\end{aligned}, Following the same procedure detailed above for the short-circuit case, we find \[\boxed{ Z_{in}(l) = -jZ_0 \cot \beta l } \label{m0088_eZstubOC} \]. The unit Neper comes from the name of the \(\text{e} (= 2.7182818284590452354)\) symbol (written in upright font and not italics since it is a constant), which is called the Neper. The formula for S11 treats the transmission line as a circuit network with its own input impedance, which is required when considering wave propagation into an electrically long circuit network like a transmission line. Since the characteristic impedance for a homogeneous transmission line is based on geometry alone and is therefore constant, and the load impedance can be measured independently, the matching condition holds regardless of the placement of the load (before or after the transmission line). Remarkably, the transmission line has essentially transformed the short circuit termination into an open circuit! If the transmission line is lossless, the characteristic impedance is a real number. see Figure \(\PageIndex{2}\)). To convert from \(\text{dB}\) to \(\text{Np}\) multiply by \(0.1151\). This is because more of the EM field is in the substrate. This is because once again the variation with length is due to the interference of incident and reflected waves. Finally, note that the argument \(\beta l\) appearing Equations \ref{m0087_eZin1} and \ref{m0087_eZin2} has units of radians and is referred to as electrical length. Thus \(\alpha = x\text{ dB/m} = x\times 0.1151\text{ Np/m}\). Key Takeaways The input impedance of a transmission line is the impedance seen by any signal entering it. Thus a mode can support a propagating mode only if the wavenumber is greater than the cut-off wavenumber, i.e. These are precision. The important result here is that a voltage wave (and a current wave) can be defined on a transmission line. Remarkably, the transmission line has essentially transformed the short circuit termination into an open circuit! (b) Open-circuit termination (\(Z_L \rightarrow \infty\)). If the transmission line is lossy, the characteristic impedance is a complex number given by equation (10). n A: The input impedance is simply the line impedance seen at the beginning (z=A) of the transmission line, i.e. Above that is air. This is undesirable because two modes will travel at different speeds and a propagating signal divides energy between the two modes and, since the modes have different group velocities, the signal will become garbled. The electrical length can also be expressed in terms of wavelength noting that \(360^{\circ}\) corresponds to \(2\) radians, which also corresponds to \(\lambda\). This in turn can cause a reactive pulse of high voltage that can destroy the transmitter's final output stage. \(V(z)\) is a phasor and \(v(z, t) = \Re\{ V(z)e^{\jmath\omega t}\}\). Z For a coaxial line, the electric fields extend in a radial direction from the center conductor to the outer conductor. It is caused by the physical dimensions of the transmission line and its downstream circuit elements. This is true for all transmission line structures supporting the minimum variation of the fields corresponding to a TEM mode. i The energy transferred is in the traveling waves. Now employing expressions for \(\widetilde{V}(z)\) and \(\widetilde{I}(z)\) from Section 3.13 with \(z=-l\), we find: \begin{aligned} This is very useful to keep in mind because it means that all possible values of \(Z_{in}(l)\) are achieved by varying \(l\) over \(\lambda/2\). Since the characteristic impedance for a homogeneous transmission line is based on geometry alone and is therefore constant, and the load impedance can be measured independently, the matching condition . The periodic variation of the \(E\) field transfers energy from the EM field to mechanical vibrations. A transmission line has the \(RLGC\) parameters \(R = 100\: \text{/m},\: L = 80\text{ nH/m},\: G = 1.6\text{ S/m}\), and \(C = 200\text{ pF/m}\). The magnetic field is circular, centered on the center conductor, so the current on the conductor can be calculated as the closed integral of the magnetic field. In general, it is double the odd-mode impedance, which is the value we care about for differential signaling, as it is used in high-speed PCB design. Transmission line loss is due to the resistance of conductors, which is described by \(R\), and loss in the dielectric described by \(G\). If you are curious, the development is done for a parallel plate waveguide and a rectangular waveguide in Chapter 6 and for a coaxial line in Section 2.9. The losses due to input impedance (loss) in these circuits will be minimized, and the voltage at the input of the amplifier will be close to voltage as if the amplifier circuit was not connected. \(\alpha:\:\gamma =\alpha +\jmath\beta = \sqrt{(R + \jmath\omega L) (G +\jmath\omega C)};\:\quad\omega = 12.57\cdot 10^{9}\text{ rad/s}\), \(Z_{0} = (R +\jmath\omega L)/\gamma = (100 + \jmath\omega\cdot 80\cdot 10^{9})/(17.94 + \jmath 51.85) = (17.9 + \jmath 4.3)\:\Omega\). What is of particular interest now is that at \(l=\lambda/4\) we see \(Z_{in}=0\). First, why consider such a thing? The semirigid cable shown at the bottom of Figure \(\PageIndex{5}\)(a) must be bent using a bending tool, as shown in Figure \(\PageIndex{5}\)(c), and in. This is precisely as expected from standing wave theory (Section 3.13). Multiplying numerator and denominator by \(e^{+j\beta l}\) we obtain \[Z_{in}(l) = Z_0 \frac{ e^{+j\beta l} - e^{-j\beta l} }{ e^{+j\beta l} + e^{-j\beta l} } \nonumber \] Now we invoke the following trigonometric identities: \begin{array}{l} n Loss is incorporated in the imaginary parts of \(\varepsilon\) and \(\mu\) for TEM modes. where \(\omega = 2\pi f\) is the radian frequency and \(f\) is the frequency with the SI units of hertz (\(\text{Hz}\)). If one were to create a circuit with equivalent properties across the input terminals by placing the input impedance across the load of the circuit and the output impedance in series with the signal source, Ohm's law could be used to calculate the transfer function. l Calculation of the damping factor and the damping of impedance bridging, Interconnection of two audio units - Input impedance and output impedance, https://en.wikipedia.org/w/index.php?title=Input_impedance&oldid=1142596906, "Aortic input impedance in normal man: relationship to pressure wave forms", JP Murgo, N Westerhof, JP Giolma, SA Altobelli, An excellent introduction to the importance of impedance and impedance matching can be found in, This page was last edited on 3 March 2023, at 08:43. Accessibility StatementFor more information contact us atinfo@libretexts.org. Specifically, \[\label{eq:37}-\omega^{2}\mu_{\text{eff}}\varepsilon_{\text{eff}}=(R+\jmath\omega L)(G+\jmath\omega C) \], If the medium is lossless (\(\mu\) and \(\varepsilon\) are real and \(R =0= G\)), then, \[\label{eq:38}\mu_{\text{eff}}\varepsilon_{\text{eff}}=LC \]. i The loss tangent of the dielectric medium and the loss tangent of the transmission line may not be the same as the EM fields may not be confined just to the medium, e.g. \(\Re\{w\}\) denotes the real part of \(w\), a complex number. When the power transfer is optimized the circuit only runs at 50% efficiency. So when \(\alpha\ell = 1\text{ Np}\), where \(\ell\) is the length of the line, the signal has decreased to \(1/\text{e}\) of its original value, and the power drops to \(1/\text{e}^{2}\) of its original value. If \(Z_L\) is equal to the characteristic impedance \(Z_0\) of the transmission line, then the input impedance \(Z_{in}\) will be equal to \(Z_L\). \end{array}, Employing these identities, we obtain: \[Z_{in}(l) = Z_0 \frac{ j2\left(\sin\beta l\right) }{ 2\left(\cos\beta l\right) } \nonumber \] and finally: \[\boxed{ Z_{in}(l) = +jZ_0 \tan \beta l } \label{m0088_eZstubSC} \]. (Sometimes p.u.l. (a) Short-ciruit termination ( Z L = 0. more than one mode. Z The input impedance of a short- or open-circuited lossless transmission line alternates between open- (\(Z_{in}\rightarrow\infty\)) and short-circuit (\(Z_{in}=0\)) conditions with each \(\lambda/4\)-increase in length. However, the convention is to introduce the dimensionless quantities Neper and radian to convey additional information. One consequence of Equation \(\eqref{eq:44}\), and noting that \(C\) is approximately independent of frequency, is that if \(G\) is known at one frequency then its value at another frequency can be quickly determined. Calculated using EM simulation. Then the line can be replaced by an impedance equal to the characteristic impedance of the line. Such a transmission line is sometimes referred to as a stub. The results of field simulations of the effective permittivity of lines of various widths and with various substrate permittivities are shown in Figure \(\PageIndex{9}\), where it can be seen that the effective relative permittivity, \(\varepsilon_{e}\), increases for wide strips. When a propagating electrical signal enters an integrated circuit, waveguide, transmission line, or other electrical network, it "sees" a certain impedance that resists generation of current in the electrical network. The phasors of the traveling voltage waves are, \[\label{eq:16}V_{0}^{+}(z)=|V_{0}^{+}|e^{\jmath\varphi^{+}}e^{-\jmath\beta z}\quad\text{and}\quad V_{0}^{-}(z)=|V_{0}^{-}|e^{\jmath\varphi^{-}}e^{\jmath\beta z} \], \[\begin{align}\label{eq:17} \text{Characteristic impedance:} \quad&Z_{0}=\sqrt{(R+\jmath\omega L)/(G+\jmath\omega C)} \\ \label{eq:18}\text{Propagation constant:} \quad&\gamma =\sqrt{(R+\jmath\omega L)/(G+\jmath\omega C)} \\ \label{eq:19} \text{Attenuation constant:} \quad&\alpha =\Re\{\gamma\} \\ \label{eq:20}\text{Phase constant:} \quad&\beta=\Im\{\gamma\} \\ \label{eq:21}\text{Wavenumber:} \quad& k=-\jmath\gamma \\ \label{eq:22}\text{Phase velocity:}\quad&v_{p}=\omega/\beta \\ \label{eq:23}\text{Wavelenght:}\quad&\lambda=\frac{2\pi}{|\gamma|}=\frac{2\pi}{|k|}\end{align} \]. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. t The Neper is used in calculating transmission line signal levels, as in Equations \(\eqref{eq:8}\) and \(\eqref{eq:9}\). Substituting Equation \(\eqref{eq:8}\) in Equation \(\eqref{eq:3}\) results in, \[\label{eq:10}I(z)=\frac{\gamma}{R+\jmath\omega L}\left[ V_{0}^{+}e^{-\gamma z}-V_{0}^{-}e^{\gamma z}\right] \], Then from Equations \(\eqref{eq:10}\) and \(\eqref{eq:9}\), \[\label{eq:11}I_{0}^{+}=\frac{\gamma}{R+\jmath\omega L}V_{0}^{+}\quad\text{and}\quad I_{0}^{-}=\frac{\gamma}{R+\jmath\omega L}(-V_{0}^{-}) \], The characteristic impedance is defined as, \[\label{eq:12}Z_{0}=\frac{V_{0}^{+}}{I_{0}^{+}}=\frac{-V_{0}^{-}}{I_{0}^{-}}=\frac{R+\jmath\omega L}{\gamma}=\sqrt{\frac{R+\jmath\omega L}{G+\jmath\omega C}} \], with the SI unit of ohms (\(\Omega\)). If the speed at which information moves varies with frequency, then a signal such as a pulse will spread out. The complete development of transmission line theory is presented in Section 2.2.2, and Section 2.2.3 relates the RLGC transmission line model to the properties of a medium. The result also depends on the length and phase propagation constant of the line. \(C\) describes the ability to store electrical energy and is mostly due to the properties of the dielectric. Figure 3.16. Spatial variation of the fields stores additional energy in the \(E\) and \(H\) fields, affecting \(\gamma\) as well as \(Z_{0}\). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Therefore, the input impedance of the load and the output impedance of the source determine how the source current and voltage change. The. This term is often used by power system engineers to quantify power transferred across a transmission line and seen at a load. Modes that have all the fields in the transverse plane (perpendicular to the propagation direction) is called a transverse EM (TEM) mode and these modes exist at DC. The total voltage and current at a point on the line is the sum of the traveling voltage and traveling current waves, respectively, but the total voltage/current view is not sufficient to describe how a transmission line works. This page titled 3.15: Input Impedance of a Terminated Lossless Transmission Line is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. This section develops the theory of signal propagation on transmission lines. This article explains how to use a Smith Chart to determine the input impedance to transmission line at a given distance from the source or the load. Only a few geometries permit analytic solution of the fields so in general a numerical field solution is required and \(Z_{0}\) and \(\gamma\) derived. At \(1\text{ GHz}\), \(\beta_{0} = 20.958\text{ rad/m}\) and \(\lambda_{0} = 29.98\text{ cm}\) (use \(\lambda_{0}\approx 30\text{ cm}\) at \(1\text{ GHz}\) as a reference). connectors used in some microwave measurements. Applying Kirchoffs laws applied to the model in Figure \(\PageIndex{1}\)(b) and taking the limit as \(\Delta z\to 0\) the transmission line equations are, \[\begin{align}\label{eq:1}\frac{\partial v(z,t)}{\partial z}&=-Ri(z,t)-L\frac{\partial i(z,t)}{\partial t} \\ \label{eq:2}\frac{\partial i(z,t)}{\partial z}&=-Gv(z,t)-C\frac{\partial v(z,t)}{\partial t}\end{align} \]. Consider a traveling wave at \(2\text{ GHz}\) on the line. Figure \(\PageIndex{3}\): Coaxial line adaptors: (a) N-type female-to-female (N(f)-to-N(f)); (b) APC-7 to Ntype male (APC-7-to-N(m)); (c) APC-7 to SMA-type male (APC-7-to-SMA(m)); SMA adapters: (d) SMA-type female-to-female (SMA(f)-to-SMA(f)); (e) SMA-type male-to-female (SMA(m)-toSMA(f)); (f) SMA-type male-to-male (SMA(m)-to-SMA(m)); and (g) SMA elbow. Since the argument of the complex exponential factors is \(2\beta l\), the frequency at which \(Z_{in}(l)\) varies is \(\beta/\pi\); and since \(\beta=2\pi/\lambda\), the associated period is \(\lambda/2\). The controlled bending radius ensures minimal change in the characteristic impedance and propagation constant of the cable. Transmission line theory is expressed in terms of traveling voltage and current waves and these are akin to a one-dimensional form of Maxwells equations. Re { Z i n } is always zero. The transmission line parameters from Equations \(\eqref{eq:12}\) and \(\eqref{eq:18}\)\(\eqref{eq:23}\) are then, \[\begin{align}\label{eq:45}Z_{0}&=\sqrt{\frac{L}{C}} \\ \label{eq:46}\beta&=\omega\sqrt{LC} \\ \label{eq:47}\lambda_{g}&=\frac{2\pi}{\omega\sqrt{LC}}=\frac{v_{p}}{f} \\ \label{eq:48}\alpha&=0 \\ \label{eq:49}v_{p}&=1/\sqrt{LC}\end{align} \]. A TEM mode has \(k_{c} = 0\) so a signal, other than DC can always propagate on the line in a TEM mode. the three lines have different characteristic impedances. &=Z_{0} \frac{1-e^{-j 2 \beta l}}{1+e^{-j 2 \beta l}} In this section, we determine a general expression for \(Z_{in}\) in terms of \(Z_L\), \(Z_0\), the phase propagation constant \(\beta\), and the length \(l\) of the line. Legal. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Signals cannot propagate on the line if the frequency is below \(f_{c}\). 1: Input reactance ( Im { Z i n }) of a stub. Most microwave substrates have negligible conductivity so dielectric relaxation loss dominates. In AC circuits carrying power, the losses of energy in conductors due to the reactive component of the impedance can be significant. Thus the transverse dimensions of the cable determine its upper frequency limit for reliable signal transmission. There are also many types of coaxial cables, as shown in Figure \(\PageIndex{5}\)(a). The values of the input and output impedance are often used to evaluate the electrical efficiency of networks by breaking them up into multiple stages and evaluating the efficiency of the interaction between each stage independently. At high frequencies, and hence short wavelengths, large internal dimensions of a large diameter cable can support more than one spatial variation of the EM fields, i.e. This is the pattern of the strip where (b) shows three lines of different width. The input admittance (the reciprocal of impedance) is a measure of the load network's propensity to draw current. What is the characteristic impedance of the line? So, if Zo = RL then the input impedance to the line will be Zo irrespective of length. Thus the attenuation constant \(\alpha\) has the units of Nepers per meter (\(\text{Np/m}\)) and the phase constant \(\beta\) has the units radians per meter (\(\text{rad/m}\)). Input impedance is primarily used in RF design, but it can be used to develop transfer functions in high speed design, which then can be used to predict impulse responses using causal models. The dimensions of some of the quantities that appear in transmission line theory are discussed in Section 2.2.4. \(\text{e}\) is sometimes called Eulers constant. In this scenario, the reactive component of the input impedance cancels the reactive component of the output impedance at the source. When calculating the forward voltage wave of a line that is infinitely long (or there are no reflections from the load). There are various coaxial lines with different diameters and different levels of attention given to the uniformity of the lines. What is the complex propagation constant of the transmission line? The Thvenin's equivalent circuit of the electrical network uses the concept of input impedance to determine the impedance of the equivalent circuit. The important takeaway from this section is that a signal moves on a transmission line as forward- and backward-traveling waves. Explore Solutions \(G\) describes loss in the dielectric which derives from conduction in the dielectric and from dielectric relaxation. Nepers and radians are dimensionless units, but serve as prompts for what is being referred to. Mathematically it can be shown that if you know the inductance (L), capacitance (C), resistance (R) and conductance (G) per unit length, Z0 Z 0 is: - R + jL G + jC R + j L G + j C Accessibility StatementFor more information contact us atinfo@libretexts.org. The characteristic impedance or surge impedance (usually written Z 0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. when \(k>k_{c}\). As the relative permittivity of the line increases, the characteristic impedance of the line reduces. If the capacitance of the line is \(100\text{ pF/m}\) and the conductive loss is zero (i.e., \(G = 0\)), what is the characteristic impedance of the line? This technique requires two measurements: the input impedance Z i n when the transmission line is short-circuited and Z i n when the transmission line is open-circuited. as the imaginary part of When a transmission line is referred to as having an impedance of \(50\:\Omega\), this is referring to the line having a characteristic impedance of \(50\:\Omega\), the line cannot be replaced by a \(50\:\Omega\) resistor. When a device whose input impedance could cause significant degradation of the signal is used, often a device with a high input impedance and a low output impedance is used to minimize its effects. Figure \(\PageIndex{7}\): Microstrip transmission line. A microstrip line is shown in Figure \(\PageIndex{7}\)(a). Solving for the fields in the region between the center and outer conductors yields the following formula for the characteristic impedance of a coaxial line (the derivation is presented in Section 2.9): \[\label{eq:50}Z_{0}=138\sqrt{\frac{\mu_{r}}{\varepsilon_{r}}}\log\left(\frac{b}{a}\right)\:\Omega =60\sqrt{\frac{\mu_{r}}{\varepsilon_{r}}}\ln\left(\frac{b}{a}\right)\:\Omega \]. When there is no reactive component this equation simplifies to In analog video circuits, impedance mismatch can cause "ghosting", where the time-delayed echo of the principal image appears as a weak and displaced image (typically to the right of the principal image). o In the previous section the telegraphers equations for a transmission line modeled as subsections of \(RLGC\) elements was derived. If the conductor and dielectric are ideal (i.e., lossless), then \(R =0= G\) and the equations for the transmission line characteristics simplify. The two most common impedances that are . In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. (The most common exception is when the dielectric is silicon as there can be appreciable conduction in silicon.) ratio for maximum voltage breakdown is \(2.7\), corresponding to \(Z_{0} = 60\:\Omega\) for an air-filled line. To minimize electrical losses, the output impedance of the signal should be insignificant in comparison to the input impedance of the network being connected, as the gain is equivalent to the ratio of the input impedance to the total impedance (input impedance + output impedance). \cos \theta=\frac{1}{2}\left[e^{+j \theta}+e^{-j \theta}\right] \\ Equations \(\eqref{eq:5}\) and \(\eqref{eq:6}\) are second-order differential equations that have solutions of the form, \[\label{eq:8}V(z)=V_{0}^{+}e^{-\gamma z}+V_{0}^{-}e^{\gamma z} \], \[\label{eq:9}I(z)=I_{0}^{+}e^{-\gamma z}+I_{0}^{-}e^{\gamma z} \], The physical interpretation of these solutions is that \(V^{+}(z) = V_{0}^{+}e^{\gamma z}\) and \(I^{+}(z) = I_{0}^{+}e^{\gamma z}\) are forward-traveling waves (moving in the \(+z\) direction) and \(V^{}(z) = V_{0}^{}e^{\gamma z}\) and \(I^{}(z) = I_{0}^{}e^{\gamma z}\) are backward-traveling waves (moving in the \(z\) direction). Equation \(\eqref{eq:50}\) is an exact formulation for the characteristic impedance of a coaxial line. Z Recall the two typical circuit models of a transmission line [1]. A non-homogeneous line has two or or more dielectric mediums, such as air and a dielectric. Also worth noting is that Equation \ref{m0087_eZin1} can be written entirely in terms of \(Z_L\) and \(Z_0\), since \(\Gamma\) depends only on these two parameters. The diameter of the outer conductor and the type of internal supports for the internal conductor determines the frequency range of coaxial components. 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Short-Circuit Terminations, Virginia Polytechnic Institute and State University, Virginia Tech Libraries' Open Education Initiative, source@https://doi.org/10.21061/electromagnetics-vol-1. The decrease in signal level represents loss and the units of decibels per meter (\(\text{dB/m}\)) are used with \(1\text{ Np} = 20 \log \text{e} = 8.6858896381\text{ dB}\). So expressing \(\alpha\) as \(1\text{ Np/m}\) is the same as saying that the attenuation loss is \(8.6859\text{ dB/m}\). The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. \(R\) is due to ohmic loss in the metal more than anything else. e These losses manifest themselves in a phenomenon called phase imbalance, where the current is out of phase (lagging behind or ahead) with the voltage. Equations \(\eqref{eq:8}\) and \(\eqref{eq:9}\) can be rewritten as, \[\label{eq:13}V(z)=V_{0}^{+}e^{-\gamma z}+V_{0}^{-}e^{\gamma z} \], \[\label{eq:14}I(z)=\frac{V_{0}^{+}}{Z_{0}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{0}}e^{\gamma z} \], \[\label{eq:15} v(z,t)=|V_{0}^{+}|\cos(\omega t-\beta z+\varphi^{+})e^{-\alpha z}+|V_{0}^{-}|\cos(\omega t+\beta z+\varphi^{-})e^{\alpha z} \], where \(\varphi^{+}\) and \(\varphi^{-}\) are phases of the forward- and backward-traveling waves, respectively. The unique feature of this connector is that it is sexless, with the interface plate being spring-loaded. This is a commonly used transmission line, as it can be cheaply fabricated using printed circuit board techniques. {\displaystyle Z_{in}=Z_{out}} &=Z_{0} \frac{1+e^{-j 2 \beta l}}{1-e^{-j 2 \beta l}} u So the wider the strip and the higher the substrate permittivity, the lower the characteristic impedance of the line. A line cannot be replaced by a lumped element except as follows: The analytic calculation of the characteristic impedance of a transmission line from geometry is not always possible except for a few regular geometries (matching orthogonal coordinate systems). Using the coordinate system indicated in Figure \(\PageIndex{1}\), the interface between source and transmission line is located at \(z=-l\). If the dielectric filling the coaxial line is polyethylene (which is most common) with \(\varepsilon_{r} = 2.3\), the characteristic impedance of the minimum loss line is \(50.6\:\Omega\). The development does not go into much detail, as the derivation is involved and can only be derived analytically for a few regular transmission line structures. ((e) Copyright 2012 Scientific Components Corporation d/b/a Mini-Circuits, used with permission [4].). Very often equations are curve fit to the numerical solutions but the structure of the equations have a theoretical foundation. 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