VOLTAGE
3. WAVE PROPAGATION & MANIPULATION
3.1 Transmission Lines
A functional circuit is designed by assembling discrete components that each impart some desired property to the network, and one can express the 'laws' of a network's behavior in terms of constant variables such as resistance, capacitance, and inductance. The simplicity of the analysis follows from the fact that one can ignore the properties of the electromagnetic field at low frequencies because the wavelength, Aocf -l, is very large (approaching infinity in the limit of DC signals) and the dimensions of the circuit elements is comparatively small. Under such circumstances a network can be mathematically described by discrete elements, but as the frequency increases, the wavelength becomes smaller and eventually becomes comparable to the size of the circuit elements. At this point the geometry of a circuit element and the circuit layout become design factors.
Also problematic at high frequency is the fact that common parameters that are used to define a circuit's response are no longer applicable. For example, voltage is ordinarily defined as the difference in electrical potential between two points that may be arbitrarily defined. The formal definition of voltage, however, is a work function associated with traversing the region between two points, that is, a line integral V=
J
E ·ds. At low frequency this integral is path-independent and V=
En - EA , but at high frequency the integral is path-dependent and the ordinary means of defining and measuring Vare invalid. At high frequency a circuit is therefore defined in terms of its wave properties, and the transition frequency between distributed and lumped circuits is often taken as I GHz, where A::::: 12 in (30 ern). A summary of the properties of electrical circuits and their measurement at various frequency ranges is outlined as follows (after Ginzton, 1957):Table 1. Electrical Measurements at Low and High Frequency
Frequency DC
Audio to 50 kHz lowrf high rf microwave
Voltage voltmeter voltmeter, rectifier voltmeter, rectifier
not accurate not significant
Current ammeter thermocouple thermocouple thermocouple not significant
Impedance Wheatstone Bridge Wheatstone Bridge Wheatstone Bridge
bridge, SWR bridge, SWR
1.MICROWAVE ENGINEERING FUNDAMENTALS AND SPECTROMETER DESIGN 27 A single conductor will transmit a DC current with losses that can be attributed to resistance, which is typically described as a scattering phenomenon between the conduction electrons and the atomic nuclei of the conductor material. An electromagnetic wave, by contrast, will likewise propagate along a single conductor, but its energy will be radiated away from the conductor. In other words, the single conductor device acts as an antenna and is subject to high radiative losses. A shield is required to prevent these radiative losses, and transmission lines are predominantly two-conductor structures such as parallel lines, coaxial lines, or hollow conductors (i.e. waveguides). In all cases the direction of propagation is along the axis of the transmission line, and one defines the propagating electromagnetic field by whatever component (electrical or magnetic vector) is perpendicular to the propagation vector.
An electromagnetic wave has two orthogonal components, the electric and magnetic fields, which are respectively analogous to the DC voltage and current. It follows that the propagating modes of an electromagnetic wave can be defined as transverse electrical (TE), transverse magnetic (TM), or transverse electromagnetic (both E and H, TEM). One or more of these modes may be suppressed by the geometry or the dielectric properties of the transmission line. For example, a waveguide is more restrictive with respect to allowed propagating modes than a coaxial line. The design and measurement of circuits at high frequency are therefore based upon wave phenomena, and circuit parameters R, L, and C are replaced by an all-encompassing impedanceZ , 15which can be related to the scattering behavior of waves.
Microwave circuits are commonly defined by a Voltage Standing Wave Ratio (VSWR) coefficient, and this is best explained by beginning with a mechanical model such as a wave in a fluid medium. The ideal scenario for wave propagation is one in which waves are propagated through a medium with minimal loss of amplitude, that is, energy, and this energy is delivered to a terminus with an efficiency of unity. An example would be water waves that impinge on a gradually sloping shoreline and are therefore not reflected.
One can take as a starting example the case in which a microwave source delivers an oscillatory signal to an infinite line. The electromagnetic wave will propagate through this medium presumably ad infinitum, but one finds that the signal weakens as the distance from the source increases. An observer at some arbitrary distance from the source would record the oscillations of the traveling wave: E(x)
= Eo
sine(J)t -p
x+ r/J). In this equation (J)=21C!, where! is the oscillator frequency, andP
is the phase constant of the medium, 27t1A. This equation describes a traveling wave and15Formally, circuit analysis at high frequency introduces imaginary variables to account for the time-domain response of circuit elements. Traditional DC variables constitute a real component, whereas the imaginary component is called a reactance.
applies equally to both infinitely long transmission lines and a transmission line that is terminated by a load that perfectly absorbs all the energy of the incident wave. Measurement of the time averaged signal intensity at all points along the line would yield a constant value in the absence of resistive losses. For a very long line in which losses become a factor, the signal intensity would decline linearly with distance. Losses in signal intensity vary among the types of transmission lines, being related to geometry and the material from which it is fabricated. For example, standard rectangular copper waveguide at X-band (WR-90) typically dissipates between 4.5 and 6.5 dB of power per 100 linear feet.
An infinitely long or perfectly absorbing line is one extreme case in describing the propagation of waves through a transmission line. The other extreme corresponds to the case in which the transmission line is terminated by a load that perfectly reflects the incident wave. A so-called stationary wave is set up in such a situation, and the important point is that, unlike the traveling wave whose time average amplitude is the same for all points, the stationary wave amplitude varies with position. Returning to the water wave analogy, the scenario is equivalent to the difference between waves impinging upon a vertical wall as opposed to a gradually sloping beach.
The amplitude of a stationary wave is characterized by peaks and troughs that are formed by the interference between the incident and reflected wave.
As a simple illustrative example, consider the incident wave that completes a half cycle at the terminus, where it is then reflected (the amplitude is zero).
This would correspond to a perfect short circuit in the transmission line and a perfectly reflecting load, and the reflected wave can have either positive or negative polarity (0° or 180°).Negative polarity nulls the incident wave, but positive polarity reinforces it, and the standing wave mirrors the incident wave with a boundary condition of zero amplitude; nodes (troughs with amplitude zero) occurs atx
=
lhn A.A transmission line that is terminated by a partially reflecting load likewise supports a stationary wave, but true nodes are. not created. In the case of the perfectly reflecting wave, nodes are created because the incident and reflected wave have equal amplitudes, and the out of phase waves null one another. When the amplitude of the reflected wave is less than the amplitude of the incident wave, the waves do not null one another, and even if the reflected wave is 180° out of phase with the incident wave, there is still some signal amplitude left at x
=
lhnA. The measured signal intensity at some point x is the sum of all possible interference effects, and a partially reflecting terminus therefore creates troughs whose minima differ from zero.It follows that the relative magnitude of the stationary wave maximum and minimum provides a measure of the wave reflection. We define a reflection coefficient
r =
Emax / Emin, where 1.0s r s
00; a perfectly reflecting waveI.MI CROWAVE ENGINEERING FUNDAMENTALS AND SPECTROMETER DESIGN 29 nulls the incident wave E
max /
Emin~ 00, and a perfectly absorbing termination reflects no wave, creating no nodes, so thatEmax / Emin =
1.0.Complete reflection represents an infinite impedance to wave propagation, whereas the perfect absorber 'matches' the transmission line by absorbing all the incident power. The ratio of the wave maximum and minimum, which has been defined in the preceding paragraph and is called the Voltage Standing Wave Ratio (VSWR), therefore provides a measure of the match between a microwave circuit device and the transmission line source of electromagnetic radiation. Impedance is therefore only defined with respect to the transmission line; one does not need to be concerned with the absolute value of the impedance. The important design factor is the ratio Z / 20 , where Z is the impedance of the device and 20 the impedance of the transmission line feed.