown in Fig. 8-47. With this configuration, a transformer is usually wound on a toroid, and the numbers of primary and secondary turns are equal, giving the transformer a 1:1 turns ratio and a 1:1 impedance-matching ratio.

The dots indicate the phasing of the windings. Note the unusual way in which the windings are connected. A transformer connected in this way is generally known as a balun (from balanced-unbalanced) because such transformers are normally used to connect a balanced source to an unbalanced load or vice versa. In the circuit of Fig. 8-47(a), a balanced generator is connected to an unbalanced (grounded) load. In Fig. 8-47(b), an unbalanced (grounded) generator is connected to a balanced load.

Fig. 8-48 shows two ways in which a 1:1 turns ratio balun can be used for impedance matching. With the arrangement shown in Fig. 8-47(a), an impedance step-up is obtained. A load impedance of four times the source impedance Zi provides a correct match. The balun makes the load of 4Zi look like Zi. In Fig. 8-48(b), an impedance step down is obtained. The balun makes the load ZL look like Zi /4.

Many other balun configurations, offering different impedance ratios, are possible. Several common 1:1 baluns can be interconnected for both 9:1 and 16:1 impedance transformation ratios. In addition, baluns can be cascaded so that the output of one appears as the input to the other, and so on. Cascading baluns allow impedances to be stepped up or stepped down by wider ratios.

Note that the windings in a balun are not resonated with capacitors to a particular frequency. The winding inductances are made such that the coil reactances are four or more times that of the highest impedance being matched. This design allows the transformer to provide the designated impedance matching over a tremendous range of frequencies.

This broadband characteristic of balun transformers allows designers to construct broadband RF power amplifiers. Such amplifiers provide a specific amount of power amplification over a wide bandwidth and are thus particularly useful in communication equipment that must operate in more than one frequency range. Rather
than have a separate transmitter for each desired band, a single transmitter with no tuning circuits can be used.

When conventional tuned amplifiers are used, some method of switching the correct tuned circuit into the circuit must be provided. Such switching networks are complex and expensive. Further, they introduce problems, particularly at high frequencies. For them to perform effectively, the switches must be located very close to the tuned circuits so that stray inductances and capacitances are not introduced by the switch and the interconnecting leads. One way to overcome the switching problem is to use a broadband amplifier, which does not require switching or tuning. The broadband amplifier provides the necessary amplification as well as impedance matching. However, broadband amplifiers do not provide the filtering necessary to get rid of harmonics.

One way to overcome this problem is to generate the desired frequency at a lower power level, allowing tuned circuits to filter out the harmonics, and then provide final power amplification with the broadband circuit. The broadband power amplifier operates as a linear class A or class B push-pull circuit so that the inherent harmonic content of the output is very low.

Fig. 8-49 shows a typical broadband linear amplifier. Note that two 4:1 balun transformers are cascaded at the input so that the low base input impedance is made to look like an impedance 16 times higher than it is. The output uses a 1:4 balun that steps up the very low output impedance of the final amplifier to an impedance four times higher to equal the antenna load impedance. In some transmitters, broadband amplifiers are followed by low-pass filters, which are used to eliminate undesirable harmonics in the output.

Typical Transmitter Circuit

Many transmitters used in recent equipment designs are a combination of ICs and discrete component circuits. Here are two examples.

Short-Range Wireless Transmitter

There are many short-range wireless applications that require a transmitter to send data or control signals to a nearby receiver. Some examples are the small transmitters in remote keyless entry (RKE) devices used to open car doors, tire pressure sensors, remote control lights and ceiling fans in homes, garage door openers, and temperature sensors. These unlicensed transmitters use very low power and operate in the FCC’s industrial scientific c-medical (ISM) bands. These are frequencies set aside for unlicensed operations defined in Part 15 of the FCC rules and regulations. The most common frequencies are 315, 433.92, 868 (Europe), and 915 MHz.

Fig. 8-50 shows a typical transmitter IC, the Freescale MC33493/D. This CMOS device is designed to operate anywhere in the 315- to 434-MHz and 868- to 928-MHz ranges with the frequency set by an external crystal. It features OOK or FSK modulation and can handle a serial data rate up to 10 kbps. Output power is adjustable with an external resistor.

The basic transmitter circuit is simply a PLL used as a frequency multiplier with an output power amplifier. The internal oscillator XCO uses an external crystal. The PLL multiplies an external crystal frequency by a factor of 32 or 64 to develop a PLL VCO signal at the desired operating frequency. For example, if the desired output frequency is 315 MHz, the crystal must have a frequency of 315/32 = 9.84375 MHz. For output of 433.92 MHz, a 13.56-MHz crystal is needed.

The XCO output is applied to the phase detector along with the feedback signal from the frequency dividers driven by the PLL VCO output. The divide-by-2 divider may be switched in when divide-by-64 is needed. The BAND input signal selects the divide-by-2 divider or takes it out. If the BAND signal is low, the divide-by-2 circuit is bypassed and the overall PLL multiply factor is 32. If BAND is high, the divide-by-2 circuit is inserted and the overall multiplication factor is 64. Using a 13.56-MHz crystal gives an output 867.84 MHz when the divide-by-64 factor is used.

The PLL VCO output drives a class C power amplifier. An external resistor may be inserted in the line between REXT and the dc power source to lower the power to the desired level. The maximum output with no external resistor is 5 dBm (3.1 mW). The dc supply voltage may be anything in the 1.9- to 3.6-V range usually supplied by a battery.

Modulation is selected by the MODE pin. If MODE is low, OOK modulation is selected. The serial binary input at the DATA pin then goes to turn the class C PA dc power off and on, on for binary 1 and off for binary 0. If MODE is high, FSK is selected. The DATA input line is then used to pull the crystal frequency between the two desired shift frequencies. A 45-kHz shift is typical. Two external capacitors C1 and C2 are used to pull the crystal to the desired frequencies. Either serial or parallel pulling may be used depending upon the type of crystal.

The PA output is fed to an external LC impedance-matching network as needed to match the 50-V output to the selected antenna, as Fig. 8-50 shows. Usually, the antenna is a loop of copper on the printed circuit board holding the transmitter IC.

One other feature of this chip is the data clock DATACLK output line. This output is the crystal frequency divided by 64. For a 9.84375-MHz crystal, the DATACLK output is 153.8 kHz. With a 13.56-MHz crystal, the DATACLK output is 212 kHz. This clock can be used with an external embedded microcontroller to synchronize the data stream.

This transmitter chip is designed to be used with an external microcontroller. It gets its BAND, MODE, and ENABLE signals from the microcontroller.

Software-Defined Radio Transmitter

The typical transmitter today is part of what we call a software-defined radio (SDR). The information to be transmitted is in digital form and will be modulated on to a carrier using digital techniques. The modulation is usually by DSP; therefore, it is a form of software. Most of the circuitry in the transmitter is digital or software-derived. An example of an integrated circuit that provides most of the functions for an SDR transmitter is the Texas Instruments AFE7070, shown in Fig. 8-51.

The data to be transmitted is organized and modulated by a DSP or FPGA into in-phase (I) and quadrature (Q) components. These are in the form of 14-bit words and are applied alternately to the IC input labeled D(13:0). They are demultiplexed into the I and Q streams and applied to a numerically controlled oscillator (NCO) of the DDS type that may or may not be used for the additional signal generation or mixing operations. The I and Q streams are then sent to a quadrature modulator correction (QMC) circuit that provides a way to match their gain, phase, and offset values. The I and Q signals must be perfectly balanced to provide the optimum error-free transmission.

The I and Q signals then go to a pair of 14-bit DACs where they are converted into equivalent analog I and Q signals. These are filtered in a pair of low-pass filters and applied to a quadrature modulator that mixes them with the carrier signal from a local oscillator (LO) or a frequency synthesizer. This signal comes in on pin LO_PIN. The mixer outputs are added together to form the final modulated analog signal to be transmitted. The signal is amplified and appears at the RF_OUT pin. It is used to drive the final external power amplifier before going to the antenna.

The other circuitry in the chip supports the main transmitter circuits. The SPI is a standard serial peripheral interface used to connect the chip to other circuits. Here the SPI usually connects to an external microcontroller and memory where data for the programming registers is generated and stored. The serial data comes in on pin SDIO. All of the other circuit functions are programmed for the desired outcome by register codes. The clock input CLK_IO is the master timing signal that controls all other circuits as well. On the output, the analog signal is converted into a digital signal and divided by 1, 2, or 4 to create an external clock signal if needed.

The AFE7070 is used in low-power compact SDRs of any type and small cellular base stations. It can operate anywhere in the 100 MHz to 2.7 GHz frequency range. For further details on the chip, find the datasheet at the Texas Instruments

website: www.ti.com

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