Important concepts in Electromagnetics and Microwaves

Important concepts in High Frequency Electromagnetics

Electromagnetics is a discpline involving time varying electric and magnetic fields. Its study involves physics and mathematics, and is listed as applied physics in some of the Institutes. The basis of electromagnetics is the Maxwells equations. These equations in differential form can be combined to yield wave eqn. for field quantities E and H. The corresponding circuit quantities are voltages and currents. The solution of wave eqn in a homogeneous loss-free medium is a plane wave described by . Wave propagation is a central theme in electromagnetics and therefore microwave engineering. The solution of various problems in electromagnetics can be understood in terms of the physical aspects of wave propagation. When this propagating wave comes across a change in medium properties the associated effects of reflection, refraction, diffraction and scattering are observed. When the time variation is set to zero in Maxwells eqns the equations for E and H get decoupled resulting in separate differential eqns representing electrostatics and magnetostatics. The solution for electrostatics and magnetostatics can also be obtained by setting in the time harmonic solution of Maxwells eqns.

Full-wave solution is the one in which variation of field quantities as a function of frequency is fully taken into account. Quasi-static condition is sometimes used in electromagnetics to describe a situation which is neither full-wave nor fully static. In this case, variation of some of the field quantities is assumed to be static only over certain frequency range. In the next section we assume a full-wave solution of an electromagnetic problem. Electromagnetics is sometimes referred to as field theory aspect of solutions. The field theory solutions can also be cast in the form of circuit representation such as [S], [Z] or [Y] matrix. For this, the electric and magnetic fields at the ports are converted into rf voltages and currents employing

andThe vectors [v] and [i] are next related to [S], [Z] or [Y] matrix. The circuit based solution is generally desirable for compatibility with circuits and is much more efficient.

Block-diagram of a transceiver microwave sub-system is shown next. It is one of the very useful module for radar applications. It includes almost all types of microwave components, active and passive.

Transceiver module

The high frequency electromagnetics may be called microwave electromagnetics, and the corresponding circuit engineering as microwave engineering. At frequencies less than 10MHz, the wavelengths are much larger comparable to the size of R,L,C, diodes and transistors which constitute the circuit. For comparison, we may assume a printed circuit board of length l say 20 inch. At 10 MHz, the ratio , and is very small. At microwave frequencies, 1 GHz to 30 GHz frequency range and higher, this ratio decreases so that at 3 GHz. The high frequency circuit components printed on this board will have size comparable to the wavelength. In this frequency regime, certain special effects become prominent because the component size becomes comparable to the wavelength of operation. Some of these effects are: proximity effect leading to coupling and cross-talk, radiation, parasitics, transit time effect, etc. Another effect called skin effect in conductors leads to higher conductor resistance at these frequencies.

The central theme of high frequency electromagnetics is fast variation of phase with space and time, . This phenomenon leads to sufficient amount of coupling (or cross-talk), and radiation if the component size is comparable to wavelength. The excessive variation of phase across a lumped element necessitates use of distributed element approach for circuit design. The use of transmission lines and waveguides at high frequencies confines these signals to reduce radiation. The guided waves in these lines are employed to design circuits which otherwise cannot be accomplished using unguided waves. The transmission lines are also employed as interconnects to carry signal from the output of one component to the input of other component. Another important effect is skin effect which leads to higher conductor attenuation. The field distribution in the lines is called a mode. For the modes in waveguides, the impedance is not uniquely defined. Therefore, measurements and circuit characterization at high frequencies is carried out in terms of S-parameters and not Z- or Y-parameters which is the practice at low frequencies. When the transit time in devices such as vacuum tubes becomes comparable to time period of the signal, it gives rise to undesirable effects such as oscillations. For this reason, vacuum tubes had to undergo miniaturization for use at high frequencies or replaced by semiconductor devices. See also Sec 11.1 and 11.2 of Electromagnetic Compatibility, Second edition, by C. R. Paul. The sections are: Changing the way we think about electrical phenomena and What do we mean by the term ground?

Coupling or Proximity EffectCoupling plays a very important role in high frequency electromagnetics. Coupling between the modes of a conductor, as a basic phenomena is responsible for singularity in charge and current distribution in conductors. Coupling is used to design power dividers and combiners, and directional couplers. It also creates nuisance in the form of cross-talk leading to signal integrity issues for digital signals.

Wave PropagationWave propagation is an important important physical phenomenon in electromagnetics. Physically, it originates from Huygens principle where a point source produces a wavefront at a distance. Every point on the wavefront acts as a secondary source and produces another set of wavefronts. This process continues and generates field everywhere away from the source. Mathematically, this process is created through Maxwells equations wherein the Faradays law defined as time variation of magnetic induction produces electric field at a distance, i.e.

and the extended Amperes law defined as

describes generation of magnetic field by the time variation of D. This process when continued describes propagation of signal from the source to destination. Medium is not necessary for the propagation of electromagnetic waves. Our immediate concerns are: unguided waves such as propagation in free space; and guided waves such as in transmission lines and waveguides. We come across unguided waves in mobile communications, broadcast, antennas and scattering from objects. The transmission lines and waveguides are used as interconnect for signal transmission and circuit element for microwave circuits.

Reference planes: The equivalent circuits at microwave frequencies are enclosed between reference planes at the ports of the network. This is due to the fact that voltages and currents have spatial variations that are significant, and therefore the network parameters [Z], [Y] or [S] depend on the reference planes. Any shift from the reference planes gives rise to the change in network characteristics.

Current and charge distributions in a conductor

Almost all the antennas, microwave circuits, and scatterers (secondary sources) consist of metallized dielectrics. The electromagnetic property of these components is determined by the charge and current distributions on the metallization. For simplicity, one may neglect the effect of dielectric to obtain the broader perspective and use free space Greens function in the analysis. The effect of dielectric can be included later by employing the proper Greens function or the use of equivalence principle for inhomogeneous dielectrics. The current distribution on a conductor surface obeys skin effect and the resultant charge density can be calculated from .

Note: Describe the internal inductance and resistance per unit length of a conductor wire before taking up a lossy transmission line case. (Refer to Sec. 8.2 of Analysis of multiconductor transmission lines, Second Edition, C. R. Paul, IEEE Press)

Skin Effect (ref.: P.A. Rizzi, Microwave Engg: Passive circuits, PHI, pp. 42-45)Skin effect refers to the current concentration in conductors at high frequencies, and the associated increase in conductor resistance is called skin effect resistance.

To describe development of skin effect in a conductor we consider a plane waves incident on a metal bar as shown. Plane waves can be produced in a number of waves. The most common form is using a distant source attached to an antenna. Here, we consider an ac source connected to a TEM mode horn as shown in the figure. The plane wave described by components is incident normally on a semi-infinite conducting bar. Applying Faraday law and Amperes law to these field components gives (assuming time variation)

and

The conduction current defined by is very large in conductors compared with the displacement current . Therefore, we may neglect the displacement current by writing .

Fig. 1: (a) TEM wave incident on a conducting bar(b) Effective dimensions of metal bar to compute its skin-effect resistance

Eliminating from the above, results in the following second-order differential eqn

Its solution is

, Selecting the forwarding traveling wave in the conductor gives

The decay inside the conductor sheet is plotted in Fig.2. The decay of fields is very fast if the skin depth is very small. The corresponding current density in the conductor is given by . The magnetic field also decreases at the same rate since with , the intrinsic impedance of conductor.

Fig. 2: Field distribution in a semi-infinite metal bar

Due to the skin effect, the field decays very fast in very good conductors. At 3 skin depth, the field and current decay to 5% of its value at the surface. Since power density is proportional to, its decay is much faster and the value at 3 skin depth is 0.25% of its value at the surface. For copper, mho/m, at 10 GHz and . At microwave frequencies, we may therefore say that the current is confined to the surface of metals.

Skin effect resistanceThe impedance of the semi-infinite metal bar, also called surface impedance, is

Due to the incident field on the bar, the associated rf voltage is . The current density is . Therefore, the current normal to yz-plane is given by

AOr

The ac resistance of the semi-infinite conductor may be interpreted as the dc resistance of a rectangular bar of length t, and effective area of cross-section perpendicular to the current direction, , i.e. skin effect resistance may be computed by assuming that the current is uniformly distributed over a thickness of one skin depth. This is illustrated in Fig. 1(b). The above analysis neglects the effect of current crowding at the corners of a rectangular conductor. The internal inductance of a conductor is given by

Henry

For a conductor wire immersed in uniform electromagnetic field, the field will penetrate the conductor from all the surfaces. Therefore, only the outer shell of the wire will be responsible for rf resistance. For a round wire of radius a and length l such that a >>, its ac resistance is given by

ohm

The variation of resistance and internal inductance of an isolated conductor wire with frequency is plotted in Fig. 3 and may be modeled as

Fig.3: Variation of wire resistance and internal inductance with frequency

(The scales are different for r and )

For a rectangular strip of width W and thickness t, the values of and are given by

t W

The break frequency is the frequency at which the asymptotes and meet and is defined by

,

The skin resistance is defined as

for and t >> or

at or

For a strip with aspect ratio W/t < 10, the expression for gets modified as

The value of k depends on the aspect ratio W/t and is about 1.5 for W/t = 1.

Variation of current density distribution inside a strip for three different frequencies is plotted next.

t

at at at

As an example, consider a 10 inch long copper strip with w = 127micron and t = 50 micron. Its dc resistance is 0.69 ohm and increases to 18.75 ohm at 10GHz; about 27 times the dc resistance. Thus conductor loss is significantly higher at the higher frequencies.

The internal inductance of metal wire with frequency behaviour may be modeled as

where

Combining the above models for resistance and internal inductance we may express the internal impedance of a conductor as

This expression satisfies the requirement for the transform to represent a real valued function of time. The values of and needed are given by

and

The internal inductance of a conductor wire (H/m) is important at only because it decreases as for . The decrease in internal inductance with frequency is due to the migration of current to the surface and the consequent decrease in magnetic flux inside the conductor.

Also see: Internal impedance of PCB lands over a broadband (ref; Analysis of Multiconductor Transmission Lines, C.R. Paul, p. 410)

The closed-form expression for the dc inductance of a strip is given by (for ) [1]

H/mError < 0.36%

For a strip with square cross-section = 48.3 nH/m compared to 50 nH/m for a circular wire. [1] R. de Smedt, Addition to DC internal inductance for a conductor of rectangular cross-section, IEEE Trans. Electromag. Compat., vol. 51, pp. 875-876, 2009

Internal impedance of a strip conductor

The internal impedance of a wire of an arbitrary cross-section may be determined numerically [2]. For regular shapes such as rectangular and circular cross-sections it leads to a closed-form expression. For a rectangular wire of width w and thickness t, is given by (for ) [2]

In the method proposed, diffusion equation for the electric field

is solved subject to Dirichlet boundary condition of the impressed value of on the boundary. The above boundary value problem may be solved using the Greens function defined as

with the boundary condition = 0 at the periphery of the conductor. The Greens function is found as

Using this Greens function, the electric field inside the wire is calculated as

This yields

The total current through the wire is then given by

The internal impedance is defined as

or

[2] A. Rong, and A.C. Cangellaris, Note on the definition and calculation of the per-unit-length internal impedance of a uniform conducting wire, IEEE Trans. Electromag. Compat., vol. 49, pp. 677-681, 2007

For W = t = 50 micron, , m , n = 100

f = ( 1e5, 1e6, 1e7, 3e7, 7e7, 1e8, 3e8, 7e8, 1e9)

Ri = (6.9246, 6.9288, 7.3240, 9.4983, 13.3793, 15.4762, 24.9551, 36.8477, 43.5713)

Xi = (0.0280, 0.2797, 2.7070, 6.7006, 10.8450, 13.0581, 23.1442, 35.9601, 43.3502)

Sheet or Surface Resistivity,

For convenience, a conductor sheet is characterized in terms of its sheet resistivity, To define , we simply divide the conductor surface in squares normal to the current direction. The ac resistance of a sheet conductor of size LxW (for metal thickness) is given by

(1.1)where , ohm/square

the resistance of a conductor sheet of unit area, is called surface or sheet resistivity. It may be noted that depends on frequency and conductivity of metal; it is independent of the unit of length chosen, micron or meter. According to (1.1), sheet resistance is then times the number of divisions into which the sheet length can be divided.

The internal inductance of a conductor of unit area is similarly obtained as

H/square

The above expression implies that a conductor plate of unit cross-sectional area and thickness has the same internal inductance as an infinitely thick conductor. This model is consistent with the incremental inductance rule of Wheeler [] which is employed for the determination of skin-effect losses of conductors by using its external inductance.

External inductance of metals

At dc and low frequencies, the current inside the conductors with finite conductivity is non-zero. The magnetic flux internal to the conductor gives rise to internal inductance. As the frequency increases, the current migrates towards the surfaces; the inductance due to the magnetic flux internal to the conductor will decrease as eventually going to zero as . The total inductance L is the sum of the frequency dependent internal inductance and the external inductance i.e.. The external inductance is associated with the magnetic flux external to the conductors. The external inductance is sometimes called loop inductance and is almost constant with frequency. Therefore, total inductance is a weak function of frequency. At GHz frequencies, the internal inductance is almost zero, and .

Incremental inductance rule of Wheeler [1]

According to the incremental inductance rule of Wheeler [1], internal inductance of a conductor can be determined from its external inductance . is obtained as the incremental increase in caused by an incremental recession of metallic walls due to the skin effect. The recession of conducting walls is shown in Fig. 1 for microstrip line. The amount of recession is equal to

According to Wheeler [1],

where m is the mth surface of metal exposed to electromagnetic field, and is the permeability of the mth surface. denotes the derivative of with respect to incremental recession of wall m, and is the normal direction to this wall. The skin resistance for the conductor (t ) is therefore given bywhere is the sheet/surface resistance of wall m. The skin resistance for the line may be converted into conductor loss as follows:

neper/unit lengthThe line inductance can be expressed in terms of characteristic impedance with substrate replaced by air, and is given bywhere c is the velocity of electromagnetic waves in free space.

Fig. 1: Recession of conducting walls of a microstrip line for conductor loss calculation

[1 ] Wheeler, H.A., Formulas for the skin effect, Proc. IRE, vol. 30, pp. 412-424, 1942.

[2] Garg, R., I. Bahl, and M. Bozzi, Microstrip Lines and Slot Lines, Third Edition, Artech House, 2013.

Various recessions ( ) considered in microstrip line are as follows:

After taking into account the recessions in all the conductor walls, the expression for the attenuation constant due to ohmic losses may be written as

dB/ unit length

Closed form expressions for the characteristic impedance of various types of printed lines are available [2]. Extensive data for conductor loss of printed lines is also available there.

For a microstrip line, one may use the following expression

where

For a microstrip line shown below the resistance per unit length of the line varies from 2.5 ohm/m at 10 MHz to 150 ohm/m at 10 GHz. The inductance per unit length varies from 2.65nH/m at 1 MHz to 2nH/m at 1 GHz. At lower frequencies, the resistance and inductance are constant. The skin depth at 10MHz is about 21 micron, which is about the metal thickness in this case.

Line resistance (ohm/m) and Line inductance (nH/m) for the microstrip shown above.

Ref: R. Achar and M.S. Nakhla, Simulation of high-speed interconnects, Proc. IEEE, vol. 89, pp. 693-728, 2001.

Model for line resistance: Many a design engineers are comfortable with models so that the effect of ground size, shape and its conductivity can be accounted for. A simple model for the resistance of a microstrip line describes the effect of strip and ground plane separately as shown below.

In this model, the total skin resistance of microstrip line is split between the strip (trace) and ground plane as . We use the skin effect approximation to determine the resistance contributed by the strip and ground planes. For the strip,

For the ground plane resistance, we model it as

where is the effective width of ground plane that carries the return current. To determine we model the ground current density as described next.

Due to the proximity of strip, the current density on the ground plane is more in the vicinity of the strip and decreases gradually with distance away from the strip. (Ref: Advanced Signal integrity for high speed digital designs, by S. H. Hall and H.L. Heck, p.210)

Approximate current density distribution along the width of ground plane of microstrip line may be written as

where is the amplitude of current. The current density on the ground plane is maximum at y = 0 and decreases away from this plane. First we normalize the current density such that the total return current in the ground plane becomes unity for . For this, we integrate the current density such that

, or

or

Therefore,

Next, we determine contributing to the resistance of microstrip line. Let us assume it arbitrarily as from the centre of the ground plane. To justify this assumption we compute the fraction of current included in this width. For this we integrate J over

Therefore, about 80% of the current is included within of the signal conductor centre. Writing , we obtain

Therefore, the resistance of a microstrip line of length l is given by

A more accurate expression for the ac resistance of a microstrip line is given by Collins, based on conformal mapping (Hall, p.211),

,where

and

A very good description of the concepts for high frequency electromagnetics is provided in Sec. 11.1 Changing the way we think about electrical phenomena Introduction to Electromagnetic Compatibility, Second edition by C.R. Paul.

Physical explanation for skin and proximity effectsThe magnetic field and current are related to each other; increase in current increases the magnetic field. Similarly, the increase in magnetic field should be associated with the increase in current.

The skin effect or current crowding in conductors at very high frequencies may be described in terms of magnetic field distribution. We surmise that current density in conductors is directly linked to the magnetic field at that point (Ref.: Lee, p.118). A strong magnetic field at a point leads to a higher current density there. With this important insight let us look at the magnetic field strength in a conducting rectangular strip.

In a strip, the current density in its cross-section is expected to be uniform at very low frequencies. As frequency increases, the current withdraws from the inner portion and concentrates towards the surface. To explain this phenomenon, let us divide the conductor cross-section into 4 equal parts and assume uniform current in each of these 4 parts as shown below.

The magnetic field circulation about these currents is also plotted there. The superposition of fields gives rise to more magnetic flux at the periphery and weaker flux at the central portion. The magnetic flux can be converted into current density J by using J = n x H giving rise to higher current density at the edges and less at the centre. The above concept of mutual inductive coupling may be used to explain the proximity effect in coplanar strips. Consider coplanar strips excited in common mode as shown below.

The superposition of field should give rise to reduced flux in the middle portion and increased flux at the outer edges. Therefore, the current density, J = n x H, should increase at the outer portions as shown. Similarly, the current distribution for differential mode of excitation will have increased current density near the coupled edges as shown.

The increased current density along the inner edges results in increased charge density and therefore stronger electric field across the inner edges.

Network model for skin effect and proximity effect in conductors (Ref.: Digital Signal Integrity by B. Young, Ch.7)

In order to explain the influence of inductive coupling between current filaments leading to crowding of current, let us consider a rectangular metal strip and divide the strip along its thickness in three equal parts as shown below.

Applying 1volt across the length of the strip, the network representation for the circuit is drawn next.

The current I flowing through the original strip can be decomposed into three filamentary currents flowing in the same direction as shown below. The reduction in net flux in the space between the outer strips should give rise to decrease in current leading to skin effect.The circuit equations can be expressed in matrix form as[V] = [Z][I]The parallel connection forces all voltages to be equal to 1, i.e.

Here, R, L, are dc values. Due to the symmetry of the circuit we set . Solution of the simultaneous equations gives and , and we compute . The current crowding can be determined by examining the ratio of branch currents

This expression shows that the current ratio increases monotonically from a dc value of 1 to the asymptotic limit 1+ a/b. It means that the current in a wire is uniformly distributed at dc and crowds towards outer side as the frequency increases. It has been assumed here that >0 and > 0 which is normally true since .

The current crowding does not occur () if or i.e no mutual coupling. The current crowding due to skin effect is symmetrical about the axis.

With this sub-sectioning skin effect can only develop along the vertical direction. To enable the skin effect to develop along the horizontal direction also the strip should be discretized along the width also.

Effect of current crowding on wire resistance and internal inductance

Redistribution of current in the wire due to skin effect causes the resistance to increase and inductance to decrease. Since V = 1, the equivalent impedance for the original strip is , the equivalent resistance and equivalent inductance . The value of can be determined once R, L, are available. The equivalent resistance is given by

The variation of equivalent resistance with frequency is similar to that of current. At dc the equivalent resistance is R/3 as expected. The resistance increases with frequency monotonically such that .

The equivalent inductance is obtained as

The equivalent inductance of the wire decreases with the increase in frequency. However, the dc inductance is different from the value L/3.

Proximity Effect in ConductorsThe current crowding associated with proximity of conductors can also be explained from the discretization of current and mutual inductive coupling. For this, consider an asymmetric lay out of three strips as shown below.

Here strip number (1) represents the signal line and the return is formed by strips (2) and (3). The sub-division of the lower strip enables us to determine the proximity effect between strips (2) and (3). The current loop for the above circuit can be drawn as shown.The increase in flux between currents I1 and I2 and the decrease in flux between I2 and I3 should lead to increase in current density underneath the signal strip.

Assuming that the three strips are identical, the network representation for the circuit is shown below:

The circuit constraints are:

The voltage drop across the strips are given by

The current crowding on the lower strips due to proximity effect can be estimated from the ratio which can be determined from the last two equations i.e.

or

It is expected that due to the proximity of strips, . At dc, the currents and are equal. As the frequency increases proximity effect gives rise to i.e, current crowding under the signal conductor.

The loop resistance and loop inductance are found as before

It is observed that the equivalent resistance increases and the loop inductance decreases with frequency under the influence of mutual inductive coupling.

Finer discretization of conductors is required for better accuracy at higher frequencies and the discretization size should be of the order of at the desired frequency (Paul, Loop and Partial Inductances).Exercise: Work out the problem when the strips are excited in common mode as shown below.

Effect of surface roughness on conductor resistance

In printed realization of microwave circuits, the substrate surface which is metalized is not flat as shown in Fig. below. It is marked by random undulations in the form of peaks and valleys and the surface roughness is characterized by its rms value, . For an ideally smooth surface, The substrate roughness is transferred to the conductor surface during the metallization process.

Cross-sectional view of a rough metalized surface

The current flowing through a rough conductor surface experiences increased surface area. The consequent increased ac resistance of a conductor may be accounted for through a correction factor (Hammerstadt and Jensen, Accurate models for microstrip computer aided design, Int. Symposium on Microwave Theory and Techniques, pp. 407-409, June 1980),

ohmwhere

,

here is the rms value of the surface roughness height.

Need for distributed parameter approachA lumped element circuit consists of inductor, capacitor, diodes, transistor, etc connected through wire or transmission lines called interconnect. The bases of low frequency circuit design are the Kirchhoffs voltage and current laws, and ignore the spatial variation of voltage. The voltage variation on the interconnect may be described as

(1)

where is the maximum voltage, is the radian frequency and is the phase constant of the interconnect. At low frequencies, where the length of interconnect is negligible compared to the wavelength, the associated voltage variation given by (1) is negligible and the circuit behaves according to the design. However, as the frequency of operation is increased the length of the interconnect begins to tell on the circuit performance. The deteriorating effect of the interconnect on the circuit performance can be reduced by eliminating them as far as possible. To understand the effect of interconnect with increasing frequency, let us study the following experiment.

The experiment consists of a sinusoidal voltage source with internal resistance connected to a lumped load by a length of wire as shown in Fig. 1. The wire or interconnect length may be assumed lossless. Let this circuit be realized on a printed circuit board (PCB) with the interconnection wire in the form of a narrow strip of metallization. The strip and the ground metallization of PCB may be modeled as a transmission line of characteristic impedance . Alternatively, if the hook-up wires are used to realize the circuit, the combination of interconnection wire and ground wires may again be modeled as parallel wire transmission line of characteristic impedance . The circuit may now be analyzed according to the transmission line theory.

Fig. 1: A transmission line fed by a source at one end and terminated in a load at the other end.

For, it is found from the analysis of the circuit of Fig. 1 that the voltage across the load, , that is, interconnect affects the phase only and not amplitude. For any other combination of the voltage across the load will vary with frequency. The circuit may be analyzed as follows:

From the transmission line theory, the input impedance seen by the source is given by

(2)and the voltage and current at the source end of the line is given by

(3)The voltage and current can be translated to the load side of the line by using ABCD-parameters of line as

(4)It yields

(5)

This expression yields for as expected. For any other combination of these parameters, the load voltage is different from this value in both magnitude and phase. However, at low frequencies and and starts varying as the frequency is increased.

The effect of interconnect on the circuit performance can be included if the line is divided into a number of segments of sufficiently small lengths as shown in Fig. 2 so that the voltage or current over each segment can be assumed to be constant. Each line segment of length can be modeled by a series resistance , series inductance , and shunt capacitance . This equivalent circuit will account for voltage and current variation over the segment length . We can build the complete model for the interconnect by combining the effect of all the segments into which the interconnect is divided. An integrated approach is to consider the line as a single segment, characterize the line in terms of distributed parameters R, L, C, and G where these line constants are given in terms of unit length, and the sinusoidal voltage variation given by (1) is built into the analysis.

Lumped element equivalent of transmission line circuit(a) for sinusoidal signal propagationThe transmission line transformation

is the key equation in the transformation of load impedance by a transmission line section of length l. In this expression, the phase shift becomes large when the length l is comparable with the wavelength in the medium. The equivalent circuit for such situations can be determined more accurately, to capture the time delay and frequency variation effects, by using transmission line based models. However, lumped element models are more compatible for spice-like solvers. In order to produce transmission line effects with lumped elements we express the equivalent circuit as a cascade connection of lumped element circuits. The elements in each of these lumped circuits are synthesized in such a way that they match the frequency dependence of the elements in the p.u.l. R,L,C,G matrices. The accuracy of lumped circuit approach depends on the number of sections per wavelength. It is found that the number of sections per wavelength should be about 10 at the maximum frequency of interest. We show next the results of a simple exercise for comparison. Consider a lossless transmission line of characteristic impedance and length. It is driven by a source of input resistance and terminated at the other end by a load as shown below

Fig.2: Lumped element equivalent circuit of a lossless transmission line section

Comparison of the analytic result for the voltage at the load with that obtained from the approximate distributed lumped circuit models. Ref.: A.E. Ruehli and A.C. Cangellaris, Progress in methodologies for the electrical modeling of interconnects and electronic packages, Proc. IEEE, vol. 89, pp. 740-771, 2001. fig. 7.

For a 1-V source, the voltage at the load is given by , where l is the physical length of the line and the line is assumed to be lossless. This analytic solution is shown by the solid line. Let us assume for the simulation that L = 1/6 nH/cm, C = 1/15 pF/cm so that and . From the graph we observe that at 500 MHz; implies cm. The maximum frequency is 10 GHz which corresponds to a minimum wavelength of 3cm. The line segment lengths are thereforecm (for n = 2), mm (for n = 5), and mm (for n = 10). As expected, the deviation of the load voltage from the expected value starts occurring at lower frequency for the coarse discretization of . The magnitude of load voltage starts deviating from the expected value for f > 3.5 GHz whereas phase starts deviating at about 500 MHz for . The corresponding frequencies for are 5.5 GHz and 3.5 GHz. The minimum number of sections to realize a phase shift between the output and input voltages is therefore given by

.A similar criterion can be developed for the digital signal application.One may try sections of unequal lengths to realize an overall phase shift with a given phase error. The number of sections required may be less.

(b) Pulse propagation (Ref.: R. Achar and M.S. Nakhla, Simulation of high-speed interconnects, Proc. IEEE, vol. 89, pp. 693-728, 2001)

Since a digital signal is described by the pulse rise/fall time and pulse width, the section length should be related to one of these characteristics of the pulse. The principal criterion in this case can be derived in a manner similar to that for the frequency domain approach, i.e. the pulse shape should be preserved when fed to the equivalent lumped circuit. Also, the time delay should be the same as that produced by the transmission line circuit. Combining the two requirements, we therefore need to determine the value of time delay produced by each section such that the pulse shape is preserved. For illustration, consider a ladder type LC realization of transmission line section as shown. This circuit can be viewed as a low-pass filter.

In order to preserve the pulse shape the 3-dB cut-off frequency of this filter should be very high, ideally infinite. However, in practice we may fix the cut-off frequency as 10 times the useful highest frequency component of the pulse, i.e. . Also,

where is the length of the segment and is the delay p.u.l. For a trapezoidal pulse with rise time , the useful highest frequency component is given by

Therefore,

or

or

In other words, the delay allowed per segment is approximately . It is assumed in the above analysis that all the frequency components constituting the pulse have the same phase velocity or suffer the same time delay. The total number of segments N needed to accurately represent a total delay of and preserve the pulse shape is given by

where is the pul delay. For a lossy line case, RLCG segments have to be used to determine N.

Example: Consider a digital signal with rise time of 0.2ns propagating on a lossless wire of length 10 cm, with a delay of 70.7ps/cm. The line can be modeled with L = 5nH/cm and C = 1pF/cm. The wave velocity is and the time delay pul . If this transmission line is to be represented by lumped element segments, one needs

. Otherwise, pulse shape may not be preserved.

Broadband modeling of transmission line parametersA printed transmission line consists of a printed strip and a printed ground plane as in microstrip line, strip line, CPW, and two strips as in a CPS line. The lumped equivalent circuit for a small section of this line may be expressed as

where is the series impedance per unit length and is the shunt admittance per unit length. The series impedance is a strong function of frequency whereas shunt admittance varies slowly with frequency, of the order of 1%. We shall model the line parameters over a broad frequency range.

Resistance per unit length,

The series resistance is contributed by both the strip and the ground plane and may be written as . The variation of resistance of an isolated strip with frequency is plotted in Fig. 1 and may be modeled as

Fig.1: Variation of strip resistance and internal inductance with frequency

For a strip of width W and thickness t, the values of and are given by

The break frequency is the frequency at which the asymptotes and meet and is defined by

,

for and t >> or

at or

The skin resistance is defined as

For a strip with aspect ratio W/t < 10, the expression for gets modified as

The value of k depends on the aspect ratio W/t and is about 1.5 for W/t = 1.

The ground resistance is normally smaller than strip resistance because the width of ground plane is very large compared to the strip and the associated current density has smaller magnitude. For a transmission line, both and depend on their proximity among other factors. The series resistance for the transmission line may be determined in an integrated fashion using any of the transmission line analysis techniques such as conformal mapping or full wave technique. Since increases as at higher frequencies, the series resistance attenuates high frequency components of the digital pulse heavily.

Inductance per unit length,

The inductance l may be divided as . The internal inductance in a strip is due to the current confined within the cross-section of strip. The variation of internal inductance of an isolated strip with frequency is plotted in Fig. 1 and may be modeled as

where is obtained from the expression

The external inductance is the loop inductance due to the magnetic field between the strip and ground plane/ another strip. In general, internal inductance is very small compared to the loop or external inductance and may be neglected at microwave frequencies. For a transmission line, both and depend on their proximity among other factors. The external inductance for the transmission line may be determined using any of the transmission line analysis techniques.

Capacitance per unit length,

The capacitance pul is determined by the transmission line geometry and the dielectric constant of the substrate, . The variation of c scales with frequency in the following manner

Conductance per unit length, The conductance and capacitance pul are related by

In general, the factor is independent of frequency although both c and vary with frequency. Therefore, we may write

Therefore, increases linearly with frequency.

Example: Broadband model of strip line parameters

Consider a strip transmission line with ground plane separation of 20mil and strip dimension . The pul parameters of the line are c = 94pF/m, and .

(a) Determine and velocity of propagation for the line.

(b) To determine line losses due to imperfect conductor and lossy dielectric, the conductivity of metal is assumed as and the substrate permittivity is modeled as

, and Determine the pul r and g as a function of frequency from 10 KHz to 1 GHz.

(c) Compute newand velocity of propagation at these frequencies.

(d) Determine of the line over this frequency range and the magnitude of transfer function of 0.5m long transmission line that is perfectly matched at the input and output.

(e) A 5V, 100MHz, 1 ns rise/fall time, 50% duty cycle periodic trapezoidal pulse train is applied to this line. Comment on the output waveform shape.

For the given stripline c = 94pF/m, and . The specified value of is actually external or mutual inductance pul; it is very large compared to .

(a) At very high frequencies at which the skin effect is established fully

,We shall determine later the frequency at which these approximations can be made.(b) to determine r, l

Break frequency = 23MHz for k = 2

Therefore, one may assume for f < 23MHz and for f>23MHz

Since , 26.8nH/m. The internal inductance is maximum at dc and is about 6% of the external inductance value of . Also,

freqr (ohm/m)

1KHz3.8733.06e-3

10KHz3.8733.06e-2

100KHz3.8730.306

1MHz3.8733.06

10MHz3.87330.6

23MHz3.87370.38

100MHz8.05298

1GHz25.452896

The approximation is valid for f > 10 MHz.

To determine g,

The given values of c and g are valid at very high frequencies where the substrate permittivity reaches the asymptotic value. At low frequencies, c and g should be corrected for frequency variation of and is given by

and

with c = 94pf/m.

Based on the above models the computed values of c and g are tabulated next.

Freq

100Hz4.186.33e-84.21e-10

1KHz4.146.27e-74.2e-9

10KHz4.16.06e-64.26e-8

100KHz4.066e-54.2e-7

1MHZ4.025.9e-44.2e-6

10MHz3.9815.88e-34.2e-5

100MHz3.9405.82e-24.2e-4

1GHz3.8990.5764.2e-3

It may be observed from the Table that at all frequencies.

(c,d) To determine as a function of frequency

,

In the low frequency region, the lines are dominated by conductor loss, Also, and therefore

, ,

In the high frequency region, defined by

, and , freqr ,ohm/m,ohm/mg,S/m,S/m,ohmv,m/s,nep/m

1KHz3.8733.06e-34.2e-95.92e-725575.87e61.07e-3

10KHz3.8733.06e-24.2e-85.92e-68081.856e73.38e-3

100KHz3.8730.3064.2e-75.92e-52565.87e71.07e-2

1MHz3.8733.064.2e-65.92e-490.41.468e8-

10MHz3.87330.64.2e-55.92e-3701.519e82.9e-2

23MHz3.87370.389.66e-51.36e-2701.519e8-

100MHz8.052984.2e-45.92e-2701.519e87.2e-2

1GHz25.4528964.2e-30.592701.519e80.329

The transfer function for the transmission line is given by

or

where is the voltage input to the line (not source voltage) and is the load voltage. For l = 0.5m,

freq,nep/m

l =0.5v,m/s

1KHz1.07e-315.87e6

10KHz3.38e-30.9991.856e7

100KHz1.07e-20.9965.87e7

1MHz--1.468e8

10MHz2.9e-20.9861.519e8

100MHz7.2e-20.9651.519e8

1GHz0.3290.8481.519e8

(e)

For a pulse with rise/fall times of 1ns, the bandwidth of the signal is . Since, the prf , upto 10th harmonic is sufficient to be included for frequency domain analysis of pulse propagation through the line. Fourier series analysis of the pulse shows the amplitude of various frequency components as follows:

The 9th harmonic amplitude is about 1/10th of the principal and can be neglected. The transmission line of length l = 0.05m will offer almost the same attenuation to all the components from 100MHz to 1 GHz. Also, phase velocity is also same for these frequencies. Therefore, the pulse shape remains almost unaffected for l = 0.05m. The transfer function is 0.983 at 1 GHz.

For l = 0.5m, the attenuation is somewhat more. However, the difference in attenuation from first to 9th harmonic is not much, 0.965 to 0.848. Again the pulse shape may not change much.

Physical explanation for skin and proximity effectsThe magnetic field and current are related to each other; increase in current increases the magnetic field. Similarly, the increase in magnetic field should be associated with the increase in current.

The magnetic field distribution about the common mode currents in the strips is sketched below.

The superposition of fields should give rise to reduced flux in the middle portion and increased flux at the outer ends. Therefore, the current density should increase at the outer portions as shown. Similarly, the current directions for differential mode of excitation will increase flux in the middle portion and reduced flux at the outer ends. The current density therefore should increase near the coupled edges as shown.

The increased current density along the inner edges results in increased charge density and therefore stronger electric field across the inner edges.

Scope of the course-workTo develop equivalent circuit representation for the given 2-port circuits or planar antennas. The methodology should be general so that it can be applied to (i) complex, multi-layered geometries consisting of various types of interconnects such as planar lines, planar circuits, vias, etc. (ii) Also, the equivalent circuit should be compact because the number of lines, vias, etc may be very large in number. (iii) The bandwidth should be such that the circuit is useful for digital applications. (iv) The equivalent circuit should be able to reproduce time-domain and frequency-domain results.(v) Should be applicable to lossy circuits.

Approach: The circuits to be analyzed for their equivalent circuit representation consist of sections of transmission lines of various types including vias, planar circuit or antenna geometries, and metalizations of various shapes the infrastructure required for such a job should include (i) wave propagation aspects in transmission lines, (ii) in frequency and time domain, (iii) in lossy medium. The planar circuit or antenna geometries can be modeled as 2-dimensional transmission line resonator configurations. Vias are modeled as zero-dimensional circuits because of their small size. The metallization of various shape, which occurs frequently in ultra-wideband antennas, may be modeled as planar resonators.

Approach 1: Analytical based: Therefore, we need equivalent circuit model for transmission line sections, planar geometries, vias, discontinuities and junctions between them. One may use PEEEC, etc. Use model-order reduction to realize a compact equivalent circuit.

Approach 2: Numerical based: Use a suitable computational technique to numerically model the geometry in the form of transfer function H(f) or impulse response h(t). Model H(f) or h(t) in terms of equivalent circuit.

Complex and determination for the lossy transmission line case. Ref.: Signal integrity book by B. Young, pp. 62-70.

Impedance Boundaries, Sec. 2.1.4 of Signal integrity book by B. Young.

In this article, the author describes the reason for generation of new field components at discontinuities (to match the tangential field components at the discontinuities); specifically the discontinuity.

Suppose two transmission line sections with different values are cascaded as shown

At the junction plane, the voltage and the current must be continuous. The continuity of voltage implies

(1)and the continuity of current means

(2)These expressions imply that the forward and reverse traveling waves, which otherwise were traveling independently, are mixed at the boundary. Also, the waveforms or field components are generated as necessary to fulfill the boundary conditions. We shall verify this next.

Solution of (1) and (2) gives

(3)

(4)

For example, if only the incident wave from the left exists. Then, since is zero, a reflected wave must be generated according to (4) as

(5)The reflection coefficient is therefore

(6)

Substituting for in (3) gives the transmitted wave as

or the transmission coefficient T as

It may be noted here that the presence of discontinuity has called for (i) generation of field components corresponding to the reflected wave and (ii) the incident and reflected waves get coupled at the discontinuity giving rise to the formation of standing wave.

Pulse propagation over a lossy transmission lineThe voltage and current on a lossless line are governed, in time domain, by

or

and The solutions to these second-order differential equations are of the form:

and

where

and

The finite conductivity of conductors and non-zero conductivity of dielectric contribute to line losses. The line losses may be accounted for by writing the transmission line differential equations for time harmonic signal as

Fourier transform of these equations yield

where * denoted convolution and is defined as

,

and r(t) and g(t) are inverse transforms of and , respectively. The presence of time domain parameters r(t) and g(t) and convolution operations present significant computational problems. The implementation of convolution operation requires storage of all the previous values of r(t), g(t), I(t) and V(t), One of the simplest methods to take care of this problem is the use of time-domain to frequencydomain (TDFD) transformation method, Sec. 8.1.7 of C.R. Paul (Analysis of Multiconductor Transmission Lines), p.375.

This method is intuitive, easy to implement computationally and general because it can be applied to any linear circuit. In this method, the transmission line or transmission line based circuit is treated as a single-input, single-output linear system as shown in the figure. The input to the system is denoted as x(t), the output is denoted as y(t), and the impulse response is denoted as h(t). The response in time domain is given by y(t) = h(t)*x(t)The frequency domain response is

where is the transform of y(t) and is the transfer function of the two-port system as shown.

The transfer function can be easily obtained by applying the unit magnitude sinusoid of appropriate frequency at the input and computing the response.

Spectra and bandwidth of digital waveforms (p.52)

For applying the TD-FD (time-domain frequency-domain) method, the input pulse x(t) is decomposed into various sinusoids; Fourier series is employed if x(t) is a periodic pulse train and Fourier transform if it is a single pulse. Let us consider a periodic train of pulses with period P; its discrete spectrum consists of frequencies where i.e.

(1)

where the Fourier series is truncated after N harmonics. For a periodic pulse train of trapezoidal pulses with peak amplitude A, duty cycle , where is the pulse width between 50% points, and 0-100% equal rise/fall times of , we obtain

To determine the coefficients of (1) we multiply both sides by and integrate over a period

or

or

or

,

orwhere For the trapezoidal pulse as shown above

Therefore,

(see also 1.126 of Paul)or

andSimilarly,

(see also (8.61) of Paul)

With the decomposition of pulse train in harmonics of prf we now determine the transfer function at these harmonics as:

Next, multiply each of the transfer functions with Fourier components to obtain the frequency domain output as

with

andor

andwith

duty cycle =

The inverse transform of produces the time domain output y(t). If the transfer function includes the skin-effect losses, the output time pulse includes this effect also.

Non-periodic pulse case: The periodic pulse train analysis can be applied to the single pulse case also provided the pulse shape is identical with the periodic pulse over a period, and choose the repetition frequency low enough so that the pulse response reaches its steady state value before the onset of next pulse. For example, a ramp waveform with a rise time of 50ps may be simulated by a 10MHz periodic trapezoidal waveform with 50% duty cycle and rise/fall times of 50ps. The trapezoidal waveform would have behaved like a ramp over its rising half portion.

Limitations of the method: Because the assumption of linearity about the transmission line and loads, this method cannot be applied to non-linear situations such as corona breakdown in transmission lines as well as non-linear loads such as diodes and transistors.

Homework # 3Due on 31.1.20111. Example on time-domain analysis of high loss line. Consider the following coplanar strips line; it is a printed version of parallel wire line. Two metal strips run parallel to each other on a substrate as shown below.

The total length of the line is 20cm and is terminated with 50 ohm resistors at both the ends. The source is a ramp function rising to a level of 1volt with rise time of 50ps. The source may be modeled as a 10 MHz periodic trapezoidal waveform with 50% duty cycle D and rise/fall times of 50ps. The pul values of the line are , and dc resistance . The effective dielectric constant of the line is and . The break frequency or corner frequency at which transition to skin-effect resistance value takes place is given by and is found to be 393.06MHz. Use TD-FD method to compute and plot the signal voltage as a function of time at the source and load nodes. The range of t for the plot may be taken as 0 to 10ns. Ref.: Paul, p.439.

For the periodic trapezoidal waveform shown below one may employ the following Fourier series expansion

where

Pulse shaping by a filter (square wave fed to a LPF)Example: Square wave pulse shaped by a LPF (ref.: Hall, p. 324)

The example on pulse shaping has a number of useful features. (i) Square wave is an ideal whereas trapezoidal waveform is a realistic pulse which can be generated in the lab and utilized for characterizing the channel. (ii) It shows that a trapezoidal pulse can be realized from a square wave by passing it through a RC filter. (iii) The rise/fall time of which can be controlled by the time constant RC of the filter, . The frequency spectrum of trapezoidal pulse is related to the rise time of the pulse and is given by . (iv) By virtue of low-pass filtering realization, the trapezoidal pulse has narrow spectrum compared to square wave. Consider an ideal, square pulse of width 2ns applied to a low-pass RC filter with R = 50 ohm and C = 5pF. Assume the amplitude of square wave to be 1V.

Solution: The expected output is a pulse with slow rise/fall time compared to zero rise/ fall time for the square wave. The rise/ fall time of the output pulse is determined by the time constant of the RC filter, RC = 250ps. The pulse rise is described by . Similarly, the pulse fall is . In general,

In the frequency domain,

;where

Fig.: Spectrum of 2ns wide square wave

The filter transfer function:

Filter output in frequency domain:

Filter output in time domain

Inverse FFT gives .

Example 8.2 (ref.: Hall, p.328)

Calculate the pulse response of the 0.5m microstrip line matched at the source and load ends. Assume that the input pulse has rise and fall times of 33ps and a magnitude of 1V. The line parameters are: and .Solution:The input pulse train is given as

To determine the output in time domain we apply TD-FD method.

To determine the effect of losses on pulse propagation we characterize lossy microstrip line from 1GHz to 5 GHz. The frequency range may be changed depending upon the useful frequency content of the pulse.

,

,

R = 8.2502(1GHz) 11.6676(2GHz) 14.2898(3GHz) 16.5005(4GHz) 18.4481(5GHz) ohm/m L(total) = 0.2513(1GHz) 0.2509(2GHz) 0.2508(3GHz) 0.2507(4GHz) 0.2506(5GHz) microH/m

Frequency dependence of dielectric constant

The dielectric constant ()of a material varies slightly with frequency over a broadband. The real part may vary decrease from 3.9 at 1 GHz to 3.86 at 10GHz, while may increase from 0.0073 at 1 GHz to 0.0074 at 10 GHz. Like the resistance and reactance of a conductor, the real and imaginary parts of permittivity are related to each other. The Kramers-Kronig relation in this connection states that

,

The line capacitance C and conductance G become a function of frequency because of the frequency variation of .

The real and imaginary part of permittivity may be modeled over a broad frequency range of and by the following expressions

,

where is the variation in over the bandwidth . The main advantage of this model is that it requires specifying the values of and at one frequency point only.

For the microstrip line, the variation of with frequency occurs due to two factors; because of the decrease in with increase in frequency and because of the increase in due to dispersion. The dispersive effect is more pronounced for microstrip line e.g. for alumina substrate increases from 6.25 at 1 GHz to 6.76 at 10 GHz. For the given example we consider the contribution of the first effect only. Also, we assign the frequency variation of to . At 1 GHz,

,

We next solve the above equations simultaneously to get

Assuming we obtain . At 1 GHz, . Using these values in the above expressions give

The frequency dependence of line capacitance C may be modeled linearly as

C = 0.1500(1GHz) 0.1486(2GHz) 0.1478(3GHz) 0.1473(4GHz) 0.1468(5GHz) nF/m

The frequency dependence of conductance G is related to C as

where

G = 0.0193 0.0383 0.0571 0.0759 0.0946 mho/m

The propagation constant of the line is given by

For low-loss line we may use Taylor series expansion as

Therefore,

alpha = 0.4962 0.9286 1.3499 1.7652 2.1766 nep/m

beta = 38.5764 76.7444 114.7701 152.7043 190.5709 rad/mphase velocity , v/c: 0.543, almost independent of freq

, Length, l = 0.5m

= 0.3901 0.3143 0.2546 0.2069 0.1684

Transfer function for the transmission line of length l

, l = 0.5m

The transfer function is computed at 0,1,2,3,4,5GHzH(omega) : 1.00-0i0.7064 - 0.3314i 0.4915 - 0.3919i 0.3412 - 0.3779i 0.2393 - 0.3375i 0.1712 - 0.29i

The ordinate should be multiplied by 2. Also, since the above plot shows the envelope only, the sinusoidal variation with frequency can be obtained if a number of data points in between are computed and plotted.

Impulse response may be computed from as . For this one may sample the positive and negative frequency response, 2000 times in steps of 100 MHz, to achieve a time-resolution of

The impulse response is plotted next

The pulse response can be computed using the TD-FD method as

where

The pulse response is given by

and is plotted above for 0.5m lossy microstrip line. Observe that the output is stable , does not tend to infinity for any value of t, and appears after a delay of about 3.06ns. The transmission line therefore is causal.

Time delay, at 1 GHz and 3.03ns at 5 GHz.

Having modeled a lossy line in time domain, we now discuss the mathematical requirements on passive networks so that their time domain model is physically consistent.

Requirements for a Physical Channel (Hall, p.331)A physical channel such as interconnect (includes transmission line, via, connectors), passive networks, etc must obey certain laws to mimic the real world behaviour successfully. Specifically, it should satisfy the conditions of causality, passivity, and stability. We state these requirements for linear and time invariant electrical networks. The importance of laying down the conditions is that the models may be tested against these conditions.

CausalityThe fundamental principle that effect cannot precede the cause is called causality. Mathematically, a linear time invariant system is causal only if its impulse response h(t) obeysh(t) = 0 for t < 0

If a system, such as a transmission line, has known delay , the system is causal only if

h(t) = 0 for t <

Since and h(t) are related through Fourier transform, we may be able to translate the constraints on h(t) in time domain to in the frequency domain.

Any causal function f(t) is supposed to be zero for t < 0. However, for translating the requirement on f(t) in terms of we may express f(t) in terms of even and odd parts as

The functionsmay not satisfy causality individually because they might not be zero for t < 0. However, jointly they could satisfy f(t) = 0 for t < 0, which is possible if

for t < 0 and

for t > 0because odd and even parts of a function are related as such. Therefore, we can express the odd function as

for all t(1)where

We may write

Taking the Fourier transform of both sides

,

The second term is Hilbert transform of . Hilbert transform for g(f) is defined as

We can therefore write

(2)

Since time domain waveforms are always real, f(t) is real; its Fourier transform satisfies the following properties (i) .

(ii) If f(t) is an odd function i.e. f(-t) = -f(t), is imaginary and odd function of .

(iii) If f(t) is an even function i.e. f(-t) = f(t), is real and even function of .

Using these properties,

translates to

(3)Comparing (2) and (3) we observe that for a causal f(t)

i.e imaginary part of can be obtained from the Hilbert transform of its real part.

Applying this property of a real and causal function to impulse response h(t) of linear time invariant system we can write

This equation demonstrates two very important properties of a system that will produce a real, linear, and causal response in the time domain.(a) The imaginary part of the frequency response is determined by the Hilbert transform of the real part i.e. knowledge of the real part is sufficient to define the entire function.

(b) In the derivation, causality of the system has been enforced through condition (1). This condition translates to in the frequency domain. Therefore, causality of the system can be tested by performing the Hilbert transform of the real part and ensuring that it is identical to the negative of the imaginary part of the function.

Similarly, one can derive the real part of transfer function of a causal system from its imaginary part and prove that

The Hilbert transform of even and odd functions can be further simplified. Consider

since is an even function It may be noted that Kramers-Kronig relations for permittivity as given below

,

confirm to the above form of Hilbert transform pairs. If the dielectric models do not satisfy Kramers-Kronig relations, the system is non-causal.

PassivityA physical system is passive when it is unable to generate energy of its own. For example, an n-port network is said to be passive if

where v(t) and i(t) are the port voltage and current vectors, respectively. The above integral represents the net power absorbed by the system upto time t. In a passive system, this quantity should be positive for all values of t.

In terms of S-parameters of a network, the passivity can be defined as

where , the power absorbed by the network. The passivity condition can be expressed in terms of S-matrix of the network as

If , the passivity of the system is not guaranteed. The above condition may further be expressed in terms of the eigenvalues of the S-matrix of the system i.e.

StabilityStability is defined such that y(t) remains bounded for all bounded inputs x(t).Using this definition, the stability of a linear, time invariant system is guaranteed only if all the elements in the impulse response matrix [h(t)] satisfy

------------------------------------------------------------------------------------------------------------Electronics & Electrical Communication Engg DepartmentIndian Institute of Technology

Sub.: Microwave Networks (EC60088) Mid-sem ExamDate: 18.2.2011 Full Marks: 30

Note: Attempt all the questions. Answer to all the parts of a question should be attempted in sequence and at one place only.

1.

An L-C circuit shown below (L = 1/6nH and C = 1/15pF) is used to represent a transmission line section. Determine the equivalent of the line. What is the frequency range over which this section can be employed if the deviation from

and phase difference is not to exceed 5% . 8

2. For a causal waveform, f(t) = 0 for t 23MHz

. since and is about 50nH/m.

freqr (ohm/m)

1KHz3.8732.9e-3

10KHz3.8732.9e-2

100KHz3.8730.29

1MHz3.8732.9

10MHz3.87329

23MHz3.873290

100MHz8.052.9e3

1GHz25.452.9e4

To determine g,

The given value of c is valid at very high frequencies. For low frequencies it should be corrected for frequency variation of and is given by

Therefore,

with c = 94pf/m.

Freq

100Hz4.186.33e-84.21e-10

1KHz4.146.27e-74.2e-9

10KHz4.16.06e-64.26e-8

100KHz4.066e-54.2e-7

1MHZ4.025.9e-44.2e-6

10MHz3.9815.88e-34.2e-5

100MHz3.9405.82e-24.2e-4

1GHz3.8990.5764.2e-3

(c,d) To determine as a function of frequency

,

In the low frequency region, the lines are dominated by conductor loss, Also, and therefore

, ,

In the high frequency region, defined by

, and , freqr ,ohm/m,ohm/mg,S/m,S/m,ohmv,m/s,nep/m

1KHz3.8732.9e-34.2e-95.92e-725575.87e61.07e-3

10KHz3.8732.9e-24.2e-85.92e-68081.856e73.38e-3

100KHz3.8730.294.2e-75.92e-52565.87e71.07e-2

1MHz3.8732.94.2e-65.92e-490.41.468e82.2e-2?

10MHz3.873294.2e-55.92e-3701.519e82.9e-2

23MHz3.873290----701.519e8

100MHz8.052.9e34.2e-45.92e-2701.519e87.2e-2

1GHz25.452.9e44.2e-30.592701.519e80.329

The transfer function for the transmission line is given by

or

where is the voltage input to the line (not source voltage) and is the load voltage. For l = 0.5m,

freq,nep/m

l =0.5v,m/s

1KHz1.07e-315.87e6

10KHz3.38e-30.9991.856e7

100KHz1.07e-20.9965.87e7

1MHz2.2e-2??1.468e8

10MHz2.9e-20.9861.519e8

100MHz7.2e-20.9651.519e8

1GHz0.3290.8481.519e8

(f)

For a pulse with rise/fall times of 1ns, the bandwidth of the signal is . Since, the prf , upto 10th harmonic is sufficient to be included for frequency domain analysis of pulse propagation through the line. Fourier series analysis of the pulse shows the amplitude of various frequency components as follows:

The 9th harmonic amplitude is about 1/10th of the principal and can be neglected. The transmission line of length l = 0.05m will offer almost the same attenuation to all the components from 100MHz to 1 GHz. Also, phase velocity is also same for these frequencies. Therefore, the pulse shape remains almost unaffected for l = 0.05m. The transfer function is 0.983 at 1 GHz.

For l = 0.5m, the attenuation is somewhat more. However, the difference in attenuation from first to 9th harmonic is not much, 0.965 to 0.848. Again the pulse shape may not change much.------------------------------------------------------------------------------------------------------

Method of Characteristics (Ref: Cad of microwave circuits, sec.14.1; Paul, Sec.8.1.3)This is one of the simplest methods and can be used to determine the time domain behaviour of circuits with linear and non-linear loads. The method of characteristics seeks to transform the hyperbolic partial differential equations of transmission line into ordinary differential equations that are easily integrable. The set of partial differential equations describing TEM mode in transmission lines are:

The method of characteristics is used to transform these equations into two ordinary differential equations, each of which holds for a different characteristic direction in the z-t plane. Along these directions, the variables z and t are related such that a change in t is associated with a corresponding change in z. For a uniform line, the first set of characteristic curves is given by

(1)This characteristic is sometimes called forward characteristic. This characteristic equation may be combined with the partial differential equation of transmission lines to yield

for lossless linesThe second characteristic curve is defined as

(2)and the corresponding ordinary differential equation for the lossless lines is obtained as

The above ordinary differential equations are directly integrable. The characteristic curves are drawn next. The increments and for the curves are related by (1) and (2), for the forward and backward curves, respectively. For the lossy lines, the ordinary differential equations are given by

Electronics & Electrical Communication Engg DepartmentIndian Institute of Technology

Class Test IDate: 7.2.2011Time: one hour

Consider an infinitely long metal strip of rectangular cross-section immersed in a uniform TM field. The dimensions of the strip are: and the material is copper with . (i) Determine and plot the resistance r and internal reactance per unit length of strip as the frequency of the TM field is increased from dc to 1GHz. (ii) Determine the break frequencies at which r is likely to undergo change between and .

Solution(i) The dc resistance pul of a PCB land of dimension Wx t is given by

,W, t ,1