Design a 100-ï‰ Microstrip Transmission Line Using the Expressions

Characteristic Impedance

Introduction and terminology

Leo Beranek , Tim Mellow , in Acoustics (Second Edition), 2019

Characteristic impedance (ρ ₀ c)

The characteristic impedance is the ratio of the effective sound pressure at a given point to the effective particle velocity at that point in a free, plane, progressive sound wave. It is equal to the product of the density of the medium times the speed of sound in the medium ( ρ ₀ c). It is analogous to the characteristic impedance of an infinitely long, dissipationless electric transmission line. The unit is N·s/m³ or rayls.

In the solution of problems in this book, we shall assume for air that

${\rho }_{\text{0}}c=407\text{rayls}\text{,}$

which is valid for a temperature of 22°C (71.6°F) and a static pressure of 10⁵  Pa.

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PCB design for signal integrity

Hanqiao Zhang , ... Jeff Ou , in High Speed Digital Design, 2015

Differential Signaling

In order to reduce interference and extend the capabilities of high-speed circuits, many interfaces have adopted a differential signaling architecture over single-ended signaling. Differential signals in PCB design are more resilient against the presence of nearby coupling, allowing for longer lengths and higher speeds in ever denser designs. Differential signaling exchanges noise tolerance for more layout area, using two signal traces. Differential signaling requires strict layout rules that will lead to functional failures if not followed. A sample of such interfaces using differential signaling include USB, Serial ATA, InfiniBand, PCI Express (PCIe), XAUI, and 10-gigabit Ethernet.

Figure 2.1 illustrates the theoretical ideal noise cancellation on a differential signal. Equal and opposite signals are transmitted and absorb a noise source equally during transmission. At the receiver's differentiator, the signals are subtracted, removing the noise and doubling the signal strength. In reality, the noise cancellation is a function of how tightly the differential signals are coupled. A loosely coupled differential pair receives noise differently on each trace due to the different separations from the aggressing signal, decreasing the effectiveness of differential signaling. Maintaining a tightly coupled pair also reduces the electromagnetic interference (EMI) impact from microstrip traces. The opposing magnetic fields cancel, greatly reducing the field energy.

Figure 2.1. Differential signaling concept.

A tightly coupled differential pair moves return current that otherwise flows on the reference plane onto the opposing signal within the pair. The amount of current returned on the opposite signal depends on the geometry of the trace and the buffer design. When less return current is present on the reference plane, the differential signals are less sensitive to voltage noise appearing on the ground plane and non-idealities in the reference plane. In such cases, it may be possible to route over plane splits or voids and low-noise power planes. Layout guidance for differential signals may be found at the end of this chapter.

Impedance

Characteristic impedance is the intrinsic and instantaneous property of the geometry cross section and not a function of the traveling length. Impedance is a design variable used to minimize mismatch between transmission lines, vertical components, and transceiver terminations. Impedance terms can be calculated from voltage and current terms; however, for this application we discuss the calculations used to analyze the characteristics of transmission line models by inductance and capacitance before using them in a simulation environment. For the differential pair in Figure 2.2, self- and mutual inductance and capacitance are identified.

Figure 2.2. Inductance and capacitance terms for a differential pair.

The impedance seen by one trace within a differential pair where signals are driven in the same (common) direction is the even mode impedance. The impedance then seen as a collective differential pair is the common-model impedance, useful for understanding the impedance of noise sources common to both signals with a differential pair. Even and common-mode impedances are defined by equations (2.1) and (2.2):

(2.1) $Z_{even}=\sqrt{\frac{L_{11}+L_{12}}{C_{11}-C_{12}}}=2\times Z_{common}$

(2.2) $Z_{common}=2\times Z_{even}$

For signals driven in opposite directions within a differential pair, the impedance on one trace is described as the odd mode impedance. The differential pair impedance is twice the odd mode and describes the impedance seen by incident signal of the differential pair. Odd and differential mode impedances are defined by equations (2.3) and (2.4):

(2.3) $Z_{odd}=\sqrt{\frac{L_{11}-L_{12}}{C_{11}+C_{12}}}$

(2.4) $Z_{diff}=2\times Z_{odd}$

Inductive and capacitive terms are related to the stack-up and differential pair design and are most accurately extracted from electric field solvers. A good rule of thumb is to extract L and C terms in the frequency range of test equipment used for impedance qualification in the manufacturing process. Generally, equipment edge rates are in the range of 500   MHz to 1   GHz. The frequency choice is not critical, however, as the slope in impedance over frequency is slow. Figure 2.3 displays the differential impedance over frequency for a stripline differential pair at 85   ohms.

Figure 2.3. Differential impedance increase with frequency is negligible.

Trace width and dielectric height have the greatest role in the characteristic impedance. A design of experiment (DOE) is used to demonstrate the sensitivity of stack-up terms on differential impedance. The example shown in Figure 2.4 is a differential stripline with parametric sweeps of trace width (w), trace space (s), core dielectric height (h1), core dielectric constant (er1), prepreg dielectric height (h2), prepreg dielectric constant (er2), and trace thickness (tt). Stronger slopes of trace width and dielectric heights indicate their significant role in impedance, rooted in their influence on capacitance terms. Increases in width, spacing, and height increase the differential impedance. Increases in dielectric constant and trace thickness decrease the differential impedance.

Figure 2.4. Parametric sweep sensitivity for stripline differential impedance (mils).

In order to obtain differential impedance calculations quickly and without complex field solver simulation, equations for microstrip and stripline are offered as approximations for the design phase. Equations for odd mode and differential impedance are offered by Eric Bogatin in Differential Impedance Finally Made Simple. Odd mode impedance is offered in equations (2.5), (2.6), and (2.7) for microstrip, symmetrical stripline, and asymmetrical stripline, respectively. Coefficients for the odd mode microstrip impedance calculation are altered from the original equation to improve correlation with field solver results. The dielectric constant E _r is an effective dielectric constant and may be approximated as an average between the upper and lower dielectrics. Stack-up parameters for the equations are defined in Figure 2.5:

Figure 2.5. Stack-up parameters for stripline (left) and microstrip (right).

(2.5) $Z_{odd\_us}=\frac{100}{\sqrt{E_r+1.41}}\ln \left(7.5\frac{h}{w}\right)$

(2.6) $Z_{odd\_sym\_sl}=\frac{60}{\sqrt{E_r}}\ln \left(2.35\frac{\left(h1+h2+tt\right)}{w}\right)$

(2.7) $Z_{odd\_asym\_sl}=\frac{80}{\sqrt{E_r}}\ln \left(4.75\frac{h1}{w}\right)\left(1-\frac{h1}{4\times h2}\right)$

Equations for differential microstrip and stripline impedance are shown in equations (2.8) and (2.9), respectively:

(2.8) $Z_{diff\_us}=2\times Z_{odd\_us}\times \left(1-0.48e^{\left(-0.96\frac{s}{h}\right)}\right)$

(2.9) $Z_{diff\_sl}=2\times Z_{odd\_sl}*\left(1-0.347e^{\left(-2.9\frac{s}{\left(h1+h2+tt\right)}\right)}\right)$

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DRAM System Signaling and Timing

Bruce Jacob , ... David T. Wang , in Memory Systems, 2008

9.2.9 Transmission Line Discontinuities

In an ideal LC transmission line with uniform characteristic impedance throughout the length of the transmission line, a signal can, in theory, propagate down the transmission line without reflection or attenuation at any point in the transmission line. However, in the case where the transmission line consists of multiple segments with different characteristic impedances for each segment, signals propagating on the transmission line will be altered at the interface of each discontinuous segment.

Figure 9.14 illustrates that at the interface of any two mismatched transmission line segments, part of the incident signal will be transmitted and part of the incident signal will be reflected toward the source. Figure 9.14 also shows that the characteristics of the mismatched interface can be described in terms of the reflection coefficient ρ, and ρ can be computed from the formula ρ = (Z_L − Z_S)/(Z_L + Z_S). With the formula for the reflection coefficient, the reflected signal at the interface of two transmission line segments can be computed by multiplying the voltage of the incident signal and the reflection coefficient. The voltage of the transmitted signal can be computed in a similar fashion since the sum of the voltage of the transmitted signal and the voltage of the reflected signal must equal the voltage of the incident signal.

FIGURE 9.14. Signal reflection at an unmatched transmission line interface.

In any classroom discussion about the reflection coefficient of transmission line discontinuities, there are three special cases that are typically examined in detail: the well-matched transmission line segments, the open-circuit transmission line, and the short-circuit transmission line. In the case where the characteristic impedances of two transmission line segments are matched, the reflection coefficient of that interface ρ is 0 and all of the signals are transmitted from one segment to another segment. In the case that the load segment is an open circuit, the reflection coefficient is 1, the incident signal will be entirely reflected toward the source, and no part of the incident signal will be transmitted across the open circuit. Finally, in the case where the load segment is a short circuit, the reflection coefficient is – 1, and the incident signal will be reflected toward the source with equal magnitude but opposite sign.

Figure 9.15 illustrates a circuit where the output impedance of the voltage source is different from the impedance of the transmission line. The transmission line also drives a load whose impedance is comparable to that of an open circuit. In the circuit illustrated to the figure, there are three different segments, each with a different characteristic impedance. In this transmission line, there are two different impedance discontinuities. Figure 9.15 shows that the reflection coefficients of the two different interfaces are represented by ρ (source) and ρ (load), respectively. Finally, the figure also shows that the transmission line segment that connects the source to the load has finite length, and the signal flight time is 250 ps on the transmission line between the mismatched interfaces.

FIGURE 9.15. Illustration of signal reflection on a poorly matched transmission line.

Figure 9.15 shows the voltage ladder diagram where the signal transmission begins with the voltage source driving a 0-V signal prior to time zero and switching instantaneously to 2 V at time zero. The figure also shows that the initial voltage V_S can be computed from the basic voltage divider formula, and the initial voltage is computed and illustrated as 1.33 V. The ladder diagram further shows that due to the signal flight time, the voltage at the interface of the load, VL, remains at 0 V until 250 ps after the incident signal appears at V_s . The 1.33-V signal is then reflected with full magnitude by the load with the reflection coefficient of 1 back toward the voltage source. Then, after another 250 ps of signal flight time, the reflected signal reaches the interface between the transmission line and the voltage source. The reflected signal with the magnitude of 1.33 V is then itself reflected by the transmission line discontinuity at the voltage source, and the re-reflected signal of –0.443 V once again propagates toward the load.

Figure 9.16 illustrates that the instantaneous voltage on a given point of the transmission line can be computed by the sum of all of the incident and reflected signals. The figure also illustrates that the superposition of the incident and reflected signals shows that the output signal at V_L appears as a severe ringing problem that eventually converges around the value driven by the voltage source, 2 V. However, as the example in Figure 9.16 illustrates, the convergence only occurs after several round-trip signal flight times on the transmission line.

FIGURE 9.16. Signal waveform construction from multiple reflections.

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Data Transmission Media

John S. Sobolewski , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

XII Interconnection Considerations for Electrical Conductors

It has been mentioned in Section X that electrical conductors are and will continue to be the predominant transmission medium for interconnecting components within a system. As bandwidths and bit rates increase, it becomes increasingly important to take into account the stray inductance and capacitance that influence the performance of these media. These stray reactances are strongly dependent on the geometry, conductor length, and proximity to the ground plane. By carefully positioning the conductor we can reduce one stray component, but only at the expense of increasing the other. Because of the relatively slow rate of change of stray inductance and capacitance with the distance from the ground plane, the conductor length is the major factor determining the stray reactances.

The current or voltage propagation delay T _d in a conductor is given by:

(8) $\text{T}\text{d}=\text{l}\sqrt{\text{L}\text{C}}$

where l is the length of the conductor in meters, L is its distributed inductance, and C is its distributed capacitance per unit length. The conductor is said to be "short" if its propagation delay T _d is much less than the rise and fall times of the pulses it is expected to transmit. If its length is such that its T _d is comparable to or longer than the rise and fall times, or greater than 0.35/f where f is the greatest frequency per second for which it will be used, the conductor is termed as being "long," in which case transmission line theory should be used to predict its characteristics.

The transmission line parameters relevant to our discussion are the characteristic impedance and the reflection coefficient. The characteristic impedance Z ₀ is the impedance seen at one end of an infinitely long line and is approximately by:

(9) ${\text{Z}}_0=\sqrt{\text{L}\text{C}}$

The reflection coefficient r is the ratio of the reflected voltage v _r to the incident voltage v _i if a conductor is not terminated in its characteristic impedance and is given by:

(10) $\text{r}=\frac{{\upsilon }_{\text{r}}}{{\upsilon }_{\text{i}}}=\frac{{\text{Z}}_{\text{L}}-{\text{Z}}_0}{{\text{Z}}_{\text{L}}+{\text{Z}}_0}$

where Z _L is the terminating or load impedance. If a long conductor is not terminated or matched in its characteristic impedance, reflections will be present and ringing will occur if pulses with sharp leading and trailing edges are transmitted over the line. Such reflections and ringing increase the inherent electrical noise of the system in which they occur. The impedance ranges of transmission lines that are usually encountered in practice are given below. Note that a strip line is a rectangular conductor over a ground with the width of the conductor begin much greater than its thickness. This type of conductor is encountered in printed circuits, for example.

Types of transmission line	Characteristic impedance (Ω)
Wire over ground	80–400
Twisted pair	80–200
Coaxial pair	40–120
Strip line	20–140

The above relations are useful in determining the length and placement of lines. In high-impedance circuits, it is important to minimize capacitance by making the line as short as possible and placing it as far as possible from the ground. In low-impedance circuits, it is important to minimize the inductance by making the line as short as possible and placing it as close to ground as possible. For long lines, the lines should be terminated in their characteristic impedance whenever possible to minimize reflections and hence circuit noise.

The presence of stray electromagnetic fields in a system causes cross-talk voltage or current to be induced in neighboring conductors. This cross-talk may be predominantly capacitive or inductive. Capacitive cross-talk is due to stray capacitance between conductors and is predominant in circuits with large voltage swings and small currents (i.e., high-impedance circuits). The capacitive cross-talk voltage v _c is given by:

(11) ${\upsilon }_{\text{c}}={\text{k}}_{\text{c}}\epsilon (\mathrm{voltage\; swing}\times \mathrm{length\; of\; line}\times \mathrm{function\; of\; spacing})/\mathrm{rise\; time\; of\; voltage}$

where ɛ is the dielectric constant of the medium between the conductors and k _c is a constant. In general, the closer the lines the greater the v _c . Spacing the lines farther apart and away from the ground reduces v _c but has the effect of increasing inductive cross-talk. When the voltage swings are small and the currents are large (i.e., low-impedance circuits), inductive cross-talk usually predominates. It arises chiefly because of mutual inductance coupling between conductors and is generally given by:

(12) ${\text{i}}_{\text{i}}={\text{k}}_{\text{i}}\left(\mathrm{current\; swing}\times \mathrm{length\; of\; line}\times \mathrm{function\; of\; spacing}\right)/\mathrm{rise\; time\; of\; current}$

where i _i is the cross-talk current, which can be reduced if the lines are farther from each other and as close as possible to the ground.

If more than one line is driven at the same time, the inductive currents in the passive line are additive. This is not the case with capacitive cross-talk. An upper bound is reached for v _c if the passive line is surrounded by active lines. Both types of cross-talk may be reduced by using special lines. Twisted pairs, coaxial lines, and multiple shielded lines all have zero mutual impedance in theory and therefore have zero inductive cross-talk. In practice, the following numbers are useful:

Type of wiring	Mutual inductance (μH/m)
"Neat" in bundles	1–7
Point-to-point over ground	0.6
Twisted pairs	0.06
Coaxial cable	0.006
Multiple shielded lines	As small as necessary

In summary, it is impossible to eliminate all the stray inductance and capacitance of conductors. It is possible only to trade one against the other by varying the characteristic impedance. In interconnecting components or systems, the goal should be to use point-to-point connections over a ground plane. All lines should be coaxial or twisted pairs and matched at least at the receiving end. Practical considerations may not permit this, but the more we approach this type of interconnections, the less trouble will be experienced with system noise and cross-talk and therefore with getting a high-speed system to operate successfully.

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Acoustic components

Leo Beranek , Tim Mellow , in Acoustics (Second Edition), 2019

Dynamic density

To simplify the expressions for the wave number k and characteristic impedance Z ₀, we can use the following shorthand known as the dynamic density where 〈u〉 is given by Eq. (4.202) so that

(4.211) $\begin{array}{c}\rho =-\frac{1}{j\omega 〈\tilde{u}〉}\frac{\partial \tilde{p}}{\partial z}=\frac{{\rho }_{\text{0}}}{1-〈F\left(k_V,a,B_u\right)〉}\\ ={\rho }_{\text{0}}{\left(1-\frac{Q\left(k_{V}a\right)}{1-\text{0}\text{.}5B_u{k}_V^2{a}^{2}Q\left(k_{V}a\right)}\right)}^{-1}\text{.}\end{array}$

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Busses, Interrupts and PC Systems

William Buchanan BSc (Hons), CEng, PhD , in Computer Busses, 2000

Cable characteristics

The main characteristics of cables are attenuation, cross-talk and characteristic impedance. Attenuation defines the reduction in the signal strength at a given frequency for a defined distance. It is normally defined in dB/100  m, which is the attenuation (in dB) for 100   m. An attenuation of 3   dB/100   m gives a signal voltage reduction of 0.5 for every 100   m. Table 2.2 lists some attenuation rates and equivalent voltage ratios; they are illustrated in Figure 2.6. Attenuation is given by

Table 2.2. Attenuation rates as a ratio

dB	Ratio	dB	Ratio	dB	Ratio
0	1.000	10	0.316	60	0.001
1	0.891	15	0.178	65	0.000 6
2	0.794	20	0.100	70	0.000 3
3	0.708	25	0.056	75	0.000 2
4	0.631	30	0.032	80	0.000 1
5	0.562	35	0.018	85	0.000 06
6	0.501	40	0.010	90	0.000 03
7	0.447	45	0.005 6	95	0.000 02
8	0.398	50	0.003 2	100	0.000 01
9	0.355	55	0.001 8

Figure 2.6. Signal ratio related to attenuation

$\mathrm{Attenuation}=20{log}_{10}\left(\frac{V_{\mathrm{in}}}{V_{\mathrm{out}}}\right)\mathrm{dB}$

Calculation of attenuation from input and output voltages

For example if the input voltage to a cable is 10   V and the voltage at the other end is only 7   V, then the attenuation is calculated as

$\mathrm{Attenuation}=20{log}_{10}\left(\frac{10}{7}\right)=3.1\mathrm{dB}$

Coaxial cables have an inner core separated from an outer shield by a dielectric. They have an accurate characteristic impedance (which reduces reflections), and because they are shielded they have very low cross-talk levels. They tend also to have very low attenuation, (such as 1.2   dB at 4   MHz), with a relatively flat response. UTPs (unshielded twisted-pair cables) have either solid cores (for long cable runs) or are stranded patch cables (for shorts run, such as connecting to workstations, patch panels, and so on). Solid cables should not be flexed, bent or twisted repeatedly, whereas stranded cable can be flexed without damaging the cable. Coaxial cables use BNC connectors while UTP cables use either the RJ-11 (small connector which is used to connect the handset to the telephone) or the RJ-45 (larger connector which is typically used in networked applications to connect a network adapter to a network hub).

The characteristic impedance of a cable and its connectors are important, as all parts of the transmission system need to be matched to the same impedance. This impedance is normally classified as the characteristic impedance of the cable. Any differences in the matching result in a reduction of signal power and produce signal reflections (or ghosting).

Cross-talk is important as it defines the amount of signal that crosses from one signal path to another. This causes some of the transmitted signal to be received back where it was transmitted. Capacitance (pF/100   m) defines the amount of distortion in the signal caused by each signal pair. The lower the capacitance value, the lower the distortion.

Typical cables used are:

•

Coaxial cable – cables with an inner core and a conducting shield having characteristic impedance of either 75 Ω for TV signal or 50 Ω for other types.

•

Cat-3 UTP cable – level 3 cables have non-twisted-pair cores with a characteristic impedance of 100 Ω (±   15 Ω) and a capacitance of 59   pF/m. Conductor resistance is around 9.2 Ω/100   m.

•

Cat-5 UTP cable – level 5 cables have twisted-pair cores with a characteristic impedance of 100Ω (±   15Ω) and a capacitance of 45.9   pF/m. Conductor resistance is around 9 Ω/100   m.

The Electrical Industries Association (EIA) has defined five main types of cables. Levels 1 and 2 are used for voice and low-speed communications (up to 4 Mbps). Level 3 is designed for LAN data transmission up to 16 Mbps and level 4 is designed for speeds up to 20 Mbps. Level 5 cables, have the highest specification of the UTP cables and allow data speeds of up to 100 Mbps. The main EIA specification on these types of cables is EIA/TIA568 and the ISO standard is ISO/IEC 11801.

Table 2.3 gives typical attenuation rates (dB/100   m) for Cat-3, Cat-4 and Cat-5 cables. Notice that the attenuation rates for Cat-4 and Cat-5 are approximately the same. These two types of cable have lower attenuation rates than equivalent Cat-3 cables. Notice that the attenuation of the cable increases as the frequency increases. This is due to several factors, such as the skin effect, where the electrical current in the conductors becomes concentrated around the outside of the conductor, and the fact that the insulation (or dielectric) between the conductors actually starts to conduct as the frequency increases.

Table 2.3. Attenuation rates (dB/100   m) for Cat-3, Cat-4 and Cat-5 cable

Frequency (MHz)	Attenuation rate (dB/100   m)
	Cat-3	Cat-4	Cat-5
1	2.39	1.96	2.63
4	5.24	3.93	4.26
10	8.85	6.56	6.56
16	11.8	8.2	8.2

The Cat-3 cable produces considerable attenuation over a distance of 100   m. The table shows that the signal ratio of the output to the input at 1   MHz, will be 0.76 (2.39   dB), then, at 4   MHz it is 0.55 (5.24   dB), until at 16   MHz it is 0.26. This differing attenuation at different frequencies produces not just a reduction in the signal strength but also distorts the signal (because each frequency is affected differently by the cable. Cat-4 and Cat-5 cables also produce distortion but their effects will be lessened because attenuation characteristics have flatter shapes.

Table 2.4 gives typical near-end cross-talk rates (dB/100   m) for Cat-3, Cat-4 and Cat-5 cables. The higher the figure, the smaller the cross-talk. Notice that Cat-3 cables have the most cross-talk and Cat-5 have the least, for any given frequency. Notice also that the cross talk increases as the frequency of the signal increases. Thus, high-frequency signals have more cross-talk than lower-frequency signals.

Table 2.4. Near-end cross-talk (dB/100   m) for Cat-3, Cat-4 and Cat-5 cable

Frequency (MHz)	Near end cross-talk (dB/100   m)
	Cat-3	Cat-4	Cat-5
1	13.45	18.36	21.65
4	10.49	15.41	18.04
10	8.52	13.45	15.41
16	7.54	12.46	14.17

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Silicon-Based Millimeter-wave Technology

Guennadi A. Kouzaev , ... Natalia K. Nikolova , in Advances in Imaging and Electron Physics, 2012

3.1.1 Quasistatic Model of TFML

The wave parameters of the microstrip transmission line are the propagation constant $k_z$ and the characteristic impedance $Z_c$ . For computation of the propagation constant $k_z=\frac{\omega }{c}\sqrt{{\epsilon }_e}$ we need the effective permittivity ${\epsilon }_e$ , where $\omega$ is the angular frequency and c is the light velocity. The effective permittivity is an equivalent parameter, and it shows the slowing of the mode due to the dielectric filling of the microstrip cross section. One of the popular and accurate quasi-static formulas is given in Hammerstad and Jensen (1980) derived as follows:

(1) ${\epsilon }_e=\frac{{\epsilon }_r+1}{2}+\frac{{\epsilon }_r-1}{2}{\left(1+\frac{10}{u}\right)}^{-ab},$

where

$\begin{array}{l}a=1+\frac{1}{49}\ln \left[\frac{u^4+{\left(u/52\right)}^2}{u^4+0.432}\right]+\frac{1}{18.7}\ln \left[1+{\left(\frac{u}{18.1}\right)}^3\right],\\ \text{and}b=0.564\left(\frac{{\epsilon }_r-0.9}{{\epsilon }_r+3}\right),\text{with}u=\frac{w}{h}.\end{array}$

The characteristic impedance $Z_c$ of the microstrip transmission line is computed by Hammerstad and Jensen (1980):

(2) $Z_c=\frac{60}{\sqrt{{\epsilon }_e}}\ln \left[F_1/u+\sqrt{1+4/u^2}\right],\Omega ,$

where

$F_1=6+\left(2\pi -6\right)\exp \left[-{\left(30.666/u\right)}^{0.7528}\right].$

The accuracy of these formulas can be improved taking into account the thickness of the strip $t_1$ according to Hammerstad and Jensen (1980).

The maximum error in ${\epsilon }_e$ is about 0.2% for ${\epsilon }_r<128.$ At low frequencies, where the dispersion is not high, the characteristic impedance $Z_c$ is computed with the error no more than 1%.

At higher frequencies and for narrow microstrips, the frequency dependence of the effective permittivity and characteristic impedance is rather strong, and this effect should be taken into account by EM simulations. The skin effect also influences both the phase constant and the characteristic impedance.

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Transmission line fundamentals

Hanqiao Zhang , ... Jeff Ou , in High Speed Digital Design, 2015

Traveling waves

Traveling waves travel free of reflection interference on a transmission line. When load impedance matches transmission line characteristic impedance there is no reflection on a transmission line. Under this condition, the voltage and current on the transmission line can be calculated by using equation 1.69 and by eliminating the reflected terms of the equation:

(1.70) $\{\begin{array}{l}V(z)=\frac{(V_0+Z_0{I}_0)e^{-j\beta z}}{2}=V_0^{+}{e}^{-j\beta z}\hfill \\ I(z)=\frac{(V_0+Z_0{I}_0)e^{-j\beta z}}{2Z_0}=I_0^{+}{e}^{-j\beta z}\hfill \end{array}$

Traveling waves' voltage and current amplitudes are constant along the line. The input impedance at any location of the transmission line can be calculated by definition:

(1.71) $Z(z)=\frac{V_0^{+}{e}^{-j\beta z}}{I_0^{+}{e}^{-j\beta z}}=\frac{V_0^{+}}{I_0^{+}}=Z_0$

The input impedance is a constant at any location on of the transmission line and is equal to the its characteristic impedance. Traveling wave is an idea condition for the operation of the high-speed system.

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Filters

D.I. Crecraft , S. Gergely , in Analog Electronics: Circuits, Systems and Signal Processing, 2002

10.7.2 Impedance scaling

It is not often, if ever, that a filter is to be designed to have the characteristic impedance of 1Ω quoted in the design tables. In the case of passive filters, it is important to match the impedance levels both at the source and at the load. Buffer amplifiers can be used with active filters at both the input and the output. The outputs are taken from the outputs of op amps which are low impedance points, so additional buffering may not be necessary here. However, impedance scaling is often used to set the capacitance of the capacitors, to a commonly available value. Since the range of available resistors is far larger than that of capacitors, the latter are chosen for a convenient value and the former are calculated according to the rules of scaling to provide the required filter.

The relationship of the resistances and reactances of the components, and therefore the transfer characteristics of the filter, can be preserved by multiplying all of them by a constant. This operation only changes the characteristic impedance and not the cut-off frequency. As before, the same procedure may be used with any existing filter which has all the required characteristics apart from the input and output impedance.

So, to change the impedance level of the filter from Z_CHold to Z_CHnew we require that

(10.62) $\frac{Z_{\text{CHnew}}}{Z_{\text{CHold}}}=\frac{R_{1\text{new}}}{R_{1\text{old}}}=\frac{L_{\text{lnew}}}{L_{\text{lold}}}=\frac{C_{\text{1old}}}{C_{\text{1new}}}$

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Transmission Lines

Krishna Naishadham , in The Electrical Engineering Handbook, 2005

4.2.1 Lossless Line

For the lossless line R = 0 = G ; hence, the attenuation constant α = 0, and the characteristic impedance Z ₀ is real. In this case, these equations apply:

(4.19) $\gamma =j\beta =j\omega \sqrt{LC.}$

(4.20) $Z_0=\sqrt{\frac{L}{C}.}$

A lossless line has these properties: (a) it does not dissipate any power, (b) it is non-dispersive (i.e., the phase constant varies linearly with frequency ω, or the velocity v_p = ω/β is independent of frequency), and (c) its characteristic impedance Z ₀ is real.

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Design a 100-ï‰ Microstrip Transmission Line Using the Expressions

Source: https://www.sciencedirect.com/topics/computer-science/characteristic-impedance

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