ECE 5260 Microwave Engineering University of Virginia Lecture 2 Review of Fundamental Circuit Concepts and Introduction to Transmission Lines Although electromagnetic field theory and Maxwell s equations are the foundation of microwave engineering (and, in fact, most of electrical engineering), much of microwave circuit analysis and design is formulated in the language of circuit theory and is based on an extension of familiar circuit concepts. In this lecture, we will review some of the important ideas from circuits that you will need to be familiar with as we study microwave circuits. Some Background: Circuit and Field Quantities and their Relations Recall that Maxwell s equations are a set of four vector field equations that describe the forces experienced by electrical charges. For each field quantity or parameter described by Maxwell s equations, there is a corresponding circuit quantity that is used to formulate circuit theory. The table below lists these quantities: Field Quantity (Units) Corresponding Circuit Quantity (Units) E, Electric Field (Volts/meter) V, Electric Potential (Volts) J, Current Density (Amperes/meter 2 ) I, Current (Amperes) ρ, Charge Density (Coulombs/meter 3 ) Q, Charge (Coulombs) B, Magnetic Flux Density (Webers/meter 2 or Tesla) Φ B, Magnetic Flux (Webers) D, Electric Flux Density (Coulombs/meter 2 ) Φ D, Electric Flux (Coulombs) H, Magnetic Field (Amperes/meter) U, Magnetomotive Force (Amperes) The last couple of circuit quantities in the table above may be unfamiliar, but they are used in the analysis of magnetic circuits (e.g., transformers) and you will probably encounter them if you take a course on electromechanical energy conversion and power. The field and circuit quantities are related by (path, surface, and volume) integrals, V = L I = S Φ D = S E d l, Φ B = J ˆn da, Q = D ˆn da, U = S L V B ˆn da ρ dv H d l... and these relations can be used to formulate circuit theory (Kirchhoff s Laws and other similar relations) directly from Maxwell s equations: lecture 2, page 1
(1) S N D ˆnda = Φ D = Q j=1 (2) C H d l = N j=1 U = I + dφ D dt (3) S N B ˆnda = Φ B = 0 j=1 (4) C E d l = N j=1 V = dφ B dt (5) S J ˆn da = N j=1 I = dq dt In the above relations (which are the integral forms of Maxwell s Equations), each integral over a field quantity corresponds to a sum of a circuit quantity. Expressions (1), (3), and (5), for instance, are integrals over closed surfaces (denoted S). In terms of the associated circuit quantities, these surface integrals correspond to a sum of the flux of a vector field through the various elements of area that comprise the closed surface. As a concrete example, expression (5), which is the continuity equation, recasts the integral of the current density over a closed surface (a field calculation) into a sum of the currents flowing on each of the circuit branches that exit the supernode that is defined by the surface S (refer to figure 1(a)). Thus, relation (5) is recognized as Kirchhoff s current law (KCL). Normally we formulate KCL by setting the right side of (5) to zero, which is valid for non time-varying charges and currents. However, in the general case, we need to include the dq/dt term which represents charge storage within the supernode and describes the stray capacitance between that node and ground. Expressions (2) and (4) are integrals of vector fields over closed paths or loops (denoted C), and in terms of its application to circuits correspond to a summation of the associated circuit quantity over elements connected in a loop (see figure 1(b)). Again, the most familiar example is expression (4) which is a statement of Kirchhoff s voltage law (KVL). The path lecture 2, page 2
integral of the electric field over an element gives the voltage across that element. Thus, the integral of E around a closed path is simply the sum of voltage drops around the loop. As before, in the usual formulation of circuit theory, the right side of (5) is usually set to zero, which is valid for static (dc) quantities. However, strictly speaking, the right side is equal to dφ B /dt, which is the rate at which the magnetic flux through the entire circuit loop is changing. This time changing flux is associated with an induced electromotive force in the loop and can be described by a parasitic loop inductance. (a) (b) Figure 1. Time-Harmonic Signals and Phasors The most important types of signals we study in electrical engineering are sinusoids or timeharmonic signals. In addition to being widely used in communications and a host of other practical applications, most arbitrary signals can be expressed as a superposition of sinusoids which is the basic idea behind Fourier Analysis. In general, we express an instantaneous (i.e., explicit function of time) sinusoidal signal in the form V (t) = V cos(ωt + ϕ) = 2V }{{} 0 cos(ωt + ϕ) =V ( ) where t is time. The second form shown above may look a bit odd because it includes an extra factor, 2. Including this term, however, is a convenience when we look at the power associated with time-harmonic signals. From expression ( ) above, the important parameters that describe a time-harmonic signal are: parameter V = 2V 0 V 0 ω = 2πf f ϕ description and units Amplitude Root-Mean-Square (rms) Amplitude Angular Frequency (radians/second) Frequency (cycles/second or Hertz) Phase (radians or degrees) lecture 2, page 3
An important class of electrical networks (including those we will focus on in this class) are linear systems or networks. A very important property of linear networks is that they preserve the frequency of any signal input to the system. That is, if the input to a linear system is at a given frequency, then the output will also be at the same frequency. This means that, given an input time-harmonic signal, we just need to keep track of the amplitude and phase to determine the output signal as these are the only quantities that can change. Phasors are a convenient way of doing this and provide a significant simplification in determining input/output relationships of linear circuits and networks. The phasor concept is based on Euler s relation from complex variables, e jθ = cos θ + j sin θ where j = 1. In the above expression, cos θ is the real part and sin θ is the imaginary part (because it is multiplied by j). From Euler s relation, we can express the time-harmonic signal ( ) in the form, 2V0 cos(ωt + ϕ) = 2 Re{ V 0 e j(ωt+ϕ)} = { 2 Re V 0 e }{{ jϕ } =V e jωt} where Re{ } denotes the real operator and selects only the real part of the argument within the brackets. The quantity, V = V 0 e jϕ = V 0 cos ϕ + jv 0 sin ϕ ( ) is a complex number known as the phasor representation of the time-harmonic signal, V (t). Note that V is a function of the amplitude (V 0 ), phase (ϕ), and possibly the angular frequency (ω) of the original sinusoid. However, phasors have no time dependence! Essentially, in expressing a signal in phasor form we have made a transformation from the time-domain to the frequency-domain. To recover the instantaneous form of the signal from its phasor representation, you need to 1. multiply the phasor by 2 and e jωt, and 2. take the real part Note that the phasor representation given in ( ) does not include the factor 2. When this factor is dropped (as we have done), the phasor represents an rms amplitude (rather than the full amplitude). Also note that the phasor in ( ) is written in both polar form (magnitude and phase) and Cartesian (or rectangular-coordinate) form with separate Real and Imaginary components. Polar form is usually most convenient, but you should recall that phasors (as well as any complex number) can be expressed either way and can be pictured as vectors in the complex plane, as shown in figure 2. Graphical representation of phasors on the lecture 2, page 4
complex plane is a useful tool for visualizing circuit design problems and we will encounter an important version of this when we study the Smith Chart later this semester. Figure 2. Impedance Another important concept from circuit theory that will play a central role in microwave circuit theory is that of impedance. Impedance (denoted Z) is a frequency-domain concept and is defined as the ratio of a voltage phasor to a current phasor. If the rms voltage and current phasors associated with a circuit element (figure 3) are given by V = V 0 e jϕ, and I = I 0 e jθ then the impedance is Z V I = V 0e jϕ I 0 e jθ = V 0 I 0 e j(ϕ θ) = Z e jψ = R + jx Impedance is a complex number with units of Ohms (Ω) and can be written in polar form with magnitude Z and phase ψ = ϕ θ, or in Cartesian form with real and imaginary parts R and X, respectively. The real part of Z, R = Re{Z}, is known as the resistance and the imaginary part of Z, X = Im{Z}, is called the reactance. From circuit theory, you should recall that for lumped elements connected in series, the total impedance of the combined elements is simply the sum of the impedances of the individual elements (figure 3(b)), N Z total = Z 1 + Z 2 + Z 3 +... = lecture 2, page 5 k=1 Z k for N series connected elements
It is also useful to define the reciprocal of impedance, known as admittance, Y : January 17, 2019 Y I V = I 0e jθ V 0 e jϕ = I 0 V 0 e j(θ ϕ) = Y e jψ = G + jb Admittance has units of Siemens (S) or, equivalently mhos. When expressed in Cartesian form, the real part of the admittance of a circuit element, G = Re{Y }, is known as conductance and the imaginary part of Y, B = Im{Y }, is called the susceptance. The concept of admittance is most useful when lumped elements are connected in parallel (or shunt). In this case, the admittance of the combination is just the sum of the admittances of the individual elements (figure 3(c)), N Y total = Y 1 + Y 2 + Y 3 +... = k=1 Y k for N parallel connected elements You need to be a bit careful in converting between impedance and admittance (and viceversa), because you are dealing with complex numbers. The admittance Y of a circuit element that has impedance Z = R + jx is so, Y = 1 Z = 1 R + jx = R R 2 + X 2 j X R 2 + X 2 = G + jb G = R R 2 + X 2, and B = X R 2 + X 2 Figure 3. lecture 2, page 6
Two-Port Networks and Network Theorems We will find that, in general, microwave circuits are usually represented as multi-port networks that have multiple pairs of terminals (serving as inputs or outputs). Most of the circuits we will encounter are two-ports that have two sets of terminals. You should recall from your circuits classes that a terminal is simply a point or node where two circuit elements can be joined. A port is a pair of terminals, across which we can apply a voltage (figure 4). Figure 4. In basic circuit theory, we can represent the voltages applied to the ports of a network as a vector whose elements are the port voltages (figure 4). Likewise we can represent the port currents as a vector. Note that, by convention, we define the port currents as flowing into the network from the terminal with positive voltage polarity and out of the network from the terminal with negative polarity. This choice is consistent with the passive sign convention rule from circuit theory power is delivered to an element or network (and carries a positive sign) if current flows into the element (network) from the terminal with positive polarity and exits the element (network) from the terminal with negative polarity. The voltages and currents at the network ports are related by a matrix known as the impedance matrix or Z-matrix, [Z]: ( V1 V 2 ) ( ) z11 z = 12 z 21 z 22 }{{} =[Z] ( I1 I 2 ) ( ) z11 z [Z] = 12 z 21 z 22 The elements of the Z-matrix, {z ij }, are known as z-parameters. We can also relate the port voltages and currents of a network using an admittance matrix or Y -matrix, [Y ]: lecture 2, page 7 ( I1 I 2 ) ( ) y11 y = 12 y 21 y 22 }{{} =[Y ] ( V1 V 2 ) ( ) y11 y [Y ] = 12 y 21 y 22
where, as you might suspect, the elements of the Y -matrix, {y ij }, are called y-parameters. From the above relations, it is straightforward to see that the impedance and admittance matrices are inverses of one-another: [Y ] = [Z] 1 = 1 z ( z22 z 12 z 21 z 11 ) and [Z] = [Y ] 1 = 1 y ( y22 y 12 y 21 y 11 ) where z = z 11 z 22 z 21 z 12 and y = y 11 y 22 y 21 y 12 are the determinants of the Z and Y matrices, respectively. In addition to using matrices to represent microwave networks, we will find Thévenin s and Norton s Theorems to be useful in simplifying networks and replacing complex networks with simple equivalent circuits. Thévenin s Theorem states that the network seen looking into a port (figure 5(a)) can be replaced with an equivalent circuit consisting of a series-connected independent (phasor) voltage source and impedance provided the network is (1) linear, (2) contains independent current and/or voltage sources, and (3) that any dependent source within the network is a linear function of its control parameter (voltage or current) and that control parameter also lies within the same network. The Thévenin voltage source in the equivalent circuit is the open-circuit voltage measured at the port of the network. Figure 5. Norton s Theorem (illustrated in figure 5(b)) is the dual of Thévenin s Theorem and states that the network seen looking into a port can be replaced with an equivalent circuit consisting of a parallel-connected independent (phasor) current source and impedance (or admittance) provided the same conditions stated above pertain. The Norton current source in the equivalent circuit is the short-circuit current measured at the port of the network when its terminals are shorted. The Thévenin impedance (or admittance) is determined by killing all the independent sources in the network and finding the resulting input impedance (or admittance) seen at the port. Because the two equivalent circuits shown in figure 5 represent the same network, the Thévenin open-circuit voltage and Norton short-circuit current are related by Z Th = 1 Y Th = V oc I sc lecture 2, page 8
Complex Power The final basic concept we need to review is power. From circuits, you should recall that the power delivered to an element is found as the product of the voltage and current, P = V I The unit for power is the Watt (1 W = 1 V 1 A). In the above expression, the current is defined as positive if it enters the element from the terminal with positive voltage polarity (+) and exits the element from the terminal with negative voltage polarity ( ). This convention (know as the passive sign convention) yields a positive value for power and indicates that power is delivered to the element. If the current enters the element from the terminal with negative voltage polarity ( ) and exits the element from the terminal with positive voltage polarity (+), then it is assigned a negative sign which yields a negative power. This minus sign indicates that power flows from the element (to the outside world) and that the element is acting as a source. The sign convention above is basically a book-keeping method for keeping track of which direction power flows in a circuit. We calculate power for time-varying and time-harmonic signals in exactly the same way. If V (t) = 2V 0 cos(ωt + ϕ) and I(t) = 2I 0 cos(ωt + θ), then P (t) = V (t)i(t) = 2V 0 I 0 cos(ωt+ϕ) cos(ωt+θ) = V 0 I 0 cos(2ωt+ϕ+θ)+v 0 I 0 cos(ϕ θ) The power above is a function of time and is called the instantaneous power. Note that the instantaneous power has both a time-dependent part and a constant (dc) part. The time-dependent component varies at twice the frequency of the voltage and current signals and this results because power is not a linear function of voltage or current. Usually, in RF and microwave engineering, we are interested in the average power delivered to a load rather than the instantaneous power. We find the average ( f ) of a time-periodic signal, f(t), that has period T by integrating the signal over a period and dividing by the period: f = 1 T t 0 +T f(t) dt t 0 Applying this to the instantaneous power (and noting the time-varying component will integrate to zero), gives us P = P avg = V 0 I 0 cos(ϕ θ) lecture 2, page 9
Recall that the amplitudes V 0 and I 0 are rms values. Thus, we can see from this expression one convenience of using rms amplitudes they eliminate the need to include a factor of 1 2 in calculating average power. The average power above can be considered to be the produce of two separate terms: V 0 I 0 this is called the Apparent Power and has units of Volts-Amperes or VA. This is obviously the same, dimensionally, as a Watt, but the VA unit is used instead to distinguish apparent power from average power (which is measured in Watts). cos(ϕ θ) this factor is called the power factor. The power factor depends on the phase difference between the voltage and current and varies from 1 when voltage and current are in phase (as for a resistor), to 0 when voltage and current have a ±90 phase difference (as for capacitors and inductors), to 1 when voltage and current are 180 out of phase (as for a power source). The average power can also be calculated directly using phasors. To see this, consider the phasor representation of the voltage and current given above V (t) = 2V 0 cos(ωt + ϕ) V = V 0 e jϕ I(t) = 2I 0 cos(ωt + θ) I = I 0 e jθ From these we see that the quantity, { Re V I } { = V 0 I 0 Re e j(ϕ θ)} = V 0 I 0 cos(ϕ θ) = P avg also yields the average power. The above expression is the basis for the definition of complex power, S: S V I = P + jq From this definition, we see that the average power delivered to a circuit element (P ) is the real part of the complex power. The imaginary part of the complex power is called reactive power, and this quantity describes the flow of energy back-and-forth between energy-storage circuit elements (inductors and capacitors). The unit for average power is the Watt (as usual), but the unit used for reactive power is the VAR (for Volt-Ampere-Reactive). Again the VAR is dimensionally the same as a Watt, but the unit is given a different name to distinguish it from average power: { } { } P = Re S = Average Power (W), Q = Im S = Reactive Power (VAR) lecture 2, page 10
The Decibel Finally, we come to the last preliminary topic we need to discuss the decibel. The decibel is a logarithmic (to the base 10) measure for the ratio of one power level, P 2 with respect to another power level, P 1. That is, this ratio expressed in db is given by P 2 P 1 (db) 10 log 10 ( P2 P 1 ) Expressing power ratios is db is useful for a couple of good reasons: For two-port networks, we are usually interested in the ratio of the output power to the input available power. This is called the transducer gain of the network. Power gain (or loss) often varies by many orders-of-magnitude (powers of 10) and such large or small numbers are more convenient to express in db (for example, a gain that might range from 0.0001 to 10000 over some frequency range can be expressed in db as ranging from 40 db to 40 db). When cascading networks, we can find the total gain or loss in db by simply summing the gains or losses (expressed in db) of each network. the power ratio above can also be written in terms of voltages. If the input power to a network (with input impedance of R in ) is P 1 and the power delivered to an output load (R L ) is P 2, then the transducer gain in db is given by P ( 2 V2 2 (db) 10 log P 10 1 R L R ) in V 1 2 Usually, in microwave circuits the input and output impedances are the same (and equal to 50 Ω), which gives us P 2 P 1 (db) 10 log 10 ( V 2 2 V 1 2 ) V 2 = 20 log 10 V 1 Notice that the ratio above is still just a ratio of powers!. The key observation is that the only difference in the expression for power ratio is that one is being expressed using a ratio of voltage amplitudes (hence the factor of 20 instead of 10) and the other as a ratio of power levels. Ultimately, however, both are just providing a measure of the ratio of output to input power level. It is useful to get some feeling for common ratios expressed in decibels and the following table gives some of ones you should remember: lecture 2, page 11
Ratio Decibels Ratio Decibels 1 0 db 1.25 1 db 0.8 1 db 2 3 db 0.5 3 db 2.5 4 db 0.4 4 db 4 6 db 0.25 6 db 5 7 db 0.2 7 db 8 9 db 0.125 9 db 10 10 db 0.1 10 db 100 20 db 0.01 20 db 1000 30 db 0.001 30 db It should be noted that in RF and microwave engineering, the decibel is also used to specify absolute power levels. When this is done, the power P is being compared to a reference power (P ref ) and the db unit is modified to indicate the reference power level. The three most common cases are when P ref = 1 µw, P ref = 1 mw, and P ref = 1 W: P ref Decibel Unit 1 µw 10 log 10 (P/1 µw) = P (dbµ) 1 mw 10 log 10 (P/1 mw) = P (dbm) 1 W 10 log 10 (P/1 W) = P (dbw) Fundamental Transmission Line Concepts Low-frequency circuit theory is based on the concept of a lumped element a circuit component in which the electromagnetic fields (and, consequently, the voltage and current) do not vary appreciably with position. At sufficiently high frequencies, we cannot treat circuit elements (including the interconnects such as wires and cables) in this way and must allow for the possibility that the fields will vary over the spatial extent of the components. Generally speaking, the microwave region (300 MHz to 300 GHz) is the frequency range where we must modify circuit theory to account for the spatial dependence of the fields. We can understand transmission lines by using concepts and methods borrowed from lumpedelement circuit theory and applying them to short-sections of transmission lines, where they are (approximately) valid. This approach leads to the incremental circuit model for a transmission line and permits us to incorporate wave propagation into conventional circuit theory. The Incremental Circuit Model In transmission line theory the length of circuit elements is comparable to or larger than a wavelength (but we usually assume the cross-sectional dimension is much less than λ). lecture 2, page 12
Circuit components for which this is true are said to be distributed. As noted above, we can often model transmission line circuits using concepts and techniques valid for lumpedelement circuits and low frequencies (e.g. Kirchhoff s laws)... provided we are careful and only consider TEM mode propagation. Using lumped circuit theory as a guide, we can model a short section of transmission line as shown in figure 1. This model is known as the incremental circuit model. In figure 1, Z is the distributed impedance (units of Ω/length and Y is the distributed admittance (units of Siemens/length). Z can be thought of as modeling the resistance and inductance per unit length of the line, while Y represents the shunt conductance and capacitance per unit length. For the short section of line, we can write the following approximate relations based on Kirchhoff s current and voltage laws: V (z) V (z + z) Z zi(z), and I(z) I(z + z) Y zv (z + z). Strictly speaking, these relations are based on simplifications of Maxwell s equations and are expected to only hold for lumped elements. However, if we take the limit as z 0, these expressions should become exact. Dividing through by z and taking this limit yields the Telegrapher s Equations: dv (z) dz = ZI(z) (1), and di(z) dz = Y V (z) (2) The Telegrapher s equations are circuit analogies to Maxwell s curl equations (Ampére s Law and Faraday s Law). The Telegrapher s equations are a set of first-order coupled linear differential equations. We can obtain expression for the voltage or current alone by taking the second derivative, d 2 V (z) dz 2 = Z di(z) dz = γ 2 V (z), and Fig. 1. Distributed circuit model for a transmission line. lecture 2, page 13
d 2 I(z) dz 2 = Y dv (z) dz = γ 2 I(z), where γ 2 = ZY. Each of these equations is known as the wave equation (or, the homogeneous Helmholtz equation ). The general solution to these equations are traveling waves and can be written as, V (z) = V + 0 e γz + V 0 eγz, and (3) I(z) = I + 0 e γz + I 0 eγz. (4) The parameter γ is known as the propagation constant and, in general, it is a complex number written as γ = α + jβ The relation, γ 2 = ZY is called the dispersion relation and it describes how a wave moves or propagates along a given transmission line. The real part of γ is called the attenuation constant (α) and it describes the change in amplitude of a wave as it propagates along the z direction. The imaginary part of γ is called the phase constant (β) and it describes the change in phase of the wave as it propagates along the z direction. To better understand the solutions given in equations (3) and (4), let s convert these to instantaneous (time-domain) expressions. Recall that to obtain an instantaneous expression from a phasor, you must multiply through by exp(jωt) and take the real part. This gives us v(z, t) = V + 0 e αz cos(ωt βz) + V 0 eαz cos(ωt + βz), and, (5) i(z, t) = I + 0 e αz cos(ωt βz) + I 0 eαz cos(ωt + βz) (6) For now, let s take a closer look at the phases of these solutions... that is the arguments of the cosines. Phase Velocity To understand the waves described by equations (5) and (6), let us examine a fixed point on a particular wave and see how it moves with time. this is depicted in figure 2. For clarity, we will assume that α = 0 for the moment. As we will later see, this value of α describes a lossless transmission line. Suppose that at a time instant t, the total phase (i.e., the argument of the cosine functions above) for the two wave solutions are, lecture 2, page 14
ωt βz = constant and ωt + βz = constant Taking the derivative with respect to time, we find that. ω β dz dt = 0 and ω + β dz dt = 0 Note that dz is just the change in position of the point in question over time interval dt. Thus, dz/dt is the velocity at which this point on the wave moving. This is called the phase velocity and it is given by, dz dt = ω β v+ p for the solution with argument ωt βz and by dz dt = ω β v p for the solution with argument ωt + βz. The negative sign indicates that the phase fronts move in the z direction. Thus, we see that the solutions, V + 0 e γz, and I + 0 e γz represents voltage and current waves propagating in the +z direction with phase velocity v p + = ω/β, and the solution V 0 eγz, and I 0 eγz represents a plane wave propagating in the z direction with phase velocity v p = ω/β. Wavelength Wavelength is another important parameter describing wave solutions. It is the spatial analogy of frequency (and it s reciprocal is sometimes called the spatial frequency or wavenumber). The wavelength (denoted by the symbol λ) is defined as the distance, at a given instant of time, over which the phase of the wave changes by 2π: βλ = 2π λ = 2π β = 2πv p ω = v p f lecture 2, page 15
Fig. 2. Illustration of wave propagation by noting how a selected point on the wave moves over time. The phase velocity can be found as v p = z/(t 2 t 1 ) = ±ω/β. where ω is the angular frequency (units of radians per second) and f is the frequency. Thus, in the above expressions for voltage and current, the first set of terms represent a (voltage or current) wave traveling along the +z direction, while the second solution represents a wave propagating in the z direction. Characteristic Impedance The voltage and current amplitudes of the traveling waves are not independent, but are related by the Telegrapher s equations: dv dz = γ(v + 0 e γz V 0 eγz ) = Z(I + 0 e γz + I 0 eγz ) The coefficients of corresponding exponentials must be the same, so V + 0 I + 0 = V 0 I0 = Z γ = Z Y Z 0 (7) lecture 2, page 16
Z 0 is called the characteristic impedance of the transmission line and it is a function of the line geometry as well as the electrical parameters of the materials from which the line is fabricated. Please note that the characteristic impedance simply gives the ratio of the voltage and current for a traveling wave on the line it is not the ratio of voltage and current for a superposition of traveling waves (also known as a standing wave ). Another important point to note is that there is a negative sign associated with the ratio of voltage to current amplitude for a negative traveling wave. This sign change must be taken into account explicitly and is a result of passive sign convention rule from circuits to represent power flow in the z direction, the sign of either the current or voltage must be reversed. We will examine this in detail a little later. lecture 2, page 17