2.3 Radio Receiver Circuit

RF, the radio frequency portion of the radio spectrum is considered to extend from 3 to 1000MHz (1GHz). At the opposite end of the spectrum, below the High Frequency (HF) band (3～30MHz), the AM radio band extends from 650 to 1650kHz. At these relatively low frequencies the effects of parasitic inductance and capacitance on circuit designs are minimal.

The radio receiver is often a student's first introduction analog electronics. However, today's receivers are amazingly complex structures. The architecture of modern receivers continues to evolve to account not only for improvements in the analog performance of devices but for advance in DSP(Digital Signal Processing) that permit more functions to be programmed in software rather than being hardwired in the circuits themselves.

1.The Superheterodyne Receiver-Analog System

The superheterodyne, or superhet, receiver, is a well-known and almost universal receiver architecture for radio receivers. We will see that such a receiver can provide both good selectivity and sensitivity, because the noise bandwidth can be limited to the channel bandwidth without compromising the receiver's ability to tune across the entire RF band. Its basic components are shown in Fig.2-28.

RF Preselector

The purpose of this component is to filter out all unwanted signals lying outside the RF band containing the possible channels to be detected. Unwanted signals can include signals fed from the transmitter itself, which might share a common antenna. In this case the preselector is either a diplexer, a filter that allows one frequency band to pass between the antenna and receiver and another (different) band to pass between the transmitter and antenna (as, for example, in CDMA mobile systems), and/or a transmit/receive switch that turns on during the receive phase and off during the transmit phase (as, for example, in GSM systems). Such a circuit thus allows full or half-duplex operation, respectively, and prevents overload of the downstream receiver components from unwanted frequencies. The function of the preselection filter is also to suppress the undesired responses at the output that arise from incoming signals lying at spurious frequencies that the receiver is not tuned to receive (which we will call the tuned frequency f _T ). We will see that such spurious frequencies can include the image frequency as well as harmonics of the incoming tuned frequency. Ideally, the preselector will also have a good impedance match in-band to avoid bandpass ripple.

RF Amplifier

The function of the input amplifier is to linearly amplify the input signal and minimize the noise added by the receiver to the signal itself. We will see that such low noise amplification can be achieved by noise matching the input of the amplifier, and is important because it can determine the overall noise matching of the entire receiver. It should have a good input and output match to avoid gain ripple. In addition, the input amplifier must not introduce distortion of the signal, because strong signals may be simultaneously present in adjacent (unwanted) channels, and any nonlinear distortion of the amplifier could swamp a weaker signal in the channel we are trying to detect. To achieve this, it will require a reasonable maximum power-handling capability indicated by its input intercept point. Of course, the RF amplifier can consist of multiple stages in order to provide the necessary gain.

Interstage Selector

The preceding amplifier will provide gain to all the channels within the RF bandwidth, and its gain is likely to roll off slowly beyond it. Furthermore, the amplifier will amplify noise across the entire band, and possibly at the image frequency as well. Therefore, this (optional) component is a filter to suppress any gain of undesired signal responses at spurious frequencies, and in particular at the image frequency. It thus maintains the system noise figure by preventing image noise from entering the mixer. It also helps to minimize LO reradiation from the RF mixer port. This component should have low in-band loss.

Local Oscillator

This is a strong signal that is normally generated by a frequency synthesizer, and is typically tuned across a bandwidth equal to the entire RF bandwidth, but offset from it, to choose any desired channel. Its function is to drive the devices within the mixer into a nonlinear regime for frequency translation (mixing). An important oscillator specification is its phase noise, since any phase fluctuation on the oscillator signal is directly superimposed on the mixer output signal. Its broadband noise should also be low so as not to raise the system noise floor. It will also require a good tuning range or bandwidth, and low spurious and harmonic content.

First Mixer

This component translates all incoming signals in the RF frequency range into signals in some intermediate frequency range, depending on the local oscillator signal frequency. The mixer translates all frequencies linearly, preserving phase information within the new range of frequencies. Within some range of RF signal amplitudes, the amplitude of the output signal is also preserved on the IF. We will also see that the choice of mixer topology (e.g., single balanced, double balanced, and so forth) is important in rejecting unwanted mixer output components. It requires low LO feed through to the RF and IF, and a large spurious-free dynamic range for the incoming RF signal. The selection of the IF frequency is important in ensuring the receiver response to unwanted spurious responses is minimized. A historic rule of thumb for HF/VHF receivers is that signals are upconverted to an IF at twice the highest RF frequency. At microwave frequencies where the tuning range is much less, signals are typically downconverted to a lower frequency. IF frequencies from 45 to 82MHz are common for mobile radio receivers in the 800MHz band, and from 110 to 300MHz for radios in the 1, 800MHz band or for both bands.

IF Filter

This component rejects the unwanted signal components generated by the mixer and other components. Its bandwidth must be sufficiently wide to pass the modulation sidebands in the desired channel without distortion. It can be high Q because it is of fixed frequency. The blocking and overload characteristics of the receiver are often determined here in combination with the following.

IF Amplifier

This component should provide adequate gain to the IF signal to drive the following stages. Because it is at a fixed frequency it can provide high gain and be well stabilized.

Second Mixer, Local Oscillator, and Second Filter

Sometimes known as the second IF strip, these components mirror the functions of the first IF strip, but across a different frequency band and with different passband characteristics. The first IF strip is designed to receive the entire RF passband and to reject spurious frequencies and the image frequency in particular, while the function of the second IF strip is to narrow in and select the desired channel from the entire passband, thereby providing additional selectivity to the receiver. For commercial AM radios, 455kHz is a fairly standard second IF frequency, while mobile phone receivers typically use frequencies around 10MHz. In some architecture, the second IF is omitted totally and a single downconversion from RF to a low IF is used instead. In that case, the (first) IF filter also performs channel selection.

Demodulator

This component extracts the modulated signal from the IF signal and converts it to baseband. In the case of an analog system, this information will be either AM or FM; for a digital system, it will typically be symbols having multiamplitude levels that are later decoded.(A symbol is the way a bit or combinations of bits are coded in the waveform.)

Baseband Amplifier

This provides output power to derive drive the relevant output device, which could typically be a speaker, fax output or video screen.

One variation on the standard superheterodyne structure is to fix the (first) LO frequency and downconvert the entire RF band to a wideband IF. This enables a fixed frequency, thus high Q , low-phase noise oscillator to be used for the RF LO, and tuning and channel selection to be performed by varying the IF (second) LO. This can overcome the problem of phase noise introduced by the LO. It can, however, result in do offsets at baseband since the second LO will be at the same frequency as the IF, but this is not a problem unique to this architecture. It does however require a second LO with a broad percentage tuning range, and also exposes the second mixer stage to strong adjacent-channel interferers since now the RF stage has limited ability to select the desired channel.

2.Receiver Characterization

We have already used two terms-receiver sensitivity and receiver selectivity-that turn out to be the two fundamental criteria in evaluating the quality of a receiver. There are other system parameters such as the receiver dynamic range and its maximum input signal, can be determined from the system architecture and a few key parameters of each of the subsystems or components within the architecture. In particular, by configuring the gain, noise figure, power capabilities, and frequency characteristics of each stage, we can derive the overall receiver performance.

Communications Channel

First, however, we will briefly consider the fundamental physical constraints and trade-offs that eventually determine the performance limits of a particular radio system.

Information theory is a complex and well-developed field in its own right. Much emphasis has been given recently to the development of new modulation and coding schemes that conserve bandwidth, power, and minimize the effects of distortion on other systems. This is important because spectrum is a limited, scarce, and consequently expensive resource that needs to be allocated sparingly and used to the benefit of all users.

A fundamental result from this theory is Shannon's equation, which predicts the capacity or information throughput of a communications channel. It can be stated in many ways, but for our purposes is a useful form. This states that the channel capacity C (measured as bits per second) is proportional to the channel bandwidth B and to the base-2 log of the Signal-to-Noise Ratio (SNR) at the detector. This equation shows why spectrum is such an expensive resource, because as the signal occupies more bandwidth-assuming signal power and SNR remain the same-more data can be squeezed through it. In practice, modern systems use sophisticated error detection involving the transmission of redundant bits and complex coding schemes in order to approach Shannon's theoretical limit.

A result that can be derived from the above is that for a given channel with some maximum information throughput, the power (or more specifically, the signal-to-noise ratio at the detector) needed to transmit that information through it can be traded off for spectral occupancy or bandwidth B according to

Thus if the spectral occupancy B ₂ can be increased, the SNR can be lower for the same information transfer rate. A good example of this is to compare the GSM cellular system with CDMA. Both use the same transmission media (free space) and are subject to similar noise sources. Yet the received GSM signal must be several decibels above the noise level to be detectable, whereas the spread CDMA signal can be detected even when buried in the noise. By clever coding that smears the signal across a large bandwidth, the CDMA signal power can be reduced to a level so that the received signal appears like noise to receivers tuned to other channels.

Receiver Noise

We have seen already that noise plays a major role in determining the overall performance of a receiver because, with the signal power, it enters as the SNR into fundamental equations that determine the data rate of the system, and the minimum signal level that can be received.

Noise enters the receiver through a number of sources:

●The channel and into the system through the antenna, where it is usually modeled as additive, white, Gaussian noise and is thermal in nature;

●The RF preselector, which bandlimits the signal but has a finite insertion loss and therefore a thermal noise contribution;

●The active devices, which contribute thermal noise, shot noise, and 1 /f noise.

These noise components produce a noise floor that sets the minimum signal level that can be detected. The noise is characterized by its power spectral density, and it can be a function of frequency. Power spectral density is the power contained within a given bandwidth, so has units of watts per hertz. Although we sometimes refer to noise voltage and noise current per hertz, in RF systems we typically measure power into a fixed reference impedance level (usually 50Ω). In this case, the thermal is a function of resistance and a fundamental result of noise power is

for the total noise power P _N , where K is Boltzmann's constant and T is the temperature in Kelvin. Clearly, the larger the bandwidth, the greater the noise power. It is for this reason that the final IF filter needs to be as narrow as possible, in order to minimize the noise power just prior to demodulation and detection. This final IF filter determines the overall noise bandwidth of the entire receiver since it will be the most narrowband component in the entire chain prior to detection.

In the radio bands below about 30MHz, the external noise is much greater than that calculated, due to both natural and man-made phenomenon. The excess above the thermal floor varies from

around 12dB in rural areas at 30MHz to as high as 76dB at a noisy urban site at low megahertz frequencies. This increase in the minimum receiver noise level needs to be considered when determining the weakest signal that can be detected.

Receiver Sensitivity

The noise floor of a receiver, just prior to demodulation and detection of the signal, determines how strong the input signal must be to be correctly interpreted, either as a “1” or a “0” in a digital system, or as a high-quality analog waveform in an analog system. This minimum input signal strength needed to produce a good quality output signal is referred to as the receiver sensitivity. However, just as there are many definitions of what constitutes acceptable quality, there are equally numerous definitions of sensitivity.

The noise floor at the input is accounted not only for thermal noise at the input, but all noise added by the system itself. If we now assume that an input signal must at least equal the noise level in order to be detected, then this input-referred noise floor is sometimes referred to as the Minimum Detectable Signal (MDS). However, most audiophiles believe that a signal-to-noise ratio of at least 10dB is necessary for acceptable sound quality, so that an output signal-to-noise ratio S _o /N _o of 10dB is used to measure the system sensitivity.

System Nonlinearity

The components in any system ultimately reach a point at which they become nonlinear, essentially a component is nonlinear when its output amplitude or phase is no longer linearly proportional to its input amplitude or phase. In most changes active devices, this results predominantly from amplitude-induced in the device transconductance, but the variation of device resistances and capacitances with voltage amplitude also contributes to nonlinear behavior. Gross nonlinearity arises from cutoff and saturation effects that occur as the device exceeds the limits of its normal active momentarily. As a result, we observe effects such as gain distortion and cross modulation such as AM-to-PM conversion.

Receiver Dynamic Range

We have said that at low signal levels the minimum signal that can be detected is limited by the noise floor that results from thermal and other noise sources in passive and active devices. At large signal levels, harmonic and intermodulation distortion components arise, causing compression and interference that ultimately limits the largest signals the receiver can handle. The difference between the minimum detectable signal and the maximum detectable signal is known as the receiver dynamic range. One interpretation of this definition is the difference between the 1dB compressed output power and the output noise floor. But this has limited usefulness because it assumes a single channel system. Instead, the Spurious Free Dynamic Range (SFDR) is a more useful measure. It is defined as the range of input power levels from which the output signal just exceeds the output noise floor, and for which any distortion components remain buried below the noise floor.

In order to increase the useful range of a receiver, the gain of a system can be automatically controlled in order to decrease the gain when strong signals cause overload or distortion. However, the impact of doing this is not always obvious and will require some care in considering the signal-to-noise ratio and making a number of trade-offs. For instance, consider AGC applied to an LNA. Adding attenuation in front of the LNA will hurt noise figure, but it will help reduce the effect of large signals (which may be interferers rather than the desired signal itself). After the LNA an attenuator prevents overload of following stages but reduces the third-order intercept point. Ideally, we would like to reduce the gain, increase power handling capability, and minimally affect the noise figure. Thus, it is necessary to consider both the gain and input intercept point in tandem, since these are the principal trade-offs in setting an AGC level. The effect on system noise figure can then be derived.

AGC is typically applied in an analog receiver as shown in Fig.2-29. As the desired signal increases in input power, the gain of the system is reduced in order to minimize distortion. The gain of the output stages is ideally reduced first to avoid raising the system noise figure and reducing sensitivity, and as earlier stages begin to distort, AGC is progressively introduced ahead of them. Therefore, the noise floor will eventually rise at the same rate as the input handling capability, since the noise floor is principally set by the loss in the front end.

The stronger signals may be present as interferers in other channels, and these out-of-band signals must be detected properly by the AGC circuits to sense the distortion and reduce the gain. Because such signals are present in the radio front end and are removed by the selectivity of later stages, it is important that they be detected at the front end, before they are eliminated by filtering downstream. For this reason, dual-stage AGC is sometimes used as shown to ensure proper detection of signal levels not just in the desired channel but also in interfering channels.

For example, consider two strong signals only a few kilohertz away from the desired signal. The receiver's first IF filter may not eliminate these since they are too close to the tuned, or desired signal. As a result they may cause distortion in the RF and IF front end and their third-order intermodulation product will lie on top of the desired signal. However, because adjacent channel signals and their harmonics will be eliminated in the much narrower second or final IF filter, they would not be detec ted if AGC were controlled only by the power at the output of the receiver. Such interfering signals need to be detected in the first IF stage and reduced prior to any active device in the receiver, possibly by inserting attenuation in the front end. In some older short-wave radios, switches were used to manually insert RF attenuation into the signal flow to reduce strong unwanted signals from saturating the early stages of the radio.

Spurious Responses

Minimizing spurious responses in a receiver is one of the key design criteria behind selecting its frequency architecture. Spurious responses are outputs that arise from unwanted frequency components. In this context, a frequency component that is undesired is one different to that for which the receiver is tuned. For instance, if we desire to demodulate a channel whose carrier is 895MHz, there could be a signal at 890MHz that creates a response in the receiver that interferes with our desired channel at 895MHz. The signal at 890MHz is labeled a spurious frequency if it creates a spurious response. Although modern coding schemes can still detect the desired signal in the presence of interfering signals at the same frequency, spurious responses remain a problem since they reduce the sensitivity to the desired signal.

Spurious responses are caused by distortion products produced in the receiver when the receiver is excited at the spurious frequency. The responses are primarily created by nonlinearities in the mixer and amplifier, although other components such as quartz filters are also nonlinear. The responses can be generated by signals entering, or already within, the receiver. For instance, they can result from leakage paths from the transmuter or from signals in components at the back end of the receiver feeding back to the input. The excitations can occur at the IF itself; its subharmonics, the image frequency, and from higher order mixing products; they all have the property that their spurious responses fall onto the IF frequency. Although we cannot quantitatively predict their amplitude from linear analysis, we can at least predict the spurious frequencies that can cause problems, and take steps where possible to eliminate any voltage or current at these frequencies to avoid them.

NEW WORDS AND PHRASES

parasitic inductance 寄生电感

analog electronics 模拟电子技术

Digital Signal Processing （DSP）数字信号处理

superheterodyne （superhet） n . [电]超外差式收音机； adj .超外差的

selectivity n . 选择性

sensitivity n . 敏感，灵敏（度），灵敏性

preselector n . 预选器，调谐器

transmitter n . 转送者，传达人发报机，话筒，发射机

the receive phase 接收相位

harmonics n . 谐波

impedance match 阻抗匹配

undesired signal 干扰信号

Local Oscillator （LO）本机振荡器

Intermediate Frequency （IF）中频

modulation sideband 调制边带

overload characteristics 过载特性

demodulator n . 解调器

sophisticated error 经验误差

additive，white，Gaussian noise n . 加性高斯白噪声

power spectral density 功率谱密度

megahertz n . [物]兆赫（MHz）

Spurious Free Dynamic Range （SFDR）无杂散动态范围

AGC abbr . 自动增益控制

spurious responses （接收机）杂散响应

NOTES

1)We will see that such a receiver can provide both good selectivity and sensitivity, because the noise bandwidth can be limited to the channel bandwidth without compromising the receiver's ability to tune across the entire RF band.

我们可以看出该接收机具有良好的选择性和敏感性，因为噪声带宽受信道带宽的限制，不影响整个RF波段内的接收机的性能。

2)This states that the channel capacity C (measured as bits per second)is proportional to the channel bandwidth B and to the base-2 log of the Signal-to-Noise Ratio (SNR)at the detector.

这个公式说明信道容量 C （以每秒多少位计量）与信道带宽 B 和检波处的信噪比的以2为底的对数成正比。

3)The channel and into the system through the antenna, where it is usually modeled as additive, white, Gaussian noise and is thermal in nature。

来源于信道的噪声和经由天线进入系统的噪声，通常以加性高斯白噪声的形式表示，其本质上是一种热量。

4)This component extracts the modulated signal from the IF signal and converts it to baseband.

本部分的作用是将已调信号从中频信号中分离出来，并将它转换为基带信号。

5)Minimizing spurious responses in a receiver is one of the key design criteria behind selecting its frequency architecture.

使杂散响应最小化是接收机继选择频率结构之后的一个主要的设计标准。

EXERCISES

1.Please translate the following words and phrases into Chinese.

a)parasitic capacitance

b)channel bandwidth

c)half-duplex

d)spurious frequency

e)input match

f)tuning range

g)baseband amplifier

h)noise figure

i)minimum detectable signal

j)LNA

2.Please translate the following words and phrases into English.

a）灵敏度

b）谐波

c）阻抗匹配

d）本机振荡器

e）过载特性

f）自动增益控制

g）信道

h）加性高斯白噪声

i）中频

j）基带信号

3.Fill in the blanks with the missing word (s).

a)The function of the input amplifier is to__________amplify the input signal and__________the noise added by the receiver to the signal itself.

b)We will see that such low noise amplification can be achieved by__________matching the input of the am-plifier, and is important because it can determine the overall noise matching of the entire receiver.

c)It should have a good input and output match to avoid gain__________.

d)Of course, the RF amplifier can consist of multiple stages in order to__________the necessary gain.

e)Clearly, the__________the bandwidth, the__________the noise power.

f)As the desired signal increases in input power, the gain of the system is reduced in order to__________.

g)Spurious responses are caused by__________products produced in the receiver when the receiver is excited at the spurious frequency.

4.Answer the following questions according to the text.

a)How many components does the superheterodyne contain?

b)What is the RF amplifier's function?

c)What are the receiver characterization and the system parameters?

d)Please simply describe the theory of AGC.

READING RF/Microwave Theory

RF/微波原理

1.Radio-Wave and Microwave Spectrum

1.无线电波和微波频谱

The literal meaning of RF is Radio Frequency, but this term is often use in the figurative sense of “anything related to electromagnetic signals”. If the Alternating Current (AC) is input to an antenna, an Electromagnetic (EM) field is generated suitable for wireless broadcasting and/or communications.

RF是指无线电频率，但是这个词常用来形容“任何与电磁信号有关的”。如果天线馈以交流电，就能够产生电磁（EM）场，可用于无线广播和通信。这些

These frequencies cover a significant portion of the electromagnetic radiation spectrum, extending from nine kilohertz (9kHz), the lowest allocated wireless communications frequency (it's within the range of human hearing), to thousands of gigahertz (GHz). Some wireless devices operate at IR or visible-light frequencies, whose electromagnetic wavelengths are shorter than those of RF fields.

频率覆盖了电磁辐射频谱的重要部分，从最低无线通信频率（人类听觉范围内）9kHz到几千吉赫兹。一些无线电设备工作在红外线或可见光频率范围内，它们的波长比RF波段要短。

When an RF current is supplied to an antenna, it gives rise to an electromagnetic field that propagates through space. This field is sometimes called an RF field; in less technical jargon it is a “radio wave.” Any RF field has a wavelength that is inversely proportional to the frequency. In the atmosphere or in outerspace, if f is the frequency in hertz and λ is the wavelength in meters, then

当天线中流过RF电流时，就会在空间产生能够传播的电磁场。这种场有时称为RF场，在某些非专业术语中，也称为“无线电波”。RF场的波长与传播频率成反比，在大气和外太空中，如果频率的单位是Hz，波长的单位是m，那么它们的关系表达式为

The frequency of an RF signal is inversely proportional to the wavelength of the EM field to which it corresponds. At 9kHz, the free-space wavelength is approximately 33 kilometers (km) or 21 miles (mi). At the highest radio frequencies, the EM wavelengths measure approximately one millimeter (1mm). As the frequency is increased beyond that of the RF spectrum, EM energy takes the form of infrared (IR), visi-ble, ultraviolet (UV), X rays, and gamma rays.

RF信号的频率与对应的EM场的波长成反比。当频率为9kHz时，自由空间中波长约为33km或21英里。当以最大的无线电频率传播时，电磁波的波长接近1mm。在RF频谱以外，随着频率的增加，电磁波表现为红外线（IR）、可见光、紫外线（UV）、X射线和γ射线。

The RF spectrum is divided into several ranges, or bands. With the exception of the lowest-frequency segment, each band represents an increase of frequency corresponding to an order of magnitude (power of 10). Table 2-1 depicts the eight bands in the RF spectrum, showing frequency and bandwidth ranges. The SHF and EHF bands are often referred to as the microwave spectrum.

射频频谱划分成若干个频率范围或频段。除了最低频率段外，每个频段代表一个数量级所对应的频率范围（10的次幂）。表2-1展示了RF频谱中的8个频段及其对应的频率和带宽范围。我们常把SHF（超高频）段和EHF（极高频）段称为微波频谱。

2.Transmission Line Theory

2.传输线理论

Many different types of microwave transmission lines have been developed over the years. In an evolutionary sequence from rigid rectangular and circular waveguide, to flexible coaxial cable, to planar stripline, to microstrip line, microwave transmission lines have been reduced in size and complexity. The microstrip transmission line is the technology employed in the current hyperthermia applicator studied.

近年来，我们已经开发了多种不同类型的微波传输线，从不可弯曲的矩形波导和圆形波导发展到可弯曲的同轴电缆、平面带状线和微带线，微波传输线的尺寸与复杂性已经大大减小。其中，微带线通常采用目前研究的高温蚀刻技术。

For fields having a sinusoidal time dependence and steady-state conditions, a field analysis of a terminated lossless transmission line results in the following relations (see Fig.2-30).

对于具有正弦特性和稳态条件的场来说，终端无耗传输线的场分析见图2-30。

If an incident wave of the form , where β is the phase constant or wave number given by , is inci dent from the -z direction then the total voltage on the line can be written as a sum of incident and reflected waves:

设入射波为 ，其中 β 为相位常数或波数， ，波从 -z 方向入射，传输线上的总电压可表示为入射波和反射波的叠加：

The total current on the line is

传输线上的总电流为

where Z ₀ is the characteristic impedance of the microstrip line, that is, the impedance the transmission line would have if it were infinitely long or ideally terminated. The incident wave has been written in phasor notation and the common time de-pendence factor has not been written.

式中， Z ₀ 是微带线的特性阻抗，即传输线无限长或接理想负载时的阻抗。入射波可用矢量表示，通常不写出时间因子。

反射波电压的幅度和入射波电压的幅度的比称为电压反射系数，即

The amplitude of the reflected voltage wave normalized to the amplitude of the incident voltage wave is known as the voltage reflection coefficient, which is

where Z _L is the load impedance.

式中， Z _L 为负载阻抗。

The total voltage and current waves on the line can then be written in terms of the voltage reflection coefficient as

传输线上的总电压和总电流可以由电压反射系数表示，即

From the previous equations we see that the voltage and current on the line are a superposition of an incident and reflected wave. If the system is static, i.e. if Z _L and Z ₀ are not changing in time, the superposition of waves will also be static. This static superposition of waves on the line is called a standing wave.

从上面的等式中，我们可以得出，传输线上的电压和电流都是入射波和反射波的叠加。如果系统是静态的，即 Z _L 和 Z ₀ 不随时间的变化而变化，那么波的叠加也是静态的。我们把传输线上波的静态叠加称为驻波。

Because of the complicated shape of this standing wave, the voltage will vary with position along the line, from some minimum value to some maximum value. The ratio of V _max to V _min is one way to quantify the mismatch of the line. This mismatch is called the Standing Wave Ratio (SWR) or Voltage Standing Wave Ratio (VSWR) and can be expressed as

由于驻波波形复杂，传输线上的电压会随着位置的改变而从最小变化到最大。 V _max 和 V _min 的比值可表征传输线的失配程度，这种失配称为驻波比（SWR）或电压驻波比（VSWR），公式为

The SWR is a real number such as 1 and with a perfect match SWR=1. By definition, impedance, characteristic or otherwise, is the ratio of the voltage to the current a particular point on the line. The standing waves cause the impedance to fluctuate as a function of distance from the load. The variation in impedance along the transmission line caused by the line/load mismatch can be written as

驻波比是一个实数（如1），当完全匹配时，驻波比为1。根据定义，特性阻抗及其他阻抗参数，都定义为传输线上特定点电压和电流的比值。驻波引起的阻抗是与负载的距离的函数。由此，由于传输线/负载失配引起的传输线上的阻抗变化可写为

where l is the distance from the load. If we substitute the ex-pression for in terms of the impedances, the generalized input impedance of the load plus transmission line simplifies to

式中， l 表示与负载的距离。如果用阻抗替换表达式中的变量，则带负载传输线上输入阻抗可简化为

With this equation the impedance anywhere along the line can be calculated if the load impedance and characteristic impedance are known.

如果负载阻抗和特性阻抗已知，则可用上式计算传输线上任意一点的输入阻抗。

In the most basic sense, then, if the load impedance equals the line impedance, the reflection coefficient is zero and the load is said to be matched to the line. All of the microwave impedance matching techniques can be reduced to this simple idea: minimize the reflection of the incident wave to as nearly zero as possible.

从最基本的意义上来说，如果负载阻抗等于特性阻抗，则反射系数为0，负载与传输线匹配。所有的微波阻抗匹配技术都基于这个观点：降低入射波的反射系数，使其尽可能接近0。

When the load is mismatched to the line and thus there is a reflection of the incident wave at the load, the power delivered to the load is reduced. This loss is called Return Loss (RL) and is equal to

当负载与传输线不匹配时，负载端将出现入射波的反射，馈给负载的功率降低。这种能量损耗称为回波损耗（RL），公式为

3.Microwave Transmission Lines

3.微波传输线

Three main categories of transmission lines are introduced here: coaxial cables, waveguides and microstrip.

这里介绍三类主要的传输线：同轴电缆、波导和微带线。

Coax cables have a core wire, surrounded by a non-conductive material (which is called dielectric or insulation), and then surrounded by an encompassing shielding which is often made of braided wires. The dielectric keeps the core and the shielding apart. Finally, the coax is protected by an outer shielding which will generally be a PVC material. The inner conductor carries the RF signal and the outer shield is there to keep the RF signal from radiating to the atmosphere and to stop outside signals from interfering with the signal carried by the core. Another interesting fact is that the electrical signal always travels along the outer layer of the central conductor: the larger the central conductor, the better signal will flow. This is called the skin effect.

同轴电缆 的芯线通常由非导电材料（电介质或绝缘材料）包裹，其外层是编织成网状的封闭的屏蔽层。电介质的作用是隔离铜芯和屏蔽层。最终，最外层是由聚氯乙烯（PVC）材料构成的屏蔽外层，其作用是保护同轴电缆。内导体承载射频信号，而外层阻止RF信号辐射到空气中，同时防止外部信号干扰芯线上的载波。有趣的是电信号总是沿着内导体的外表面，内导体越粗，流过的信号就越强。这就是所谓的趋肤效应。

A waveguide is a conducting tube through which energy is transmitted in the form of electromagnetic waves. The tube acts as a boundary that confines the waves in the enclosed space. The skin effect prevents any electromagnetic effects from being evident outside the guide. The electromagnetic fields are propagated through the waveguide by means of reflections against its inner walls, which are considered perfect conductors. The intensity of the fields is greatest at the center along the X dimension, and must diminish to zero at the end walls because the existence of any field parallel to the walls at the surface would cause an infinite current to flow in a perfect conductor. Waveguides, of course, cannot carry RF in this fashion.

波导 是以电磁波形式传送能量的导体管，它将电磁波限制在密封空间内。它的趋肤效应能够有效地阻止电磁感应的泄漏。电磁场通过在波导内壁的反射而传播，将波导称为理想导体。沿 X 方向中心处的电场强度最大。但是，由于与内壁表面平行的场的存在会导致在理想导体中的电流无限大，因此场强在端壁处减小为0。当然，波导不能用这种方式传播射频信号。

There are an infinite number of ways in which the electric and magnetic fields can arrange themselves in a waveguide for frequencies above the low cutoff frequency. Each of these field configurations is called a mode. The modes may be separated into two general groups. One group, designated TM(Transverse Magnetic), has the magnetic field entirely transverse to the direction of propagation, but has a component of the electric field in the direction of propagation. The other type, designated TE(Transverse Electric) has the electric field entirely transverse, but has a component of magnetic field in the direction of propagation. TM waves are sometimes called E waves, and TE waves are sometimes called H waves, but the TM and TE designations are preferred. The mode of propagation is identified by the group letters followed by two subscript numerals. For example, TE ₁₀ ,TM ₁₁ , etc. The number of possible modes increases with the frequency for a given size of guide, and there is only one possible mode, called the dominant mode, for the lowest frequency that can be transmitted.

当频率高于截止频率时，波导中的电场和磁场以无限种模式传播。每一种场分布代表一种模式。这些模式分为两大类：一类是指特定的TM波（横磁波），它的磁场与传播方向垂直，在传播方向上有电场分量；另一类是指特定的TE波（横电波），它的电场与传播方向垂直，在传播方向上有磁场分量。TM波有时也称为E波，TE波有时也称为H波，但我们更倾向于TE波和TM波。我们通常用两个下标数字来表示TE波或TM波中的传播模式，如TE ₁₀ 、TM ₁₁ 等。对于给定尺寸的波导，可能传播的模式数会随着频率的增大而增多。以最低频率传输的模式称为主模。

A typical microstrip line can be taken as a two-layer PCB, the top layer is chemically etched away to leave copper traces of width W , separated from the ground plane by a dielectric substrate of some thickness d and relative permittivity ε _r .

典型的 微带线 可以看作双层PCB，上层经过化学腐蚀形成宽度为 W 的金属贴片，通过厚度为 d 、相对介电常数为 ε _r 的介质基片与下层接地板隔离。

Because of the anisotropic dielectric geometry, the microstrip line cannot support a true TEM wave for the following reasons: a microstrip line has most of its electric field concentrated in the region between the line and the ground plane; a small fraction propagates in the air above. Because the speed of light c is different in air and dielectric the boundary-value conditions at the air-dielectric interface can not be met with a pure TEM wave and the exact fields constitute a hybrid TM-TE wave. Because the dielectric substrate is electrically very thin( d λ ), for this application, the fields are quasi-TEM. Because the fields are quasi-TEM, good approximations for the phase velocity, propagation constant, and characteristic impedance can be obtained from the static solution.

由于各向异性电介质的几何构型，这种微带线不能传播TEM波，原因如下：这种微带线的绝大多数电场分布在贴片和接地板之间；一小部分电场散射在上层的空气中。因为在空气中和在介质中光的传输速度 c 不同，空气与介质分界面的边界条件与纯TEM波不同，其场确切地说是TM-TE混合波。因为介质基片非常薄（ d λ ），可以近似将其看作准TEM波。因为该场是准TEM波，所以可以用静态场分析方法近似求解其相速度、传播常数和特性阻抗。

The phase velocity in microstrip line is given by

微带线中相速度的表达式为

and the propagation constant is given by

传播常数的表达式为

where ε _e is the effective dielectric constant and is given by

式中， ε _e 是有效介电常数，其表达式为

The effective dielectric constant ε _e is the dielectric constant of an equivalent homogenous medium that replaces the air and dielectric layers.

有效介电常数 ε _e 是指将空气/介质层等效成均匀介质后的介电常数。

The characteristic impedance of a microstrip line can be calculated, given the width W and substrate thickness d with the result as

给定贴片宽度 W 和基片厚度 d ，可计算微带线的特性阻抗为

If all microstrip based circuits consisted of a proper width straight feed line terminating in a load, there would not be much need to worry about compensating for discontinuities. Even in this ideal case, the transition from microwave source to microstrip line and from the microstrip to load can be the source of large reflections. Typical microstrip discontinuities are junctions, bends, step changes in width and the coaxial cable to microstrip junction. If these discontinuities are not compensated, they introduce parasitic reactance that can lead to phase and amplitude errors, input and output mismatch, and possibly spurious coupling. The strength of a particular discontinuity is frequency dependent, where the higher the frequency, the larger is the discontinuity.

如果基于微带线的电路只包括适当宽度直馈线及终端负载，则不需要考虑其不连续性的补偿问题。即使在理想条件下，波源到微带线和微带线到负载都会造成大的反射。微带线的结点、弯曲、宽度阶跃变化和同轴线与微带线的连接处都会产生不连续性。如果不对这些不连续性进行补偿，将会产生寄生电抗，这些寄生电抗会导致相位和振幅误差、输入和输出失配以及可能的寄生耦合。一些不连续性的强度与频率有关，频率越高，不连续性越大。

4.Microwave Networks

4.微波网络

A field analysis using Maxwell's equations for many microwave problems would be hopelessly difficult. Circuit and network concepts can be extended to handle many microwave analysis and design problems of practical interest. When we treat various transmission lines and waveguide, we can derive the propagation constant and characteristic impendence. The transition between different transmission lines, or discontinuity on a transmission line, can not be treated as a simple junction between two transmission lines, but is augmented with some type equivalent circuit to account for reactance associated with the transition or discontinuity. This is called Microwave Networks Theory or Equivalent Transmission Line Theory.

利用麦克斯韦方程组的场分析方法求解微波问题非常困难。可以扩展电路和网络的概念来处理微波分析与实际应用中的设计问题。在处理各种传输线和波导时，可以获得传播常数和特性阻抗。不能把不同传输线之间的过渡或传输线上的不连续性，看成是两个传输线的简单连接，而应该当成某种与过渡或不连续性相关的等效电路来处理。这就是所谓的微波网络理论或等效传输线理论。

For an arbitrary two-conductor TEM transmission line, the voltage and the total current, U and I , can be found as

对于任意双导体TEM波传输线，电压和总电流的表达式为

A characteristic impendence Z ₀ can then be defined for traveling wave as

行波中特性阻抗 Z ₀ 的表达式为

After having defined and determined the voltage, current, and characteristic impedance, we can proceed to apply the circuit theory for transmission line to characterize this line as a circuit element.

前面定义了电压、电流和特性阻抗，我们进而把电路理论应用到传输线中，以电路元件的形式来表征传输线。

Even though there are many ways to define equivalent voltage, current, and impedance for wave guides, since these quantities are not unique for non-TEM lines, the following considerations usually lead to the useful results:

虽然等效电压、电流和波导阻抗的定义有很多种，但是因为在非TEM波传输线中这些变量不是唯一的，所以通常做以下约定：

1)Voltage and current are defined only for a particular waveguide mode, and are defined so that the voltage is proportional to the transverse electric field, and the current is proportional to the transverse magnetic field.

1）等效电压和电流只针对特定的波导模式，并且电压正比于横向电场，电流正比于横向磁场。

2)In order to be used in a manner similar to voltages and currents of circuit theory, the equivalent voltages and currents should be defined so that their product gives the power flow of the mode.

2）为了能够采用类似电路理论中的电压和电流，定义等效电压和等效电流，从而能够计算出该模式下的功率流。

3)The ratio of the voltage to the current for a single traveling wave should be equal to the characteristic impedance of the line. This impedance may be chosen arbitrarily, but is usually selected as equal to the wave impedance of the line, or else normalized to unity.

3）在行波中，电压/电流比值的大小应该等于传输线上的特性阻抗。这个阻抗值可以任意选择，但在一般情况下，通常选择与传输线阻抗大小相同的阻抗值，或者归一化阻抗值。

The most useful networks are impedance and admittance networks, ABCD networks, scattering parameters, and so on. For a two-port network, they can be established separately as

几种常用的网络是阻抗矩阵、导纳矩阵、 ABCD 矩阵网络、散射参数矩阵等。对于两端口网络来说，它们的表达式分别为

5.Quarter-Wave Impedance Transformer

5.四分之一波长阻抗变换器

A general mismatch in impedance between two points on a transmission line can be compensated with a quarter-wave transformer. The quarter-wave transformer is a very useful matching technique that also illustrates the properties of standing waves on a mismatched line. A quarter-wave transformer in microstrip is shown in Fig.2-31.

一般通过四分之一波长变换器来解决传输线上两点之间的阻抗失配。这是一种非常有用的匹配方法，而且也说明了失配传输线的驻波特性。微带线四分之一波长变换器如图2-31所示。

In a quarter-wave transformer, we want to match a load resistance R _L to the characteristic impedance Z ₀ through a short length of transmission line of unknown length l and impedance Z ₁ . The input impedance looking into the matching section of line is given by

在四分之一波长变换器中，我们通过插入一段长度 l 和阻抗 Z ₁ 都未知的短传输线来实现负载阻抗 R _L 与特性阻抗 Z ₀ 的匹配。传输线上匹配点的输入阻抗表达式为

If we choose the length of the line l = λ/ 4 then βl =π / 2, divide through by tan βl and take the limit as βl =π / 2 to achieve

如果我们选择 l = λ /4，则 βl =π/2，上式中除以tan βl 且取 βl =π/2的极限，可得到

For a perfect transition with no reflections at the interface between microstrip and load, Γ =0, so Z _in = Z ₀ and this gives us a characteristic impedance Z ₁ as

对于理想变换器来说，微带线和负载的连接处没有反射，即 Γ =0，也就是 Z _in = Z ₀ 。特性阻抗 Z ₁ 的表达式为

which is the geometric mean of the load and source impedances. With this geometry, there will be no standing waves on the feedline although there will be standing waves on the λ/ 4 matching section. Why was the value of l = λ/ 4 chosen? In fact, any odd multiple(2 n +1) of l = λ/ 4 will also work.

即负载和源阻抗的几何平均值。虽然在 λ/ 4匹配端存在驻波，但是就馈线而言，不存在驻波。为什么选择 l = λ /4？事实上，只要长度 l 为 λ /4的（2 n +1）倍即可。

When the line length is precisely λ/ 4 the reflected wave from the load destructively interferes with the wave reflected from the Z ₀ , Z ₁ interface and they cancel each other out. It should be noted that this method can only match a real load. If the load has an appreciable imaginary component, it must be matched differently. It can be transformed into a purely real load, at a single frequency, by adding an appropriate length of feed line.

当插入的传输线长为 λ /4时，来自负载的反射波和 Z ₀ 、 Z ₁ 连接处的反射波产生严重干扰，进而相互抵消。应当注意的是，这种方法只能对实负载进行匹配。如果负载有明显的虚部，必须用别的方法进行匹配。在一定频率上，可以适当增加馈线长度进而把带虚部的负载转化为实负载，再进行阻抗匹配。

NEW WORDS AND PHRASES

infrared adj . 红外线的

ultraviolet adj . （光）紫外的

spectrum n . 频谱，范围，系列

microwave n . 微波

inductance n . 感应系数，自感应

shunt v . 分流，并联

coax n . 同轴电缆

waveguide n . 波导

skin effect 趋肤效应

cutoff frequency 截止频率

coefficient n . 协同因素，折算率

wave number 波数

Voltage Standing Wave Ratio 电压驻波比（VSWR）

electromagnetic adj . 电磁的

Return Loss （RL）回波损耗

transverse adj . 横向的，横断的

dominant mode 主模

relative permittivity 相对介电常数

impendence n . 阻抗

mismatch n . 错配，失谐； v .使搭配不当

microstrip n . 微带线

quasi-TEM 准TEM波

sinusoidal adj . 正弦曲线的

phase velocity 相速度

Equivalent Transmission Line Theory 等效传输线理论

scattering parameter 散射参数

feed line 馈线

NOTES

1)Some wireless devices operate at IR or visible-light frequencies, whose electromagnetic wavelengths are shorter than those of RF fields.

一些无线电设备工作在红外线或可见光频率范围内，它们的波长比射频波段要短。

2) The inner conductor carries the RF signal and the outer shield is there to keep the RF signal from radiating to the atmosphere and to stop outside signals from interfering with the signal carried by the core.

内导体承载RF信号而外部覆盖层阻止RF信号发射到大气中以及阻止外部信号干扰芯线上的载波。

3) There are an infinite number of ways in which the electric and magnetic fields can arrange themselves in a waveguide for frequencies above the low cutoff frequency.

电场和磁场能够以高于截止频率的频率在波导中以无限种模式传播。

4)Voltage and current are defined only for a particular waveguide mode, and are defined so that the voltage is proportional to the transverse electric field, and the current is proportional to the transverse magnetic field.

等效电压和电流只针对特定的波导模式，并且电压正比于横向电场，电流正比于横向磁场。 8bvpkCU1lKRQmsCOyvMI7V9rBBroho8I53Y0P5Q8yfcz33UUc/G06oImtzx/Zy1D