Iec Tr 62152-2004.Pdf

General

Separating and testing components of a transmission chain is essential for practical applications, necessitating clear interfaces and measurement techniques with standardized terms and definitions Operating with normalized voltage waves, which correspond to the square root of power waves, is beneficial, typically using the square root of a reference impedance, usually the system's application impedance.

In this way, for instance, the scattering parameters are defined For example, S 21 is the forward operational transfer function and S 11 is the operational reflection coefficient

Using the square root of the normalized voltage or power waves is beneficial for two main reasons: first, network analyzers primarily measure voltages; second, the natural logarithm, ln, of a complex quantity can be expressed in decibels as $20 \log_{10} |z|$, while the imaginary part remains as the argument, arg(z), in radians.

Complex operational attenuation or operational propagation coefficient * B

The complex operational attenuation (complex operational loss) introduced by a two-port component, cascade of components, link, cable assembly etc into a system is defined by using the scattering parameter S 21 as

* (A.1b) where in (A.1a) ln S 21 A B > Np @ in (A.1b) 20 log 10 S 21 A B > @ dB in (A.1a) and (A.1b) arg(S 21) B B >rad@ where

A B is the operational attenuation = 20 log 10 (1/|S 21 |) (dB)

B is the operational attenuation phase constant = –arg(S ) (rad)

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The ratio of the unreflected complex power (voltage multiplied by current) entering a two-port network to the complex power consumed by the load is expressed in decibels as NOTE 1 A B Typically, the load is a resistance that matches the system's application impedance, denoted as Z N When the generator and load impedances are identical, the operational attenuation is referred to as insertion loss.

The theory of complex functions reveals that the natural logarithm of the ratio of two square roots of complex power waves can be expressed using the argument and exponential forms Specifically, we can represent this relationship as \$\ln(z) = \ln(j) + \arg(z)\$, where \$z\$ is a complex number defined by its real and imaginary components.

1 * áá ạ ã ¨¨ © § where A is in nepers and B in radians

When A is expressed in decibels, B will not be affected; it remains in radians.

Impedance

The nominal characteristic impedance $ Z_{CN} $ of a two-port device represents the resistive component of the average characteristic impedance $ Z_C $, defined with a tolerance at a specific frequency This nominal impedance $ Z_N $ pertains to the system terminals where the two-port operates Additionally, the nominal reference impedance $ Z_R $, typically equal to $ Z_N $, is utilized for measurement purposes.

Operational reflection coefficient

The operational reflection coefficient of a two-port network is represented by the scattering parameter S11 This coefficient is equivalent to the reflection coefficient rc at the input when the two-port is terminated with its reference impedances ZR, which typically match the nominal impedances of the system terminals.

Return loss

The return loss where the mismatch effects at the input and output of two-port have been eliminated (compare with the continuous wave (CW) burst measurement method)

Defining structural return loss is crucial, even though it is not directly measured from cable assemblies, as it highlights the variations among different types of return losses.

LICENSED TO MECON Limited - RANCHI/BANGALORE FOR INTERNAL USE AT THIS LOCATION ONLY, SUPPLIED BY BOOK SUPPLY BUREAU. c) Reflection loss of a junction (see Figure 2 )

2 2 2 r ln (1 S ) ln (1 S ) Np j arg( (1 S ) ) rad

* (A.4a) or * r ln (1S 2 ) 20 log 10 (1S 2 ) > @dB j arg( (1S 2 ) ) rad> @ (A.4b)

2 2 2 r 10 ln (1 ) 10 log (1 ) dB j 1arg(1 ) rad

* (A.4c) d) Mismatch loss of a junction (not recommended)

2 2 2 m ln (1 S ) ln (1 S ) Np j arg( (1 S ) ) rad

2 2 2 m ln (1 S ) 20 log10 (1 S ) dB j arg( (1 S ) ) rad

2 2 2 m 10 ln (1 ) 10 log (1 ) dB j 1arg(1 ) rad

In c) and d) S is the complex reflection coefficient of the junction

General coupling transfer function

This is distinguished between the near-end and far-end coupling transfer functions T n and T f

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P 0 is the unreflected power sent into the near end of the system (1)

Figure A.1 – Coupling between two systems

Coupling transfer function is a general term valid through the whole frequency range

It may be expressed in decibels and radians

[dB & rad] 20 log P [dB] j arg [rad]

P (A.8) and the (complex) operational transfer, coupling screening, unbalance, attenuation, etc are

A x is the (operational ) attenuation (dB);

B x is the (operational ) attenuation phase constant (rad).

Benefits of the concept of operational quantities

Measurements are always taken between well-defined resistive terminations

This means that the impedances at a reference plane between the cascaded units of the system are specified

Individual units can be specified and tested separately and made by different manufacturers

This makes open systems, networks and cabling possible.

Two-port transmission technique – Terms a) Image transfer function OUT OUT

T P V b) Image transfer attenuation or loss A= 20 log 1

Z C1 and Z C2 represent the image or characteristic impedances at the input and output of a two-port network These impedances are equivalent to the input and output impedances when the opposite port is terminated with its respective image impedance.

P V and P OUT V OUT are the input and output square root of complex powers.

When defining the image, there are reflections at the input and output; in other words, the input and output are terminated with their image impedance c) Complex image attenuation > @ > @ A jB

1 dB log 20 d) Image attenuation A log T 1 > @ dB e) Image phase shift B = arg T 1 > @ rad f) Image phase propagation time or delay p

Z g) Image group propagation time or delay g d d

Z h) Image phase velocity p p v 1 W i) Image group velocity g g v 1 W j) Complex operational attenuation * B A B jB B k) Operational attenuation ááạ ã ¨¨© §

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Two-port theory and fundamental concepts in transmission engineering 1

General

This annex aims to establish essential terms and definitions for globally describing the transmission properties of two-port or quadripole systems It connects classical quadripole theory with the scattering matrix representation, which utilizes the incident and reflected square root of power waves at the two-port's input and output Ultimately, it demonstrates that these two concepts are interconnected through fundamentally identical equations.

The two-port theory, initially designed for voice and carrier technologies, has evolved to encompass transmission and microwave applications, making it applicable across the entire frequency spectrum and diverse use cases.

In the following Clauses, we will use the term two-port exclusively.

Transfer equations for a passive two-port

For a passive impedance-symmetrical two-port (see Clause C.4 and Figure C.1), the following equations are valid r i

Figure C.1 – A quadripole or two-port

1 L.HALME: CHAPTER L4, part of English version of the L Halme´s book (Halme, L.K.: Johtotransmissio ja sọhkửmagneettinen suojaus, (Transmission on lines and electromagnetic screening, in Finnish), Parts A and B,

Otakustantamo 2nd Eddition Helsinki 1989, 605 pages), corrected by J Walling (2000-09-27)

The image impedance of a two-port network is denoted as \$Z_0\$, while the complex image attenuation or transfer constant is represented as \$A + jB\$ This constant corresponds to the image attenuation of a two-port when terminated in its image impedance The incident voltage and current waves at the input are represented by \$U_i\$ and \$I_i\$, respectively, while the reflected waves from the output are denoted as \$U_r\$ and \$I_r\$ By solving the relevant equations for these variables and substituting them into the output equations, we can determine the actual voltage and current at the output terminals.

By solving equations (C.5) and (C.6) for U 1 and I 1 we obtain

Equations (C.7) and (C.8) for input terminals can be derived from equations (C.5) and (C.6) for output terminals by interchanging the voltages, swapping the currents, and replacing * with –*.

From equations (C.5), (C.6), (C.7) and (C.8), we can also solve the currents expressed by means of the voltages, as well as the voltages expressed by means of the currents:

Chain matrix

Equations (C.7) and (C.8) can be presented in matrix form ằ ẳ ô º ơ ê ằ ằ ẳ º ô ô ơ ê ằ ẳ ô º ơ ê

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Here, the multiplier matrix is called the chain matrix and is generally expressed as: ằ ẳ º ô ơ ê ằ ẳ º ô ơ ê ằ ẳ º ô ơ ê

Where the constants A, B, C and D forming the chain matrix are called the transfer parameters They are bound to each other by the relation

The transfer parameters can be calculated by alternately considering the output of the two- pole either as short-circuited or open-circuited, whereby

The chain matrix is well suited for the examination of cascaded two-ports

An impedance-unsymmetrical two-port (see Clause C.3) can be treated as a symmetrical one by cascading it (as shown by Figure C.2) with an ideal transformer with a turns ratio of

Figure C.2 – An impedance-unsymmetrical two-port (a) with its equivalent circuit (b)

We are here concerned with the cascading (or chaining) of two-ports, whereby the calculations can be appropriately carried out by means of chain matrices

Let us suppose two two-port with the chain matrices A 1 and A 2 being interconnected as shown by Figure C.3

Figure C.3 – Two chained two-ports

The matrix equations, with the direction arrows as indicated in Figure C.3, are

The combining of equations (C.17) yields

The matrix A is hence obtained as a product between the chain matrices of the two-ports to be chained

The turns ratio of the transformer in Figure C.2 can be rewritten as

K n c and the transfer equation of the transformer is obtained in the matrix form

In accordance with equation (C.13), the chain matrix A 1 of a symmetrical two-port is equal to

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The transfer equations of an impedance-unsymmetrical two-port can be written in the matrix form as follows ằ ẳ ô º ơ ê ằ ằ ằ ằ ằ ẳ º ô ô ô ô ô ơ ê ằ ẳ ô º ơ ê

This matrix equation can also be solved for U 2 and I 2

From the matrix equations (C.22) and (C.24), we can obtain the following transfer equations for an impedance-unsymmetrical two-port:

The end results obtained can also be obtained direct from the transfer equations of an impedance-symmetrical two-port on the basis of Figure C.2

By solving equations (C.25), (C.26), (C.27) and (C.28,) currents can be expressed by means of voltages or vice-versa, resulting in the following expressions:

NOTE A short reminder on matrices:

When M 2 = K *M 1 , where M 1 , M 2 and K are matrices, then M 1 = K -1 *M 2 , where K -1 is the inverse matrix of K.

K 1 A 1 1* where the determinant is ' ADBC

The symmetries and impedances of a two-port

Let us examine the two two-ports illustrated in Figures C.4 and C.5

Figure C.4 – An impedance-symmetrical two-port with Z 1 = Z 2, , when Z A = Z B Figure C.5 – An impedance-unsymmetrical two-port for which Z 1 z Z 2 when Z A = Z B

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The two-port network depicted in Figure C.4 is classified as impedance-symmetrical or port-symmetrical, whereas the two-port network shown in Figure C.5 is identified as impedance-unsymmetrical or port-unsymmetrical.

A two-port is considered transfer-symmetrical or reciprocal if the complex composite loss in the direction A->B equals that in the direction B->A, regardless of the generator and terminating impedance values Two-ports made up of passive components, excluding gyrators, are inherently reciprocal Additionally, a two-port that exhibits no directional dependence in its properties is classified as both reciprocal and impedance-symmetrical.

A two-port network is considered longitudinally symmetrical when its input terminals exhibit earth-symmetry, meaning the admittances at each terminal relative to earth are equal This condition is known as transversal symmetry in the two-port configuration.

In addition to the complex image attenuation represented as * = A + jB, a key characteristic of a two-port network is the image impedance The image impedances Z₀₁ and Z₀₂ can be determined through short-circuit and open-circuit measurements, as illustrated in Figure C.5.

Z Z 02 Z 2k Z 2t where the subscripts k and t refer to the short-circuit and open-circuit conditions, respectively

Let us recall the equations (C.7) and (C.8) valid for a longitudinally symmetrical two-port:

Taking into account that U 2 = Z B I 2, (see Figure C.6), we obtain with equations (C.33) the input impedance of the two-port:

Figure C.6 – A two-port terminated with an impedance Z B

The input impedance $ Z_1 $ is influenced by the characteristics of the two-port network and the terminating load impedance $ Z_B $ It is evident that at high attenuation $ A $, the impact of $ Z_B $ on $ Z_1 $ is minimal According to equation (C.34), $ Z_1 $ approaches $ Z_0 $ when $ \tanh |*| \approx 1 $.

The input impedance is exclusively influenced by the characteristics of the two-port when $ A > 2 \, \text{Np} $ A two-port is classified as electrically short if $ A \ll 2 \, \text{Np} $ and $ B \ll S/2 $, while it is considered electrically long when $ A \geq 2 \, \text{Np} $ and $ B \geq S/2 $.

2 Numbers in square brackets refer to the reference documents at the end of this Annex

When the output is short-circuited (Z B = 0), we have

IfZ B = 0, but A ặ 0 and B = S/2, then Z 1k ặf This applies, for example, to a lossless short- circuited line with length O/4 When the output is open (Z B =f), we have

Further, when Aặ 0 and B ặS/2, then Z 1t ặ 0 Additionally we have *= jB Replacing B by S/2, the equation (C.34) can then be written in the following form:

The impedance $ Z_B $ can be transformed into $ Z_1 $ using the conversion (C.37), applicable only at the specific frequency where the length of the lossless line is $ \lambda/4 $, known as a quarter-wavelength transformer Additionally, equations (C.35) and (C.36) demonstrate that the characteristic impedance $ Z_0 $ and the impedance of a longitudinally symmetrical two-port can be derived from the short-circuit and open-circuit impedances.

Impedance matching

To ensure uniform transmission in cascaded two-port networks, it is crucial to match the image impedances of the interconnected sections Reflections at these interconnection points can disrupt transmission, making impedance matching essential in telecommunication engineering The characteristic impedances of the devices must closely align to achieve non-distorted transmission However, a single significant mismatch is permissible within each repeater section, as long as other mismatches remain minor, since at least two mismatches are necessary to create a signal-distorting forward echo.

By substituting the quantities U 2 = I 2 Z 0, which correspond to a proper matching (Z B = Z 0 ) into equations (C.33), we obtain

I (C.39) from which it follows that the input impedance is

Hence the input impedance is under these conditions independent of *.

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Correct matching enables the greatest possible complex power to be transmitted from a generator to the load (In the literature, the term complex power often refers to the quantity

In transmission engineering, the product of voltage and current phasors is referred to as complex power, with the quantity UI representing the apparent power.

Figure C.8 – Power matching for maximizing the effective power

The complex power obtained with the load Z B is

(C.42) which reaches a maximum when Z P = Z g , which yields g

Power matching occurs when the load absorbs the maximum effective power, achieved under the condition that the resistances $ R_g $ and $ R_p $ are equal, and the imaginary parts satisfy $ j(X_g + X_p) = 0 $ This condition is fulfilled when the impedances are complex conjugates, expressed as $ Z_g = Z^*_p $ While power matching is commonly applied in the context of matching transmitters to antennas, it is typically effective at a single frequency, known as the tuning frequency, and has limited use in broadband transmission techniques Additionally, a two-port can function as either a power source or a load.

The input or output impedance of a two-port network can be designed to be resistive and frequency-independent by utilizing a series combination of resistance (R) with either inductance (L) or capacitance (C) For instance, by connecting an impedance of $ R + \frac{1}{jZ_C} $ in parallel with an impedance of $ R + jZ_L $ and selecting $ C = \frac{L}{R^2} $, a frequency-independent resistive impedance $ R $ can be achieved.

Level concepts

The term "level" refers to both relative and absolute values, particularly in the context of power, voltage, or current within a transmission system When discussing these parameters, we specifically refer to power, voltage, or current levels.

When assessing the power, voltage, or current at a measuring point in a transmission system, we focus on a relative level compared to the feeding point In contrast, comparing these quantities to a standardized reference value yields an absolute level.

Levels are commonly expressed in decibels (dB), more seldom in nepers (Np) The use of nepers is actually restricted to some theoretical calculations The units are related by

1 dB = 0,05/lg e = 0,1151 Np or 1 Np = 20 lg e = 8,686 dB

The relative power level can be expressed using the power and voltage at the measuring point, denoted as \$P_x\$ and \$V_x\$, in comparison to the corresponding values at the feeding point, represented as \$P_A\$ and \$V_A\$.

N P (C.44) and the relative voltage level is

The relative level at the input of the system is always zero

If P 1 and V 1 are the standardized reference values, the absolute power level is given by

N P (C.46) while the absolute voltage level is

In telecommunication engineering, the reference for absolute power levels is 1 mW and the reference for absolute voltage levels is 0,775 V, which corresponds to 1 mW in a 600 : load

Nowadays, voltage levels are seldom used in telecommunication engineering, to avoid confusion There is a tendency towards an exclusive use of power levels

In conjunction with broadcast relaying, community antennas and closed-link television systems, instead, voltage levels based on a reference of 1 PV have been adopted

A reference impedance of 75 : is implied; however, in the last two systems so that one is here actually concerned with power levels

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In transmission engineering, the designations dBm (dB(mW)) and dB(PV) are used to differentiate between absolute levels referenced to 1 mW and 1 PV, respectively A nominal level is established when 1 mW is applied to the input or at a fictive reference point in the system, with this nominal level consistently represented as 0 dBm0 Consequently, the entire system can be viewed as operating at 0 dBm0, meaning that a designation like –50 dBm0 indicates a level that is 50 dB below the system's nominal level.

Figure C.9 – Absolute and nominal level in a system

In speech transmission, it is essential to weight disturbing noise signals according to the ear's sensitivity curve A psophometrically weighted noise level, such as -50 dB0p, indicates a level 50 dB below the nominal level This can also be expressed as psophometrically weighted noise power reduced to the 0 dB(mW) point, which corresponds to 1 mW and is 50 dB below that reference level.

When it is necessary to emphasise that a level is a relative level or, respectively, a voltage level, designations dBr and dBu are employed.

Attenuation and gain concepts

The complex image attenuation or image transfer constant * of a two-port is defined as a logarithmic ratio between the power P 1 = U 1 I 1 fed to the input terminals and the power

P 2 = U 2 I 2 obtained at the output, when the two-port is terminated in an impedance which is equal to the output image impedance of the two-port (see Figure C.10)

Figure C.10 – Definition of the complex image attenuation * of a two-port

A is the image attenuation and B is the image phase constant If the two-port is impedance- symmetrical (Z 01 = Z 02 ), the equations are more simple and we obtain the expression

In practical applications, a two-port network often connects terminating devices with impedances that differ from the network's image impedances, leading to the concept of operational attenuation The complex operational attenuation, or complex operational transfer constant, is defined as the logarithmic ratio of the power $ P_1' = \frac{E^2}{4Z_g} $ supplied by the generator to a load equal to its internal impedance $ Z_g $, and the power $ P_2 = \frac{U^2}{Z_p} $ delivered to the load $ Z_p $ at the output of the two-port.

Figure C.11 – Definition of the complex operational attenuation of a two-port

U is called the operational transfer constant H B

The operational attenuation, denoted as A B, and the operational phase constant, represented as B B, play crucial roles in system performance When the impedances of the generator and the load are equal (Z g = Z p), the equations simplify significantly, leading to a clearer expression for the operational transfer constant.

The complex operational gain –* is the opposite number of the complex operational attenuation:

> @ dB ln 2 > @ Np lg 2 20 dB lg 10 dB lg 10 j p

-A B is the operational gain and –B B is the operational gain phase angle

Residual attenuation refers to the portion of signal loss that remains in a transmission line after the effects of amplifiers (repeaters) have been accounted for It is calculated by subtracting the total gain of all amplifiers from the overall attenuation of the transmission path In mathematical terms, residual attenuation can be expressed as the difference between the total attenuation, denoted as $ A_k $, and the sum of the gains of the amplifiers, represented as $ S_k $ in decibels (dB).

Figure C.12 – Definition of residual attenuation

The residual attenuation of the 2/4-wire line shown by Figure C.12 is equal to

A 1…5 are the attenuations of different line sections;

A h is the attenuation of the hybrid networks;

S 1 and S 2 are the gain of the respective repeater

The reference equivalent measures the speech-transmitting capabilities of a telephone connection, defined as the additional attenuation required for a reference system to match the loudness of speech in the actual system being tested If the tested system is less sensitive than the reference system, the reference equivalent is deemed positive.

The NOSFER system, located in the laboratories of CCITT, serves as an international reference system for measuring the sending reference equivalent In this process, a speaker alternates between speaking into the microphone of the system under test and the NOSFER microphone A VU meter in the NOSFER transmitting circuit allows the speaker to maintain consistent loudness, while a listener adjusts the attenuator (A R) in NOSFER to ensure equal loudness is perceived from both systems.

Figure C.13 – Measurement of the sending reference equivalent

To measure the receiving reference equivalent, the speaker communicates into the NOSFER microphone, while the listener alternates between both systems The loudness is equalized using an attenuator, similar to the method used for measuring the sending reference equivalent.

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The standard deviation of test results from a trained testing team typically ranges from 1.5 to 2.5 dB, with a 95% confidence margin falling between 0.5 and 4 dB.

Figure C.14 – Measurement of the receiving reference equivalent

Efforts are underway to substitute subjective measurement methods with objective ones, currently being evaluated by CCITT However, the precision of results from these objective methods has not yet aligned sufficiently with those derived from subjective approaches.

Concepts related to return loss and matching

In the circuit illustrated in Figure C.15, the incident voltage wave, represented as \$U_i\$, approaches a reflection point, while the reflected voltage wave, denoted as \$U_r\$, is sent back from that reflection point.

Figure C.15 – Definition of the complex return loss

The reflection coefficient U is the ratio between the reflected and incident waves:

The complex return loss * r is correspondingly defined as

A r is the return loss and B r is the reflection phase constant

The expression (C.62) for return loss can be rewritten in the form

The normalized impedance, represented as \$z_N = \frac{Z_2}{Z_1}\$, is crucial in understanding the reflection coefficient's constancy, which also keeps the term $\frac{|z_N + 1|}{|z_N - 1|}$ constant As illustrated in Figure C.16, the numerator and denominator can be viewed as line segments that measure the distances from the endpoint P of the vector \$z_N\$ to the points (-1,0) and (1,0), respectively Points where the distance ratio from (-1,0) to (1,0) remains constant can be located by drawing an Apollonius' circle through point P This circle is defined by points maintaining a constant distance ratio from two fixed points To construct the Apollonius' circle, equal sections of length $|z_N - 1|$ are marked on either side of (1,0) along a line parallel to $|z_N + 1|$ By drawing lines from point P to the endpoints of these sections, two intersection points on the real axis are found, and the distance between these points determines the diameter of the Apollonius' circle.

(formed by the points for which the ratio of distances from the points (-1,0) and (1,0) is constant)

The points on the circumference of the circle in Figure C.16 indicate a constant return loss value, while the return loss is greater inside the circle and smaller outside Due to the circle's symmetry about the real axis, typically only half of the circle is illustrated By plotting multiple Apollonius’ circles for different return loss values, a chart similar to Figure C.17 can be created Any normalized impedance, \$z_N\$, represented on this chart directly correlates to the corresponding return loss in dB, as exemplified by \$A_r | 12 \text{ dB}\$.

A r = 20 lg |z N +1|/|z N –1| where z N =r+ jxis the normalized impedance

When substituting Z 2 = 0 (short-circuit) or Z 2 = f (open-circuit) to equation (C.62), A r will vanish When Z 2 = Z 1 (proper matching), there will be no reflections and, consequently,

A r =f. The complex reflection loss * s is

The reflection loss, denoted as $ A_s $, and the reflection loss phase angle, represented as $ B_s $, quantify the attenuation of complex power transferred to the load $ Z_2 $ compared to the unreflected power if the load were $ Z_1 $ Equation (C.67) illustrates that with optimal matching, where $ Z_2 = Z_1 $ and $ A_s = 0 $, there are additional impedance pairs that also result in zero reflection loss This relationship is visually represented in Figures C.18 and C.19, which depict a combination of circles indicating constant return loss and curves for constant values.

The right-hand side of the complex plane can be converted into a unit circle centered at the point (1,0), resulting in Smith's chart for transmission lines.

In the Smith chart, Apollonius’ circles of constant return loss are represented as concentric circles centered at (1,0), allowing for a direct interpretation of impedance variations along the line due to mismatches between the line and the load Each clockwise turn on the chart corresponds to a half-wavelength shift toward the generator In the case of a lossy line, the reflection attenuation varies as one moves toward the generator, resulting in a converging spiral pattern on the Smith chart.

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Figure C.18 – Curves for constant values of A s or A r in the complex plane

Figure C.19 – Curves for constant values of A s or A r in the complex plane

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Figure C.20 – Smith chart for transmission lines

The voltage-standing-wave ratio VSWR is the ratio between maximum and minimum values of the line voltages:

VSWR U (C.69) where the reflection coefficient

From equation (C.69) we can calculate the absolute value of the reflection coefficient

Scattering parameter

Scattering parameter of a one-port

We can characterize a port, as shown in Figure C.21, by incident (i) and reflected (r) voltage, current and square-root of power waves

In a one-port network, the voltage (U) and current (I) at the terminals are essential parameters, with R₀ representing the image impedance When comparing this to the characteristic impedance of a homogeneous transmission line, as illustrated in Figure C.22, it is beneficial for practical applications to select nominal values for the characteristic impedance, such as 50 ohms.

The impedance serves as a reference for measurements, but it may not align with the image impedance of the one-port This is because \$V_i\$ is defined as the unreflected square root of the power entering the one-port, as well as the square root of the fictive power, which is determined by matching the generator to this impedance.

Recalling that the square-root of power is

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The relation between the incident and the reflected wave can be expressed by means of the scattering parameter S: i r S V

The parameter S is here identical to the reflection coefficient U, which equally represents the ratio of the reflected voltage to the incident voltage at the reflection plane (see Clause C.8)

From the definition for V i and V r , it follows that

U (C.79) the solution of which gives

(C.80) where Z = U I is the input impedance of the one-port

If Z= R 0 , the voltage of the reflected wave is V r = 0 The inverse value of S, when expressed in dB or Np and radians, is called the complex return loss * r (compare with equation C.61):

V i at a one-port, which is fed from a generator with an internal impedance Z g equal to the image impedance of the one-port Z 0 , is: g i 2 Z

WhenU and Iare substituted into equation (C.83), expression (C.82) is obtained: o g g g g 0 g 0

Figure C.23 – One-port fed from a generator with source impedance Z g

When the impedance $ Z $ equals the reference impedance $ R_0 $, the reflected wave disappears Additionally, reflections are absent between the generator and the load when their impedances are equal, specifically when $ Z_g = Z $ This scenario indicates that the generator and load impedances are properly matched.

The maximum effective power is transmitted to the load, when Zg = Z* (see Clause C.5), whereas the maximum complex power is reached when Zg = Z In accordance with equations

(C.76) and (C.82), the maximum complex power is g

(C.84a) From the expressions (C.74) and (C.75) we obtain

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This yields the expression for the complex power absorbed by the one-port, which is represented by the load:

If the impedance Z g of the generator, that feeds the one-port, is taken as reference impedance then the maximum complex power at the load is

We obtain then the ratio between the complex power absorbed in the one-port and the actual complex power if the one-port is represented by the reference impedance Z:

Compare to equations (C.65) and (C.66) Expressed in logarithmic units, this is called the complex reflection loss

Scattering parameters and scattering matrix of a two-port

A two-port, shown in Figure C.24, can be treated as two individual one-ports, face to face

For both one-ports, the incident and reflected waves are characterized by

In this context, R 01 and R 02 denote the reference impedance at the input and output, respectively The variables V i1 and V r1 represent the square roots of the incident (unreflected) and reflected complex powers at port 1, while V i2 and V r2 correspond to those at port 2.

Complex power is defined as the product of voltage and current, represented as \$S = UI\$ Apparent power, denoted as \$S^* = UI^*\$, plays a crucial role in electrical power engineering, particularly when analyzing the phase angle between voltage and current Here, \$I^*\$ refers to the complex conjugate of the current \$I\$.

The scattering parameters S mn of a two-port are defined as follows: i2 22 i1 21 2 r i2 12 1 ù 11 r1

S S (C.93) is called the scattering matrix [4] It has the elements:

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Under operational conditions, the complex operational reflection coefficient $ U_B $ and the operational transfer function $ H_B $ are determined when the input voltages $ V_{i2} $ and $ V_{i1} $ are zero, as indicated in equation (C.53) This condition is met when the terminal impedances $ Z_A $ and $ Z_B $ match the reference impedances $ R_{01} $ and $ R_{02} $, respectively, as outlined in equation (C.90).

S 11 or U B11 is the complex operational reflection coefficient at the input;

S 22 or U B22 is the complex operational reflection coefficient at the output;

S 21 or H B21 is the operational transfer function in the forward direction;

S 12 or H B12 is the operational transfer function in the backward direction

The scattering matrix can thus be written in the form

The relationship between scattering parameters and the relevant working quantities can be established by first examining the condition $ V_{i2} = 0 $ and its effect on the reflection factor at the input of the two-port network.

Figure C.25 – Termination Z B by virtue of the stray parameters of the two-port

Let us consider the influence of the termination Z B on the parameters V i2 and V r2 of the two- port The scattering parameter of the termination is

(C.95c) where R 02 is the reference impedance at the output of the two-port The connection between

V i2 and V r2 is obtained by the relation r2 2 i2 S V

If the reference impedance is chosen to be equal to the impedance Z B of the termination, then

S 2 = 0 and V i2 = 0 Similarly, it can be shown that V i1 = 0, when R 01 = Z A

Hence, we can conclude that, if the terminal impedances Z A and Z B are selected to be equal as reference impedances, the parameters V i1 and V i2 are equivalent to zero

The input and output impedances of the two-port are

As derived above, we have V i2 = 0 for R 02 = Z B If an e.m.f E 1 with an internal impedance equal to the reference impedance R 01 is connected to the input terminals, then equation (C.82) yields

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By substituting this into the expression

The voltage ratios S 21 and S 12, adjusted for terminal impedances, represent the transfer function of a two-port network under operational conditions, which is why they are referred to as the composite transfer function Additionally, the complex composite loss of the two-port is analyzed in both the forward and backward directions.

In the passive transfer-balanced case i.e if the two-port is reciprocal, S 21 is equal to S 12

The scattering matrix is essential for analyzing the transmission properties of a two-port network, as it allows for direct measurement of composite reflection and loss coefficients at both the input and output when the two-port is connected to a reference impedance.

In the matched case, the input and output voltages, $ V_i $ and $ V_r $, represent the square roots of the complex power at the two-port, given that the reference impedance is appropriately selected.

Z A E 1 ’ is the internal electromotive-force at port 1 of the two-port V i2 and V r2 are obtained by changing the sub-index 1 o 2 and impedance Z A oZ B

In the literature [5], parameters V i and V r are defined utilizing the so-called available power

The results are identical to those previously mentioned, but only when the reference impedances are purely resistive

In this case, we have

The expressions are equivalent to (C.97) and (C.101) only when Z A and Z B are purely resistive In line transmission, utilizing available power matching is generally inadvisable due to persistent reflections (refer to Clause C.5) However, available power matching is applicable in lump-loaded lines, where reflectionless matching at the line's ends is also considered Additionally, available power matching is essential for effectively matching a transmitter to an antenna.

Examples

Example 1

Consider an ideal transformer with a turns ratio n:1 (Figure C.26) Then,

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Let us choose terminal impedances as reference impedance

Therefore, the scattering matrix of the ideal transformer is

Hence, both ports are reflectionless, and attenuation is = 0 dB, i.e the transfer function is equal to one.

Example 2

Let us determine the scattering matrix of a passive, reciprocal two-port terminated by Z A and

Z B The image impedances of the two-port are Z 01 and Z 02 , the complex image attenuation is

* and input impedances are Z 1 and Z 2

Figure C.27 – Determination of a scattering matrix of a passive reciprocal two-port

From expression (C.94) we obtain S 11 and S 22

We find the voltage U 2 to be

By substituting this into expression (C.94), we get

By substituting expressions (C.109), (C.111) and (C.112) into the matrix (C.93), we get

S S (C.112b) we obtain the scattering matrix of the two-port shown in Figure C.27.

Tiêu đề	Background of Terms and Definitions of Cascaded Two-Ports
Trường học	MECON Limited
Chuyên ngành	Electrical Engineering
Thể loại	Technical report
Năm xuất bản	2004
Thành phố	Geneva

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