talk for summer students, 2001

ELECTRONICS FOR RADIO ASTRONOMERS

Jon Hagen

Radio astronomy came out of electronics. Karl Jansky, who accidentally discovered signals arriving from space (from the "fixed stars") was a physicist, working for Bell Labs. He'd been assigned to identify the various sources of static (non man-made interference) that interfered with trans-Atlantic radio telephone links. This was 1932; long distance telephony, even over land, was fairly rare. Telegraphy was still used for long distances. Submarine cables capable of telephony weren't installed until, believe it or not, 1956 - just one year before Sputnik. Radio astronomy really got started after WWII by physicists turned war-time engineers, who, in developing radar, had advanced radio engineering into the age of microwaves, semiconductors, and dish antennas. Some of these physicists were: Edward Purcell, Robert Dicke, Thomas Gold, Bernard Lovell, Martin Ryle, Anthony Hewish, and Ronald Bracewell. (An exception was Grote Reber, a radio ham and engineer, who singlehandedly built the first modern dish antenna in his backyard made the first radio maps of the sky).

Today radio astronomy is no longer the new astronomy; the atmosphere's radio window is only one of several used by astronomers, who, while well trained in physics, are pretty much left on their own to pick up the theory and practice of optical engineering and/or radio engineering. Are they - you - still expected to pick up this knowledge? Of course. Astronomers are experimentalists; they go to observatories because they know the tools and techniques; they know best how to get the data they want, how to design and modify data taking setups, how to calibrate, evaluate, etc. You do need to know how all the equipment works. Fortunately, you have the math and physics background needed to pick up electronics in short order. (You probably like to know how things work - and have found answers as to the workings of your tv set, your cd player, radio propagation, your cell phone, electric power transmission, etc).

Nearly any electronics system can serve as an example to present the basic circuit elements and circuit principles. So, as an example system, we'll choose a radio telescope and, in particular, its receiving system - the "radio" part. Figure 1 shows a conventional radiotelescope receiver.

This receiver is called a radiometer, because its final stage, is a square-law detector, which computes a fairly long term (e.g. 1 second) running average of the received power. Here computes simply means that the output voltage of the detector is proportional to the average of the square of the input voltage. Note that signal flow is from left to right. A convention used in block diagrams and circuit diagrams (a.k.a. schematic diagrams) is that signal flow should be from left to right and top to bottom.

This receiver would be an ordinary AM broadcast receiver if the detector were replaced by a circuit that computes a short term (e.g. 0.1 msec) average of the absolute value of the input voltage.

The various blocks of the receiver perform the following functions:

BPF1: This band-pass filter prevents strong out-of-band signals from overloading the RF amplifier. An overloaded amplifier becomes non-linear and produces spurious signals. Any ohmic loss (inefficiency) in this filter degrades the receiver's sensitivity, so it omitted when possible.

RF Amp: This first power amplifier makes the signal much stronger than whatever noise is added by subsequent circuit blocks. The noise introduced by this amplifier is, therefore, the factor that determines the sensitivity of the receiver.

Mixer: This block, better called a frequency converter, is used to translate the input frequency band (determined by BPF1 or BPF2). The new center frequency is known as the IF or Intermediate frequency. Most of the amplification in the receiver takes place in the IF amplifier.

The mixer is a multiplier; its output voltage is proportional to the product of its two input voltages. If the inputs are sinewaves at FRF and FLO, there will be two outputs, one at FRF - FLO and the other at FRF + FLO. One of these frequencies will be at FLO, the intended intermediate frequency. The other frequency is unwanted, and will be blocked by BPF4, the IF filter.

Operation of the mixer can be described by the multiplication of two sinusoids:

sin( R t)sin( L t) = 1/2 [ cos(( R - L) t) - cos(( R+ L) t) ]

The Local Oscillator (LO) is a low power sinewave source - a typical mixer requires about 10 mW of LO power. The term local indicates that a superheterodyne receiver contains an internal or local oscillator. Receivers usually are designed to minimize radiation of the signal from the local oscillator. Nevertheless, most receivers do radiate a detectable signal from their LO. Tune an AM radio to, say, 600 KHz. Its local oscillator frequency will then be 600 + 455 = 1055 kHz. This is a frequency in the AM band. Place a second receiver very close to the first and tune it to 1055 kHz. You will easily detect the LO of the first receiver. TV rating services used to run trucks up and down neighborhood streets, sniffing LO radiation from TV antennas to see what channels were being watched. Battlefield receivers should have little or no LO radiation for obvious reasons. You must turn off radio receivers to disable LO radiation when taking off or landing in an airliner.

You are already familiar with frequency conversion. Your cable television box converts the selected channel to Channel 3 or Channel 4. Your television set always remained tuned to this new IF channel. This kind of receiver system that includes one or more frequency conversions is called a superheterodyne receiver. The television set is itself a superheterodyne receiver. The selected channel (selected input band) is translated to an IF band centered around 45 MHz. AM and FM receivers are superheterodynes that use IF bands centered at 455 kHz and 10.7 MHz, respectively. The advantage of this system is that the sharp IF bandpass filter needs no tuning when a new station is selected. Only the local oscillator and the relatively wide filters BPF1 and BPF2 need to be adjusted. Before the invention of the superheterodyne, receivers had many tuning knobs; each amplifier had to be carefully tuned to the frequency of the desired station.

BPF3: This IF passband filter is usually the narrowest filter, so it determines the overall pass band of the receiver. It also blocks the unwanted output frequency of the mixer.

BPF2: This filter is present for image rejection. The mixer will produce an output at FLO if it has an input signal either FIF above the local oscillator frequency or FLO below the oscillator. Thus there are two input bands that will be converted into the IF band. BPF2 eliminates any signals from the undesired input band.

Square Law Detector: This circuit includes a load resistor to properly terminate the IF amplifier. The voltage developed across the load resistor causes current to flow in the diode. The analysis is a little bit subtle. The I/V law for the diode is I(V) = Constant x [exp(eV/kT) -1]. (It does not have a "square law region"). The capacitor C1 is large enough that the voltage across it will be dc. The op amp expanded, ignoring the V3 and higher terms. The RC averager will eliminate the linear voltage feed-through because this linear term is high-frequency ac (at the IF frequency). The net result is that the output voltage from the detector is given by

VDET = e<V2 >/(2kT).

AC Circuit Analysis

The block diagram of Figure 1 is drawn as a chain rather than as a circuit. If we draw in the missing ground symbols, we'll see that it is a circuit containing many loops and nodes.

Figure 2a shows a very simple radio circuit. The receiver consists of an antenna which drives a load resistor through a series LC circuit. This series LC circuit acts as a short circuit at the frequency f= (2pi (LC)-1.

Putting aside for now the operation of a diode detector, suppose our detector is a sensitive thermometer with which we can monitor the temperature of this resistor. We can collapse the two antennas and the intervening space to get the equivalent circuit of Figure 2b. Here we have a simple series circuit, excited by an equivalent source (a Thévenin generator). You may have learned how to analyze this circuit using a method of complex variables:

I( ) = Re[ Vej t/(RGEN + RLOAD + j L -j/( C)].

The output voltage is just the current times RLOAD. Let us graph the output voltage vs. frequency for some made-up values of L, C, Rgen and Rload:

This was a simple example of circuit analysis - a one-loop circuit. In circuit analysis, the goal always is to find the response of the circuit to a stimulus. The response is completely specified by the current in each branch and the voltage at each node. The stimulus can be a voltage or current generator producing dc or a sinewave or any arbitrary waveform, V(t) or I(t). Sinewaves are not arbitrary waveforms because they are the eigenfunctions of linear systems. A linear system contains circuit elements such as resistors, capacitors, and inductors. These components are linear components in that they have voltage/current relations that can be written as linear differential equations. From this it follows that when a linear system is driven by a sinewave source, the response will be sinusoidal. Every voltage and every current will be sinusoidal and have the same frequency as the source. Their amplitudes and phases will be functions of the element values and the circuit topology.

A linear circuit allows superposition: the response to a collection of voltage and current generators is the sum of the individual responses obtained when these generators are activated one at a time. (A non-active voltage generator is replace by a short circuit and a non-active current generator is replaced by an open circuit). Superposition allows us to put two sources in series, cos( t) and sin( t), giving them amplitudes, respectively, of 1 and j, where j2 = -1. The amplitude j is not physical, but the equations don't care. This complex stimulus, cos( t) +j sin( t) = ej t, will therefore produce a complex (non-physical) response. The real part will be the response for a stimulus cos( t), and the imaginary part will be the response for a stimulus sin( t). With the complex stimulus, all the voltage and currents will be of the form Vej t and I ej t, where the amplitudes V and I are (complex) constants. This trickery is useful because it turns differential equations into linear equations. For an inductor, for example, V(t) = Ld/dt I(t). With the complex stimulus, V(t)/I(t) = j L, which is a constant, just as for a resistor V/I = R. The differential equation for a capacitor is I= C d/dt V(t). For the complex stimulus, this becomes V/I = 1/(j C). These 'generalized resistances' are called impedances, and are designated by the symbol Z.

ZR = R ZL = j L, and ZC = -j/( C).

Without this convenient complex variable method, we would have to carry along separate cosine and sine amplitudes for every current and voltage. Using the complex source, the response is proportional to the source function. The constant of proportionality is known as the circuit's transfer function. For the circuit of Figure 2b, the transfer function is

Transfer function ( ) = complex load voltage/complex generator voltage

= RLOAD/(RGEN + RLOAD + j L -j/( C).

The magnitude of the transfer function is the ratio of amplitudes. The phase of the transfer function is the difference between the output phase and the generator phase.

Figure 3a shows a more complicated three-section bandpass filter.

This circuit has more than one loop, but, because of its simple topology, we can analyze it without resorting to simultaneous equations. The trick is to assume that the output voltage is unity and then work backward to calculate the generator voltage. For a given generator voltage, the output will be the given voltage divided by the calculated voltage. Referring to Figure 3b,

the steps would be:

1. Calculate ILOAD

2. I3 = ILOAD + 1/Z3

3. V2 = 1 - I3 Z2

4. I1 =V2/Z1

5. Igen = I1 + I3

6. Vgen = V2 - Rgen Igen

This kind of topology, a ladder network, is found in nearly every filter and amplifier.

Amplifiers

The term amplifier really means power amplifier. In a situation where we want only to step up voltage or current, we could use a transformer. When a generator is followed by an amplifier, the combination is normally capable of supplying more power than the generator alone. Conservation of energy requires that an amplifier be connected to an external energy source - usually one or more dc power supplies. A power steering system is an example of a mechanical amplifier. The added energy comes from a hydraulic pump driven by the engine.

Most of the amplifier in the microwave receivers at the observatory use Field Effect Transistors or FETs. A FET, like any other type of transistor, is a voltage-controlled resistor. Very little current is drawn by its control input (gate), so the input power is very low. The output current and hence the output power can change appreciably, so the power gain is high.

Figure 4 shows a one-stage FET RF amplifier. Current is supplied to the drain via the inductor.

This current flows through the channel of the FET and out the source, to ground.

The magnitude of this bias current is determined by the dc bias voltage impressed on the drain through the inductor L2. This gate bias can be varied by R1 to achieve the desired drain current (some value of current that puts the device in the middle of its active region so, i.e. at a value where dID/dVG is high. The value of this derivative is known as transconductance, and is given the symbol gm. The capacitors C1 and C2 used as DC blocks. Their values are high enough to make their impedance, -j/( C), negligible, so they don't block the signal at all. The inductors act as RF chokes, they have high enough values of inductance that their impedances, j L, is high, so that the signal is not shorted to ground by the DC bias supplies.

The voltage gain of the amplifier is easy to calculate. If the input voltage changes by VIN, the drain current will change by an amount gm Vin. Since L2 is a large inductor, this change in drain current must flow through the load resistor, RLOAD. Therefore VOUT = -gm VIN RLOAD and the voltage gain is VOUT / VIN = -gm RLOAD.

In this simple analysis, the transistor is represented by the simple equivalent circuit of Figure 5a. Since no gate current flows, the amplifier's power gain is infinity. Also, the voltage gain is unlimited since RLOAD can go to infinity. A more realistic model for the FET, shown in Figure 5b, includes a resistor, rDS in parallel with the current generator and an input resistor, rG, in series with a gate-to-source capacitance, cGS. See if you can derive an expression for the power gain of the amplifier when the transistor is represented by this fairly realistic equivalent circuit. Note, amplifier designers using CAD programs often work with equivalent circuits containing as many as 20 circuit elements.

Other basic topics:

Amplifier noise - Today's low-noise amplifiers are about an order of magnitude too noisy in that the noise added by the amplifier is comparable to the background sky noise. Astronomers need to know how to get the best performance from their amplifiers. Amplifier designers need to know how to get the best performance from available transistors. Transistor designers need to understand the physical sources of noise in transistors.

Amplifier stability - Any circuit containing active devices (e.g. transistors) and a source of power has what is needed to oscillate. It is important that amplifiers not break into oscillation. It is also important that oscillators be designed to produce the purest possible waveforms.

Backend equipment - detectors, correlators, Fourier Transform machines, pulsar processors, etc.

Transmission lines - Because of the finite speed of light, circuit interconnections become important when travel times are comparable to the time scale of current and voltage changes in the circuit. You should understand the operation of transm ission lines, reflections, terminations, etc.

Microwave circuitry - The microwave part of the spectrum can be defined as the frequency range over which waveguides and waveguide components are used. The analysis of these components is quite interesting and ordinary circuit theory can be extended to apply to waveguide circuits.

Antennas - a bigger step into applied electromagnetics.