Fig.L1.1. Photographs of Hertz s original equipment: (a) first coaxial cable; (b) tunable frame antenna which received the first radio wave.

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1 ECEN 5004, Spring 2018 Active Microwave Circuits Zoya Popovic, University of Colorado, Boulder LECTURE 1 INTRODUCTION AND BACKGROUND REVIEW L1.0. SHORT HISTORY The history of microwaves started with Maxwell's theory in the nineteenth century. Maxwell mathematically showed that electromagnetic wave propagation exists, and that light is an electromagnetic wave. Not many people understood Maxwell's theory at the time. Two people, however, did: Heinrich Hertz, who verified the theory with a series of ingenious experiments, and Oliver Heaviside, who developed a mathematical language for Maxwell's theory that most engineers could understand and use. Heaviside introduced vector notation and provided foundations for guided wave and transmission line theory. He also introduced the transform that today goes by the name of the Laplace transform. Hertz was the first true microwave engineer. Between 1887 and 1891 he performed a large number of experiments at wavelengths between 6cm and 6m. Probably his most important experiment was the following. He used a high voltage spark (rich in high harmonics) to excite a half-wave dipole antenna at about 60MHz. This was his transmitter. The receiver was an adjustable loop of wire with another spark gap. When he adjusted the resonance of the receiving antenna to that of the transmitting one, Hertz was able to show propagation of waves for the first time. Hertz demonstrated first reflector antennas, finite velocity of wave propagation in coaxial transmission lines, standing waves, and a number of microwave and RF techniques. Unfortunately, he died at an early age of 36 from a tooth infection. Hertz s original equipment is now in Bon, Germany, and it is kept functional (see Fig.L1.1). (a) (b) Fig.L1.1. Photographs of Hertz s original equipment: (a) first coaxial cable; (b) tunable frame antenna which received the first radio wave. The next important discovery in the development of microwave engineering was the metallic waveguide, discovered independently by Southworth at AT&T and Barrow at MIT. Southworth made his invention in 1932, but could not talk about it because of company policies until a meeting in Barrow was at the same time working on antennas, and came to a conclusion that a hollow 1

2 metal tube could guide electromagnetic waves. His first experiments in 1935 were not successful, because he did not understand cutoff in waveguides, and tried to guide a 50-cm wave through a 4.5- cm tube (which is well below cutoff at =50 cm). He soon understood the problem, and repeated his experiment successfully with a tube 18 inches in diameter. Before the Second World War, a high power microwave source was invented - the magnetron. This triggered development of radar (RAdio Detection And Ranging), which was under way simultaneously in Great Britain, the United States and Germany, but the first radar was built in Britain and played an important role in the victory of the Allies. In the US, the microwave field prospered at that time at the MIT Radiation Labs. A famous and never out-dated series of books on microwave theory and techniques resulted from this work. Most of the work done used waveguides and coaxial lines as the transmission medium. Waveguide can handle high power levels, but is narrow band, whereas coax is broadband, but limited in power and achievable circuit complexity. In the early 50's, planar transmission lines, such as strip line and microstrip, were developed. Microstrip lines are currently used for many microwave applications, since they are planar, low cost, compact and allow a large variety of circuits on a relatively small area. Other planar transmission lines on dielectric substrates, such as coplanar waveguide with finite ground and backed by a ground plane, are still a research topic today. Soon after the invention of the magnetron in 1937, the klystron tube was invented. These tubes can work as both amplifiers and oscillators. Another often used tube is the TWT (Travelling Wave Tube) invented in the 50's. All the tube sources are bulky and require large additional equipment such as power supplies and refrigerators. There was a clear need for smaller and cheaper active devices, which came with the development of semiconductor devices. A microwave MESFET was first made by Carver Mead at Caltech in 1965, but he failed to patent it. The most commonly used device today at microwave frequencies are the GaAs high-electron mobility transistor (HEMT), GaN HEMT, GaAs or InP Heterojunction Bipolar Transistors (HBTs) and SiGe and Si bi-cmos devices. In this class, you will use single transistors (die and packaged) for several active circuit designs, and at the end of the semester you will design a Microwave Monolithic Integrated Circuit (MMIC) in the Qorvo 250-nm GaAs process. Fig.1.2 shows examples of previous class projects. (a) (b) Fig.L1.2. (a) Design of a distributed broadband amplifier in the Triuint MMIC process (by Mike Elsbury, UCB). (b) Photograph of a fabricated MMIC impedance tuner designed at UCB (by Luke Sankey) and fabricated by Triquint. 2

3 In order for all of us to use the same vocabulary, we will need to review some fundamental components and concepts: Transmission-line theory. This is review material at roughly the junior level that you will need to be very familiar with throughout the class. For example, Chapter 18 from Introductory Electromagnetics, Popovic & Popovic, or Chapter 3, Microwave Engineering, D. Pozar. S-parameters and basic multi-port network theory Impedance matching review Some passive circuits that we will need for active circuit design L1.2. SCATTERING PARAMETERS S-PARAMETERS In a coaxial cable, voltages and currents make sense and are measurable (how?), but, for example, in a waveguide, they do not make much sense. It is hard to measure voltages and currents in a transmission line, since any probe presents some load impedance, which changes what we are measuring. The standard quantities used at microwave frequencies to characterize microwave circuits are wave variable and scattering parameters. Usually, in a microwave circuit, we talk about ports, which are not simple wires, but transmission lines connected to a circuit. These transmission lines support waves traveling into and out of the circuit. A microwave circuit, in general, can have many ports. You can think of this in the following way: we send a wave into an unknown N-port circuit, and by measuring the reflected wave at the same port (like an echo) and the transmitted wave at some other port, we can find out what the microwave circuit is. The problem is, if you look at an N-port, and you send a wave through port 1, when measuring what gets reflected at port 1 and transmitted, say, at port 3, what you measure will depend on what loads were connected onto the other ports during the measurement. The convention is to terminate all other ports with the characteristic impedances of the transmission lines connected to the ports, so that there is no reflection from these ports. In other words, you can think of an s-parameter as a generalized reflection or transmission coefficient when all other ports of a multi-port circuit are matched. Let us look at the generalized N-port microwave circuit. The ports are indexed by the subscript i that goes from 1 to N. The normalized voltage waves ai and bi are defined as Vi Vi ai bi Z Z i Zi is the real normalizing impedance, usually the characteristic impedance of the transmission line connected to port i. The ai and bi 's are complex numbers and are often called wave amplitudes. The waves going into the circuit are called incident, and the ones coming out are called scattered. The magnitudes of the wave variables are related to power in the following way. The currents and voltages in terms of ai and bi are ai bi Vi ( ai bi) Zi and Ii Zi Assuming we are using RMS quantities, the power going into port i is equal to P { VI } a b * 2 2 i i i, i 3

4 where the asterisk denotes the complex conjugate of a complex number. This formula means that we can interpret the total power going into port i as the incident power minus the scattered power. This formula can be extended to calculate the power flowing into the entire circuit: P N N P aa bb i1 i * * where a* is now the Hermitian conjugate, that is the complex conjugate of a. Here a is a column vector of order N consisting of all the a's. Usually this is defined as an input vector, and the vector b is defined as the output vector of a microwave network, and they are related by b=s a, where S is called the scattering matrix. We can measure the coefficients of the scattering matrix by terminating all the ports with their normalizing impedance, and driving port j with an incident wave aj. All the other incident waves will be zero, since the other terminations are matched and have no reflection. The scattering coefficients sij are then s ij bi. a In general, a scattering matrix has many parameters that need to be determined for a specific network. For example, for a two-port network such as an amplifier, 8 numbers need to be determined (4 complex numbers). Fortunately, in many cases it is possible to reduce the number of unknown coefficients when some of the properties of the network are known. One important property is reciprocity. A network is reciprocal if the power transfer and the phase do not change when the input and output are interchanged. This means that for reciprocal networks, the scattering matrices are symmetrical. In order for a network to be reciprocal, it has to be linear, time invariant, made of reciprocal materials, and there cannot be any dependant voltage or current sources in the network. For example, a transistor amplifier is not reciprocal because of the dependant current source, and you know from your circuits classes that an amplifier usually does not work well backwards. A nonreciprocal device used commonly in microwave engineering is an isolator, which contains a nonreciprocal material called a ferrite. In this case there is a static magnetic field that gives a preferred direction to the device. Isolators typically have a low loss in one direction, about 1dB, and a very high loss in the other direction, usually 20dB or more. Isolators are, often used to protect a transmitter, just like you used one at the output of the sweepers in the lab. For example, if you have a radar that is producing a MWatt, in case of an open circuited output, you do not want the power to reflect back into the transmitter. Another simplification that can be made in a scattering matrix is when the network is lossless, which means it absorbs no power. This means that the scattered power is equal to the incident power, or mathematically this is equivalent to saying that the scattering matrix is unitary, which is written as SS*=I, where I is the identity matrix. A matrix is unitary when the inner product of each column with itself gives unity, and the product of each column with any other column gives zero. This expression tells us that the columns of the scattering matrix form an orthonormal set, which cuts down the number of s-parameters by a factor of two. j 4

5 1.3. IMPEDANCE MATCHING REVIEW Lumped-element matching The best review of lumped-element matching is given in the paper by Rhea in two parts, posted on the web page. Please read the paper. The simplest lumped-element matching circuits use a single inductor and capacitor, such as shown in Figure L.1.3 for the case of a parallel C and series L. In principle, any load can be matched with one of these circuits if the elements are chosen correctly. First consider the circuit on the left in the figure. The series inductance adds a positive reactance jxl = jl, so with an inductor alone, only impedances of the form z = 1 jx can be matched (x 0). This corresponds to the bottom half of the r = 1 circle of the passive Smith chart. A capacitor is needed to bring the load impedance to this semi-circle. This can only be done if the load is either inside the r = 1 circle or above both the g = 1 and r = 1 circle circles. The strategy is to bring enough capacitive susceptance to bring the load onto the bottom half of the r = 1 circle, and then add inductance to bring the load into the center (Fig. L1.4). Figure L1.3. LC matching networks that use a series inductance and parallel capacitance. Next consider the circuit on the right of Fig.L.13. With capacitance alone, only loads of the form y = 1 jb can be matched (b 0). This is the top half of the g = 1 circle. An inductor wcan bring the load to this semi-circle, provided the load is inside the g = 1 circle or above both the g = 1 and r = 1 circle circles. The strategy in this case is to add enough inductance to bring the load onto the top half of the g = 1 circle and then add capacitance to bring the load to the center of the Smith chart. Figure L1.4. LC matching on the Smith chart. 5

6 Stub matching Stub matching is discussed in every textbook. Please review it or come ask me if you need review material. The one thing that you might want to keep in mind is that the impedances of the lines and stubs do not need to be Z0 like in textbooks. The reason for originally using only Z0 lines is that people were doing this type of matching in coaxial lines and only one characteristic impedance was available. However, in microstrip, it is easy to get a range from 20 to 90, so this gives more options for the stub matching Quarter-wave line matching A single quarter-wave long line is an impedance transformer, with an input impedance equal to Zin = Z 2 /ZL, where Z is the characteristic impedance of the line used for matching. For a lossless line, to make the input 50 (or any real impedance), it is easy to see that only real loads ZL = RL can be matched. To match a complex impedance using a quarter-wave line, one can add a length of 50- (or other Z0) line in order to make the load real and then add a quarter-wave match., Fig. L1.5. Figure L1.5. Quarter-wave line matching on the Smith chart Slug matching Cascaded sections of transmission lines with varying impedances that can also vary in position are referred to as slugs. This type of matching is used in coaxial circuits, and can be easily varied mechanically. One type of slug matching is shown in Fig. L1.6 where two dielectric cylinders can be moved up and down a section of air coaxial line. For a fixed load, they can then be glued into place after the load is matched. The impedance in the dielectric slug is reduced by r from the impedance of the rest of the line, which is usually 50. The tuner covers the widest range when the electrical length of the slugs is 90 at the operating frequency. First we can determine the region of the Smith chart that can be matched. Since we can make the length of the 50- line in front of the load anything we want, the phase of the load does not matter. This means that the boundary of the region of the Smith chart that we can reach is a circle about the origin. The magnitude of the reflection coefficient that we match is determined by the spacing between the slugs. It varies from zero when the slugs touch each other up to its largest value when 6

7 the separation is 90. The maximum can be calculated by cascading quarter-wave section transformations. The impedance looking back through the two slugs is 50/r 2. This means that we can match a reflection coefficient that satisfies 2 r 1 s 2 r 1 In terms of the matching procedure, the spacing between the slugs adjusts the magnitude that can be matched, while the line in front of the load adjusts the phase. Figure L1.6. Slug matching with two quarter-wave long dielectric slugs inserted in an air coaxial line. Slug matching is also used in nonlinear device (and circuit) characterization, usually to build empirical device models. In the load-pull and source-pull approach, air coaxial tuners are used to provide as large of a range of impedances to a device under varying input power and DC conditions. In this case, the slugs are actually metal pieces that move coaxially and slide along the z-axis. Fig. L1.7 shows a photo of a Focus Microwave fundamental frequency tuner, with a detail of the slug geometry. Fig.L1.7. Mechanical load-pull tuner with slug geometry and function shown on a Smith chart Single-section line matching If an impedance lies in a certain area of the Smith chart, it is possible to do a single-line matching section. In this case both the impedance and the electrical length need to be determined, and this type of match is relatively broadband. First, we need to decide what the possible range of impedances that are available could be. For example, in microstrip, it is usually about

8 Fig.L1.8 shows the range of impedances that can be matched with a single section of line they lie in the r = 1 and g =1 circles (why?). Fig.1.8. Matching using a single section of transmission line. The Smith chart shows a possible method of determining the required Z 0 and in order to match a normalized impedance z. A possible method for determining the required characteristic impedance and line length is sketched in Fig.1.8. First, r1 is determined using a bisector of the line between the normalized load impedance and the center of the chart. That determines ZT. Now we need to renormalize Z to ZT, which gives z. Moving to the real axis by gives r2 which now renormalized to 50 gives the desired line Z0. In your homework, you will go over a different approach using the simulator Broadband matching The general idea in making a broad-band matching circuit is to transform the impedance in geometric steps. Typically the bandwidth is proportional to the number of steps. The transformation can be done with lumped elements, quarter-wave sections or stub networks. On standard textbook method is to use an exponentially-tapered nonuniform line, or a Klopfenstein taper (see Pozar s discussion for more information). This method can only be used to match purely real impedances. Another method that can be used to transform real impedances are coaxial transformers, the simplest is illustrated in Fig. L1.9 (left). At the input, the outer conductor of the top transmission line is connected to the inner conductor of the bottom line, and the two inner conductors are connected at the output. This results in the total voltage at the input being the sum of the two line input voltages (in series), and the total current at the output is the sum of the two inner conductor currents (in parallel). Thus, the voltage and current transformation ratios are equal to 2 and the impedance transformation ratio is 4:1. This means that such a transformer matches a load to a 50- line. The interesting thing is that this is true over a very large bandwidth (often several decades) if implemented in coaxial line. Notice that there is nothing specified for the electrical length, except that the two lines are exactly the same length, giving the same delay. For very short lines, they are a lumped element effectively, and the coaxial lines are in that case wound around a magnetic core which is used to suppress unbalanced currents in the shield. At the higher frequency end, the line length is not the limitation, but rather the parasitics of the connections at the input and output, which 8

9 will effectively unbalance the delays. The line impedance is Z0/N, so for a 4:1 transformer as in the figure on the left, Z=25. Fig.L1.9. Coaxial 4:1 impedance transformers. Left: 4:1 impedance transformer with Z=25, Middle: 2.25:1 with Z=33.3. Right: photo of a 3-30MHz transformer balun. Other transformation ratios are also possible, but it turns out that N needs to be a rational number. For example, 2:1 is not possible, because 2 is not a rational number, but 2.25 is possible and it is close enough. The design of a 2.25:1 transformer is shown in Fig.L1.9 in the middle. A reference for these transformers is given at the end of this lecture. These types of transformers can also be implemented in non-coaxial media, usually at the expense of bandwidth and with some grounding issues that need to be solved. 9