We are interested in finding the potential V (−→r ) at apoint on the z-axis P = (0, 0, z) for a uniform charge density ρS distributedon a disk of radius r = a lying in the xy-plane and centred around theorigin, assuming the reference is chosen as infinity.

(a) Draw a clear sketch of this problem to analyze the geometry. Yoursketch should clearly indicate the field point P , the field positionvector −→r , and the source position vector −→r ′.

(b) Solve for the potential V (−→r ) at a point on the z-axis P = (0, 0, z)

Note: You will probably require one of the following integrals in orderto solve this problem

∫1√

x2 + a2dx = ln

(x+

√x2 + a2

) ∫x√

x2 + a2dx =

√x2 + a2∫

1

x2 + a2dx =

1

atan−1

(xa

) ∫x

x2 + a2dx =

1

2ln(x2 + a2

)

TOPIC: Electrostatics and Magnetostatics (ENEL475)

Q.1

An infinitely long coaxial cable has a hollow innerconductor of radius a, which has a cladding of radius b which is a magneticmaterial of permeability µ, as depicted in the figure below. The innerconductor carries the total supply current I, and the outer conductorcarries the total return current I; both are distributed uniformly aroundtheir respective cylindrical surface in the directions indicated in the figure.

(a) Choose an appropriate coordinate system and, using Ampere’s law,show that the magnetic field everywhere in the region a < ρ < b isgiven by

~H =I

2πρaφ

A

m

(b) Using the given field in part (a), calculate the stored magnetic energyin the cladding region for a length d of the cable and for a current I.

(c) If the relative permeability of the cladding is µr = 20, what is theinductance of a length d of this cable?

Q.2

Starting from Gauss’law in integral form, producea derivation of the concept of Divergence by showing that

∇ · ~D = lim∆v→0

∮S

~D · d~S

∆v

Starting from a simple microscopic/atomic basis,explain how each of the following macroscopic properties is manifest:

(a) Conductivity σ

(b) Permittivity ε

(c) Permeability µ

Q.3

Q.4

Consider the loop containing a resistor as shown below. The loop is placed in a magnetic flux density described by:

B=-20 cos(100t-/3)az mWb/m2

a) Find the EMF (Vemf)b) Calculate the induced current in the loop. Indicate the direction of current flow during the first

quarter period on the figure above.

A ground penetrating radar system is modeled as a uniform plane wave in free space impinging on the

ground at normal incidence. The incident electric field (in free space, so properties are o, o, =0) is given by:

Ei(x,t)=10 cos(109t-3.3x)ay V/m

a) Find the wavelength.

The ground has properties of r=4, r=1, and =0.1 S/m.

b) Calculate the reflection () and transmission (T) coefficients.c) Find an expression for the reflected electric field (Er(x,t)).d) Find an expression for the transmitted electric (Et(x,t)) and magnetic fields (Ht(x,t)).

0.2 m

0.4 m

15

y

x

TOPIC: Electromagnetic waves and applications (ENEL476)

Q.1

Q.2

A distortionless transmission line has R=5 /m, L=20 H/m, C=30 nF/m and is operated at 10 MHz. Calculate the following quantities: a) G

b) the impedance of the line, Zo

c) the attenuation of the line,

d) the phase constant of the line,

e) the wavelength on the line,

A load of ZL=30-j40 is attached to 6 cm of the transmission line.

f) Using the appropriate equation, find Zin for the section of transmission line terminated by the load.

A load of impedance ZL=30-j60 is attached to a transmission line with 75 characteristic impedance

(Zo=75 ). The frequency of operation is 5 GHz and the wavelength on the line is 6 cm. Use the Smith Chart to solve the following questions.

a) Find the reflection coefficient at the load ().b) Find the standing wave ratio (s). Verify with the appropriate equationc) Find the input impedance Zin for a line of length of 5.0 cm attached to the load.d) Find the admittance of the load (YL).e) Find the distance from the load to the first voltage minimum.f) Find the shortest distance to a purely resistive load.

A load of impedance ZL=70+j25 is attached to a transmission line with 100 characteristic

impedance (Zo=100 ). The frequency of operation is 900 MHz and the wavelength on the line is 67 cm. To match the load to the line, design a series stub tuner with an open termination on the stub, and a shunt stub tuner with a short termination on the stub.

Q.3

Q.4

Q.5

Determine the S parameters of two port network consisting of a series resistance R terminated at its input and output ports by the characteristic impedance Zo.

Z0 = 50 Ω

Z0 = 50 Ω

Z0 = 50 Ω

50 Ω

50 Ω

Input matching Network

Output matching Network

Z0 = 50 Ω

ΓS = 0.8 -170°

ΓL = 0.7 80°

a) Design the input matching network by giving the lengths of 50 Ohms transmission lines asfunction of the wavelength , λ as shown in the above figure to produce the source reflectioncoefficient ΓS = 0.8 @ -170° at 1 GHz.

b) Design the output matching network by giving the lengths of 50 Ohms transmission lines asfunction of the wavelength , λ as shown in the above figure to produce the load reflectioncoefficient ΓL = 0.7 @80° at 1 GHz.

An amplifier is driven by modulated signal having a 20 MHz bandwidth is constituted by the cascade of three amplifiers A1, A2 and A3 and having the following characteristics:

Amplifier A1: G1=23 dB, IP3_1=23 dBm, NF1=2 dB.

Amplifier A2: G2=17 dB, IP3_2=42 dBm, NF2=3 dB.

Amplifier A3: G3=13 dB, IP3_3=50 dBm, NF3=5 dB.

a) The parameters IP3_N are specified at the output of each amplifier and the reference noisetemperature To is 290°K.

b) Determine the equivalent noise figure NFe of the power amplifier as well as its equivalentnoise temperature Te.

c) Calculate the third order interception point IP3 at the output of the power amplifier. Deducethe P1dB of the power amplifier.

Q.1

Q.2

Q.3

TOPIC: RF/Microwave Active Circuits (ENEL574)

d) Determine the carrier to third order intermodulation products ratio (C/IMD3) at the outputthe power amplifier when it is driven with a two-tone signal spaced by 10 MHz and having atotal input power of -8 dBm

The scattering and noise parameters of a GaAs FET transistor at 2 GHz are:

S11 = 0.9∟-60°, S21 = 3.1∟140°, S12 = 0.02∟62° and S22 = 0.8∟-27°

Fmin = 1.5 dB, Γopt = 0.7∟55°, rn = 0.95

a) Study the stability of the device and draw the input and output stability circles in the SmithChart,

b) Can the device be considered unilateral?c) Draw the operating power gain circle for GP = 20 dB.d) Determine the source and load reflection coefficients required to design an amplifier to have an

operating power gain of 20 dB. Explain your choices of ΓS and ΓL.

Q.4

An amplifier is attached to an antenna through a 9 cm long

section of a coaxial cable. Assume the coax line has no loss, and

its parameters are C=96 pF/m and L=240 nH/m. The antenna has

a radiation impedance of 50 Ω (you can assume the antenna is

your generator, and its radiation impedance the generator

internal impedance), however the amplifier is not matched to the

system. Measurements showed that the amp has an input

impedance of (25+j20) Ω. The system operates at 10 GHz.

You have been asked to improve the system performance. Here

are your tasks:

1. Assume that the antenna generates voltage Vg. Compute the

voltage and the current at the input terminal of the

transmission line (so from the antenna side, remember, your

antenna acts as a generator here), find out the amplitudes

of forward and backward travelling waves.

2. Compute available power. What percentage of available

power at the generator is delivered to the amp?

3. Now match the system using lumped elements. What

percentage of available power is delivered to the amp?

Remember to calculate the actual values of necessary inductances

or capacitances.

Q.1

TOPIC: RF/Microwave Passive Circuits (ENEL575)

Consider a rectangular waveguide with the following dimensions: a=22.0mm, b=10.0mm. The waveguide walls are made of a perfect conductor, waveguide is filled with air (lossless). Frequency is 10 GHz

a. How much the TE20 mode will be attenuated at thatfrequency over a distance of 1 cm (expressed in dB)

b. The waveguide is loaded with an impedance of(154.6+j154.6)Ω. Match the system using double stubserial tuner. The tuners are short-circuited, andseparated by 5/4λ (lambda). You are allowed to add 1or 2 sections of λ/8 section of a waveguide betweenthe tuner and the impedance (only if you find itnecessary). Draw the structure, pay attention to thefeasibility of your structure (that is, could youactually build it? Show all dimensions (in mm). Inparticular, show how the serial connections of thewaveguide and the tuners look like.

Q.2

Design a three-port resistive divider for an equal power split and 75 ohm system impedance (lets’ call this system A). Derive SA matrix of this system. Now terminate port 3 of the 3-port system A with a 150 ohm resistor – effectively turning the initial 3 port structure into a 2 port structure with matrix SB. Compute SB

21 in this new system.

Q.3

Field of Study Examination, Feb 24 2017

Subject area: Radio Frequency Systems and Applied Electromagnetics

This question paper has 12 pages (not including this cover page).

This question paper has 4 questions.

Answer a minimum of one question and at most three questions from this subjectarea.

Use a separate booklet (i.e., blue booklet) for the answers to questions in thissubject area.

1. This question has 4 parts (a)-(d).

A wireless receiver is constituted by a low noise amplifier (LNA) with gain G = 35 dB and noise

figure NF = 3 dB; and P1dB = 13 dBm, a band pass filter having 1 dB insertion loss (IL), and a

passive mixer having a conversion loss (CL) of 4 dB as shown in the lineup below. The receiver

parameters are: noise reference temperature: T0 = 290 K; signal bandwidth: B = 1 MHz; output

third-order intercept point: IP3 = 27 dBm. Boltzmann constant k = 1.38× 10−23J/K.

Low noise amplifier

Mixer

G = 35 dBNF = 3 dB

Bandpass filter

IL = 1 dB CL = 4 dB

Si, Ni So, No

(a) Calculate the overall noise figure (NF ), the gain and the equivalent noise temperature (Te) of

the receiver.

(b) Calculate the noise power at the output of the receiver.

(c) Calculate the dynamic range (DR) of the receiver.

(d) Find the carrier to the third-order intermodulation products ratio (C/IMD3) at the output

the receiver when it is driven with a two-tone signal having a total power of 10 dBm.

1/12

2. This question has 12 parts (a)-(l).

At a sufficient distance from an antenna, the fields radiated by the antenna may be represented

using a uniform plane wave. Assume that the fields are traveling in a source-free region of free space

(ε = ε0, µ = µ0, σ = 0). The electric field is given by:

E(x, t) = 100 cos(6× 109t− βx

)ayV/m

Find the:

(a) frequency (f)

(b) wavelength (λ)

(c) phase constant (β)

(d) magnetic field (H(x, t))

The antenna system is being used for thru-wall inspection (i.e. waves travel through a wall and

detect objects on the other side). The wall has εr = 3, σ = 0.1 S/m and µr = 1. Given that the

wave is normally incident on the wall, find the:

(e) attenuation coefficient (α)

(f) phase constant (β)

(g) intrinsic impedance (η)

(h) reflection coefficient (Γ)

(i) transmission coefficient (T )

(j) velocity at which the wave travels in the wall

(k) For the incident field given above, find an expression for the electric (Er(x, t)) and magnetic

(Hr(x, t)) fields reflected from the wall

(l) For the incident field given above, find an expression for the electric (Et(x, t)) and magnetic

(Ht(x, t)) fields associated with the wave transmitted into the wall

2/12

3. This question has 2 parts (a)-(b).

A capacitor is made up of 2 concentric spheres with the inner sphere having radius a and a charge

+Q coulombs. The outer sphere has a radius b and a charge −Q coulombs. NOTE that a, b and Q

are constants.

The gap between the capacitors is divided into 3 equal spaces with permittivities of ε1, 2ε1 and 3ε1.

(a) Using field analysis, derive an expression for the capacitance C of this capacitor.

(b) If the permittivity is ε1 = ε0εr with εr = 2 and a = 5 mm and b = 9 mm, what is the value of

the capacitance of the concentric spheres.

3/12

4. Design a quarter wave transformer using rectangular waveguide technology. The transformer is to

operate at 10 GHz and be positioned between a section of a waveguide with dimensions a=22.86 mm

× b=10.16 mm (air filled), and a load with 800 Ω impedance (wave impedance).

Design the transformer using a change in a broad dimension a. Ignore effects at the corners/junc-

tion between the transformer section and regular waveguide, in your design, but describe them

qualitatively. Draw the structure, show all dimensions of the transformer.

4/12

Maxwell’s Equations

integral form point formI

~E · d~L = −

ZZ

~B · d~S Faraday’s Law ~∇× ~E = −~B

I

~H · d~L = +

ZZ

~D

· d~S Ampere’s Law ~∇× ~H = ~J +

−→D

I

~D · d~S = =

ZZZ

Gauss’ Law for Electric Fields ~∇ · ~D = I

~B · d~S = 0 Gauss’ Law for Magnetic Fields ~∇ · ~B = 0

Electric Fields and Potential

Coulomb’s Law

~F 12 =12

40

¯~R12

¯2 ba12Electric Field

~E(~r) =

ZZZ 0

(~r0)

40

¡~r − ~r0¢¯~r − ~r0

¯30Work

= −Z

~E · d~L

Potential Difference

= −Z

~E · d~L

Potential field of distributed charges

(~r) =

ZZZ 0

(~r0)

40¯~r − ~r0

¯ 0 Electric Field from Potential

~E = −~∇

Potential of a Dipole

(~r) =~p · ¡~r − ~r0¢40

¯~r − ~r0

¯3Fields from Sheet of Charge

~E(~r) =2ba

Fields from Line Charge

~E(~r) =2

baMagnetic Fields, Forces, and Torque

Biot-Savart Law

~H(~r) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Z

d~L× ¡~r − ~r0¢4¯~r − ~r0

¯3ZZ

~K × ¡~r − ~r0¢4¯~r − ~r0

¯3 0

ZZZ 0

~J × ¡~r − ~r0¢4¯~r − ~r0

¯3 0Lorentz Force Equation

~F = (~E + ~v × ~B)

Magnetic Force on Currents

~F =

I

d~L× ~B (line current)

~F =

ZZ0

~K × ~B0 (surface current density)

~F =

ZZZ 0

~J × ~B 0 (volume current density)

Torque

~T = ~S × ~B = ~m× ~B

Fields from Line Current

~H(~r) =

2ba

Fields from Sheet of Current Density

~H(~r) =1

2~K × ba

5/12

Current and Conductors

Continuity of Current

integral form

I

~J · d~S = −

point form ~∇ · ~J = −

Current Density

~J = ~E

Material Relations and Boundary Conditions

0 = 8854× 10−12 Fm 0 = 4 × 10−7 HmDielectric Material Relations

~D = 0 ~E + ~P = ~E

= (1 + ) 0 = 0

Magnetic Material Relations

~B = 0

³~H + ~M

´= ~H

= (1 + )0 = 0

Electric Boundary Conditions

normal³~D1 − ~D2

´· ba21 =

tangential³~E1 − ~E2

´× ba21 = 0

Magnetic Boundary Conditions

normal³~B1 − ~B2

´· ba21 = 0

tangential ba21 × ³ ~H1 − ~H2

´= ~K

Angles at Interface

tan 1

tan 2=

1

2¯~D2

¯=¯~D1

¯scos2 1 +

µ2

1

¶2sin2 1¯

~E2

¯=¯~E1

¯ssin2 1 +

µ1

2

¶2cos2 1

Polarization Charge Relations

bound surface charge density =~P · ba

bound volume charge density = −~∇ · ~P

Circuit Parameters, Power and Energy

Resistance

=

=

−Z

~E · d~L

ZZ

~J · d~S

Power Loss (Joule/Ohmic Loss)

=

ZZ Z

³~J · ~E

´

Capacitance

=

=

I

~D · d~S

−Z

~E · d~L=2

2

Energy Stored in Electric Field

=1

2

ZZ Z

³~E · ~D

´ =

1

2

ZZ Z

Inductance

=Ψ

=

ZZ

~B · d~S

I

~H · d~L=2

2

Energy Stored in Magnetic Field

=1

2

ZZ Z

³~B · ~H

´

6/12

Cylindrical Coordinates = cos

= sin

=

⇐⇒ =

p2 + 2

= tan−1

=

Field Component Transformations⎡⎢⎢⎣

⎤⎥⎥⎦ =⎡⎢⎢⎣

cos sin 0

− sin cos 0

0 0 1

⎤⎥⎥⎦⎡⎢⎢⎣

⎤⎥⎥⎦⎡⎢⎢⎣

⎤⎥⎥⎦ =⎡⎢⎢⎣cos − sin 0

sin cos 0

0 0 1

⎤⎥⎥⎦⎡⎢⎢⎣

⎤⎥⎥⎦Differential Elements

Differential d~l = d~S = one of =

Elements ba + ba + ba () ba () ba () ba

Note: figures on this page are reproduced from Electromagnetic Fields and Waves, 2nd Edition, Iskander, from

Waveland Press.

3

7/12

Spherical Coordinates = sin cos

= sin sin

= cos

=p

2 + 2 + 2

= cos 1 p2 + 2 + 2

= tan 1

Field Component Transformations

=

sin cos sin sin cos

cos cos cos sin sin

sin cos 0

=

sin cos cos cos sin

sin sin cos sin cos

cos sin 0

Di erential Elements

Di erential d~l = d~S = one of =

Elements ba + ba + sin ba¡2 sin

¢ba ( sin ) ba ( ) ba 2 sin

Note: Þgures on this page are reproduced from Electromagnetic Fields and Waves, 2nd Edition, Iskander, fromWaveland Press.

3

8/12

Vector OperatorsCartesian Cylindrical

Gradient ~∇ =

ba +

ba +

ba

ba + 1

ba +

ba

Divergence ~∇ · ~F =

+

+

1

()

+1

()

+

Curl ~∇× ~F =

¯¯¯ba ba ba

¯¯¯

1

¯¯¯ba ba ba

¯¯¯

Laplacian ∇2 = 2

2+

2

2+

2

21

µ

¶+1

22

2+

2

2

Spherical

Gradient ~∇ =

ba + 1

ba + 1

sin

ba

Divergence ~∇ · ~F = 1

2¡2

¢

+1

sin

(sin )

+

1

sin

Curl ~∇× ~F =1

2 sin

¯¯¯ba ba sin ba

sin

¯¯¯

Laplacian ∇2 = 1

2

µ2

¶+

1

2 sin

µsin

¶+

1

2 sin2

2

2

4

9/12

ȝo = 4ʌ x 10-7 H/m 1/Po = 8 x 105 m/H Ko = 120S : Ho=8.85x10-12 F/m ar ax =sin ș cos I aș ax =cos ș cos I aij ax =-sin I ar ay =sin ș sin I aș ay =cos ș sin I aij ay =cosI

ar az =cos ș aș az = -sin ș aij az =0

ax aȡ =cos I ax aI = -sin I ax az =0 ay aȡ =sin I ay aI = cos I ay az =0

az aȡ = 0 az aI 0 az az =1 x = ȡ cos I y = ȡ sin I

Cartesian Cylindrical Spherical

dxax+dyay+dz az dȡaUȡdIaIdzaz dr ar+ r dș aT + r sinș dIaI

(dydz)ax (ȡ dI dz)aU (r2 sinș dș dIar (dxdz)ay (dȡ dz)aI (r sinș dr dIaT (dxdy)az (ȡ dȡ dI)az (r dr dș)aI

dx dy dz ȡ dȡ dI dz r2 sinș dr dș dI

r = 2 2 2x y z x = r sin ș cos I

ș = cos-1 2 2 2

z

x y z

y = r sin ș sin I

I = tan-1 y

x

z = r cos ș

aaa ***zyx z

VyV

xVV

ww

ww

ww

aa1

a ***zz

VVVVww

Iww

UUww

IU

asin1

a1

a ***IT Iw

wTTw

www V

rV

rrVV r

»»¼

º

««¬

ª

ww

w

w

ww

zyzyx EE

xEE

*

»»

¼

º

««

¬

ª

ww

w

w

ww

z

zEE1E1EIUU

UU

IU*

»»»

¼

º

«««

¬

ª

ww

w

w

ww

ITT

TT

IT Esin1Esin

sin1E1E

2

2 rrrr

rr*

»¼

º«¬

ªww

ww

ww

2

2

2

2

2

22

zV

yV

xVV

»¼

º«¬

ª

w

ww

www

ww 2

2

2

2

22 1)(1

zVVVV

IUUU

UU

»»»»

¼

º

««««

¬

ª

w

w

w

w

w

w

ww

ww

2

2

22

22

22

sin1

)(sinsin1)(1

IT

TT

TTV

r

Vrr

Vrrr

V

zyx AAA

Azyx

aaa zyx

ww

ww

ww

u

z

1

AAA

A

IU

IU

U

IU

U

U z

aaa z

ww

ww

ww

u

IT

IT

T

IT

T

T

AAA

A

sin

sin

sin1

2

rr

r

arraa

r

r

r

ww

ww

ww

u

10/12

Magnetostatics materials

³u

L

R

R

alIdH 24S

*&&

T=m x B I

NL <

2

12112 I

NM < ³ BxlIdF

&&&

Pr=1+Fm

Lorentz force Continuity

Electrostatics Materials

C=Q/V

)/( SlR V

Hr=1+Fe

Boundary conditions D1n-D2n=Us Et1=Et2 B1n-B2n=0 a21 x (H1-H2)=Js

a21 · (D1-D2)=Us a21 x (E1-E2)=0 a21 · (B1-B2)=0 a21 x (H1-H2)=Js

Js is surface current (also denoted as K)

Maxwell’s equations

³ x³ xww

Ls

emf ldBxvsdtBV

&&&&&

)(

Note: for static fields, time derivatives are zero. In phasor form, time derivatives become jZterms.

Time-varying fields: UPW

Vector wave equations for time-harmonic fields in lossy medium. With lossless medium or free space, D=0.

ESO 2

oror HHPPZE

fT /1

or

orHEHHPPK ||/||

EZ

pv

Uniform plane wave in lossless medium. For free space, Pr=1 and Hr=1.

xz

o azteEtzE &&)cos(),( EZD

yzo azteEtzH &&

)cos(||

),( KD TEZ

K

One example of E and H fields in lossy medium.

»»¼

º

««¬

ª»¼

º«¬ª 11

2

2

ZHVPHZD

»»

¼

º

««

¬

ª»¼

º«¬ª 11

2

2

ZHVPHZE

4/121

/||

»»¼

º

««¬

ª¸¹·

¨©§

ZHV

HPK

ZHVTK 2tan

ESO 2

G D

Parameters describing UPW in lossy medium.

2ZPVED

o45 VZPK

good conductor: (VZH>>1)

))()(Re(21)( * zHzEzP ssavg

&&&u

P(z,t)=E(z,t) x H(z,t)

Poynting vector

12

12KKKK

* 12

22KK

K

T Transmission and reflection coefficients: normal incidence

it

itTKTKTKTK

coscoscoscos

12

12||

*

it

iTTKTK

TKcoscos

cos2

12

2||

ti

tiTKTKTKTK

coscoscoscos

12

12

*A

ti

iTTKTK

TKcoscos

cos2

12

2

A

Transmission and reflection coefficients: oblique incidence

ri TT

22

11

sinsin

HPHP

TT

i

t

VE &

Crr

dvr

r

³ |'|4')'(=)rV(

o

v&&

&&

HSHU

³ x CldEV&&

R2ro

v aR4

=)r(E *&&³

HSHU dv

³ ³ v

vr dvdE UHHs

o s*&

³³³ xww

x xssc

sdDt

sdJldH &&&&&&

Bt

E&&

ww

u

vD U x&

Dt

JH&&&

ww u

0 x B&

³ x dvDEWE&*

21

³ x dvBHWM&&

21 MJM

&& u

tJ v

ww

xU&F=q(E+vxB)

EP oe&&

HF

HM m&&

F

0 ³ xs

sdB &&

022 ss EE&&

J

022 ss HH&&

J

EDJ j

HUvV 2 02 V

pvP U x&

psnaP U x&&

³ ³ xww

xc s

sdBt

ldE &&&&

11/12

Waves and T/R – continued

||1||1

**

s

21

2sinHH

HT

B

1

2tan HHT B

1

2sin HHT c

)/( wLRac VG Transmission lines:

TD cos2

|| 22

z

o

oave e

ZV

P

Waveguides:

2

1 ¸¹

ᬩ

§

ffc

gOO

2

1

'

¸¹

ᬩ

§

ff

uuvc

pp

2

1' ¸¹

ᬩ

§

ff

uuv cgg

2'uvv gp

Distortionless transmission lines: R/L=G/C

RG D LCZE

GRRo

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