1- A quarter-wave monopole (as shown in the figure below) situated above a perfectly conducting ground plane is excited by a sinusoidal source at its

Size: px

Start display at page:

Download "1- A quarter-wave monopole (as shown in the figure below) situated above a perfectly conducting ground plane is excited by a sinusoidal source at its"

Elmer Roy Boone
5 years ago
Views:

1 1- A quarter-wave monopole (as shown in the figure below) situated above a perfectly conducting ground plane is excited by a sinusoidal source at its base. Find its radiation pattern, radiation resistance, and maximum directivity. Z = j1.5 Ω R = Ω, D = 3.8 in r

2 3- Consider two half-wave linear wire antennas aligned, so that their axes are offset by 60. If the transmit antenna radiates 10 W of power at an operating frequency of 1 GHz, find: (a) the max power that will be available at the receiver antenna when they are 1 km apart, (b) the magnetic and electric field strength at the position of the receive antenna.

3 6- A linear infinitesimal dipole of length L and constant current is placed vertically a distance h above an infinite ground plane. Find the first five smallest heights (in ascending order) so that a null is formed (for each height) in the far-field pattern at an angle of 60 from the vertical.

4 6- Yarıçapı a=0.001λ olan bakır (σ = 5.7 x 10 7 S/m) telden yapılan ve 175 MHz te çalışan yarım dalga boyu bir dipol anten 300 ohm luk bir iletim hattına bağlanmıştır. Antenin 3 kazancını (G 0 ) hesaplayın. Antenin radyasyon yoğunluğu U( θ) = Asin θ olsun. 6- A resonant half-wave dipole with radiation resistance R r =73 ohm is connected to a transmission line with characteristic impedance is 300 ohm. The dipole is made of copper with σ=5.7x10 7 S/m and f=175 MHz. Calculate the dipole gain if its field pattern can be 3 approximated by U( θ) = Asin θ. The dipole radius is a=0.001λ. λ = = = = = m, L 0.86 m, a 0.001λ m π fµ 0 L 0.86 Rs = = RL = Rs = R s = Ω σ πa π Γ = = 0.6 e = 1 Γ = 0.64, e = 1 e = e e = r cd 0 r cd 73 + RL π π π 3 4 Prad = A sin θsinθdθdφ πa sin θdθ A 0 = = 0 0 4πU 16 D = = G = e D = = 1.08 max Prad 3π 3π 4 1- Yarıçapı 6.35 mm alüminyum (σ = 3.5 x 10 7 mhos/m) telden yapılan ve 100 MHz te çalışan yarım dalga boyu bir dipol anten 50 ohm luk bir iletim hattına bağlanmıştır. Antenin radyasyon, yansıma ve toplam verimini hesaplayın. 1- A half-wave dipole made of aluminum wire (σ = 3.5 x 10 7 mhos/m) with a diameter of 6.35 mm (0.5 in) is operating in free space at 100 MHz. The antenna is connected to a transmission line with a characteristic impedance of 50 ohm. Determine the radiation, reflection and the overall efficiency and the gain of the antenna. ωµ 3 L λ f = 100 MHz λ=3 m, Rs = = Ω RL = Rs = Rs = 0.165Ω σ πa 4πa Rrad 73 ecd = = = = 99.87% 1 Rrad + Rohmic Z Z 73 + j % A 0 Γ = = Γ = er = Γ = = ZA + Z j e = e e = = 86.5% G = e D = (0.9987)(1.643) = 1.64 =.148 db 0 cd r cd 0

5 5- A z-directed electric dipole of current moment II resides on the y axis at y = y. In the far zone, this source causes an electric field given below. Derive an expression for the total power radiated by this dipole. Set up the proper expression for power, simplify it, but it is not necessary to perform the last integration unless you wish to do so. jkr e jky sinθsinφ Eθ = jkηil e sin θ. 4πr 4- The magnetic radiation field of a short dipole oriented in the z direction, with dipole moment m is given below. Find the radiation resistance for length λ 50 l λ Dipol momenti m olan ve manyetik alanı aşağıda verilen z yönündeki kısa dipol antenin λ 50 l λ10 iken radyasyon direncini hesaplayın. jkr ˆ msinθ e Hr ( ) = H ( ) j ˆ φr φ = φ. λ r l m = I( z) dz = I0l, Pt = ηda = I0R l H S m π π m I0l m π 3 1 πη l t = φ sin θ sinθ θ π sin θ θ 0 r r 0 = = = 0 0 P d r d d I R R λr 4λ 6 λ r

6 1- A linear dipole antenna of length 10 cm is oriented with its axis along z-direction and driven with a peak current of 10 A at a frequency of 1 MHz. a) Determine whether this is a Hertzian or short dipole antenna at this frequency. l c l 4 = 10 cm, λ = = 300m = Hertzian Dipole! f λ b) By what factor does the radiated electromagnetic field propagating through free space change (if at all) between two locations situated at R1= 50 km and R= 150 km from the antenna? S 1 1 E1 R 150 but E = = = 3 R R E R 50 1 c) What is the value of power density at R1=50 km. i) along the z-direction ii) perpendicular to z-direction (in the z=0 plane) 15πI0 l S( r, θφ, ) = sin θ r λ i) For θ = 0 S = 0, (along the z-direction) ii) For θ = 90 S = 0.1 pw m d) Is the state of polarization the same for waves emitted at arbitrary angles θ? If so, state what it is. If not, indicate why not. The radiation is linearly polarized in all directions. The axis of polarization lies perpendicular to ˆk in the kz-plane. e) What is the total (average) power radiated by the antenna? π π15πi 0 l l π 3 TOT 0 S P = S( r, θφ, ) dω = sin θr sinθdθdφ = 30π I sin θdθ = W r λ λ

7 1- Answer the following questions: a) Why is a half-wave dipole more useful than a short-length dipole, even though the directivities of both antennas are nearly the same? Greater efficiency because: impedance of half-wave dipole easier to match to common system impedance and Rrad>>Rdiss for half-wave dipole. b) Roughly sketch the current distribution along each arm of a dipole whose total length is 5λ 8. c) A 5 mm long z-oriented dipole is driven by a 3 GHz current of 1.5 A. What is the amplitude of the E field at a distance of m along the + x axis? What is the corresponding power density and direction of power flow at that location? λ 4π l = m, λ = 0.1 m l = short dipole, kr = 16 >> 1 Farfield 0 λ π l E 10π Vm 66.3mW m along the -axis λ 8πr η0 E = I = P = = + x

8 - Calculate the radiation efficiency for a λ/80 Hertzian dipole operating at 3 MHz in free space. Assume that the dipole is made of copper wire (σ = S/m) with a = 1.04mm. R r dl 1 = 80π 80π = = 0.13 Ω, λ m ( ) 4 dl Ω s π µσ L s R = f = Ω R = R = = Ω πa π( m) Rr 0.13 ecd = = = = 58.3% R + R r L - A Hertzian dipole of length L=m operates at 1MHz. Find the radiation efficiency if for copper σ=57ms/m and radius a=1mm δ = = =, πµ fσ π 4π L RL = = = = 0.084Ω σa σπaδ π 10 (10 15) loss o ( l) I av ( l) I av l I av µ β β Rr = 0 80π 80 π ( ) εo 6π I = o 6π I = = = Ω o λ I, o 1 Prad Prad Rrad erad = = = = = 0.9, or 9%. seemingly, this has Pin Prad + Ploss Rrad + Rloss 0.1 nothing to do with the condition of max power in the antenna circuit

9 3- Serbest uzayda yarim dalga boyu dipol anten 100 V luk (peak) ve 50 ohm luk bir kaynağa bağlanmıştır. Dipol antenin giriş empedansı 73+j4 ohm dur.antenden yayılan toplam gücü ve r = 100 m, θ = 45 için elektrik alanı hesaplayın. 3- A half-wave dipole in free space is connected to a 100 V (peak) source having a 50 ohm impedance. The input impedance of the dipole is 73+j4 ohm. Find the total radiated power and the magnitude of the E- field at r = 100m, θ = 45. Magnitude of the current into the halfwave dipole at the feed point is I I = = A(peak), Prad = = 1.58 W j4 jk1000 π e cos( cosπ 4) Eθ( r = 100m, θ = 45 ) = j60i0 = 0.90Vm 1000 sinπ cm uzunluğunda olup z-yönünde ve merkezi orijinde olan bir dipol antendeki fazor akım A ve λ = 7.6 m dir. r = m, θ = 70, φ = 135 noktasında elektrik ve manyetik alan değerlerini ve güç yoğunluğunu hesaplayın. 1- A 38 cm long dipole oriented and centered on the z-axis is driven by a phasor input current of A at λ = 7.6m. Find the E & H fields at r = m, θ = 70, φ = 135. Also determine the time average Poynting vector at this location. l λ = = 0.05 = use triangular current distribution D (0.38) 0.038m m λ = 7.6 = r = m, θ = 70, φ = 135 is in the far field π π () ( ) 7.6 jkr e ki0le E sin θ = jη θ = j sin 70(0.38) = V m 8πr 8 π() H φ jkr ki0le Eθ 1 = j sinθ = = A m S = Re( E H ) = aˆ r W m 8πr η

10 1- The expression for the far electric field of two Hertzian dipoles at right angles to each other (below figure), fed by equal-amplitude currents with 90 phase difference, is given below. Find the far zone magnetic field, the radiation intensity, the power radiated, the directive gain, and then directivity. jβη( Idl) jβr E = e [(sinθ jcosθcos φθ ) ˆ + ( jsin φφ ) ˆ] 4πr

11 6- A half-wave dipole placed symmetrically along the z-axis. a) Determine the vector effective length (in terms of λ) b) Determine the max value of the vector effective length and angle θ where this occurs. c) Determine the ratio (in %) of the maximum effective length to its total physical length.

12 4- Find the directivity pattern and maximum directivity of a monopole antenna that has jkr electric field given by E = sinθe r. θ 4- Consider two identical z-directed electric dipoles of current moment II which reside on the x-axis, one at x = d and the other at x = -d. Determine an expression for E θ (r,θ) under the assumption that r is very large. Begin with the expression for E θ in the far zone due to a dipole at the origin and show your steps.

13 - Find the shortest length of a center-driven dipole antenna such that there would be no radiation in the direction that makes a) 90 o with the antenna axis; Express the dipole length in terms of the operating wavelength. b) 75.5 o with the antenna axis. Express dipole length in terms of the operating wavelength.

14 - Consider a dipole antenna of length l = 5λ. a) Analytically determine the directions of the nulls in the radiation pattern. l kl kl cos ( cosθ) cos ( ) n + 1 kl = 5λ kl = 5 π, E = = 0 cos ( cosθ) = 0 cosθ = n sinθ 5 n = 0 θ = 78.5 n = 1 θ = 53.1 n = θ = 0 n = 1 θ = n = θ = 180 n = 3 θ = 0 b) Generate a plot of the normalized radiation pattern.

15 3- Assume that the antenna of a 900 MHz handheld mobile phone can be modeled as a z- directed monopole of length 4 cm on the on the infinite ground plane. The input power to the antenna is 1 W and the radiation efficiency is 80%. Find the amplitude of the radiated electric field at the base station located 1 km away from the antenna over a hill of height 300 m. l = 4cm = l = 8cm P 0.8 W. From Stutzman m d rad =, P rad = U( θφ, ) dω U( θφ, ) = U F( θφ, ), m Ω = F( θφ, ) dω P rad U m = Ω 0 Um F( θφ, ) F( θφ, ) rad E U( θφ, ) P η = r = r = Ω r E = F ( θφ, ) η0pr ad r Ω For a center-fed dipole antenna Eθ = jωa z ( r )sinθ A z kl kl cos cosθ cos µ I e, k πr sin θ jkr 0 F( θφ, ) = πl πl cos cosθ cos λ λ sinθ f = 900 MHz λ = 33.33cm, l = 8cm=0.4 λ, r = (1000) + (300) = m, θ = tan = 73.3 Fθ ( = 73.3 ) = π Ω =, Dmonopole( l = 4cm) = Ddipole( l = 8cm) D monopole Directivity increases as the length of a dipole increases from l << λ to l = λ. For l = λ we have D = Therefore D 1.55 D monopole = 3.1. Therefore E π = = 3mVm π

16 5- A very nice property of linear-reciprocal systems is superposition, and we can use it to calculate quite accurately the pattern of any dipole. For example, consider a λ-long centerfed dipole, oriented along the z-axis, with a peak current of I o. a) Draw the current distribution on the dipole. b) This antenna can be modeled by four distinct short dipoles located at z = ± λ 4 and z = ± 3λ Determine the current distribution of each of these 4-dipoles.. Calculate and draw (polar/linear) the E-plane pattern of the l-long dipole using the approximation of 4-distinct short dipoles. 3. What is its peak directivity at what angle? c) Compare the pattern calculated in (b) with the exact pattern of the dipole. They are identical

17 4- Compute and plot the normalized far-field pattern F(θ) of a 3λ/-length dipole antenna occupying the segment ( 3λ /4, 3λ/4) of the z-axis and has a current distribution given by I( z) = Imsin β ( 3λ 4 z ).

18 8- A linear infinitesimal electric dipole of length l and constant current is placed vertically a distance h above an infinite, perfectly conducting electric ground plane. Find the first five smallest lengths (in wavelengths, in ascending order) so that a null is formed (for each height) in the far-field pattern at an angle of 40 o from the vertical - A dipole has length of L=1 m and is operated at 00 khz. a) Find the radiation resistance assuming a linearly-tapered current distribution, i.e. I = I0 L z, L z L. (make short dipole assumption). 8 3 π L π 1 5 λ = c f = = 1500 m, Rr = η 10π = = Ω λ b) Calculate the beam solid angle Ω A. (make infinitesimal dipole assumption).

19 9- A vertical infinitesimal electric dipole is placed at a distance h = 1.8λ above an infinite, perfectly conducting electric ground plane. a) Determine the angles (in degrees from the vertical) where the array factor of the system will achieve its maximum value b) Determine the angle θ where the max of the total field will occur c) The relative (compared to its maximum) field strength (in db) of the total field at the angles determined in part (a)

20 1- An infinitesimal x-directed electric dipole is placed horizontally a distance h above an infinite, perfectly conducting electric plane (x-y plane). Determine the E and H far-zone fields.

21 jkr e 3- The vector potential due to an x directed electric dipole at the origin is A = µ x ˆIl 4πr where xˆ = sinθ cosφ rˆ + cosθ sinφθˆ sinφ φ ˆ jkr. If we define g( r) = e (4 π r) then the components of the vector potential can be written as Ar = µ Ilg( r) sinθ cosφ, Aθ = µ Ilg( r)cosθ sinφ, and Aφ = µ Ilg( r)sinφ. From the above, determine the far-zone magnetic field. (There are two ways to answer this question. One involves computing the farzone magnetic field from H = Α by performing all the partial derivatives 1 and, µ subsequently, taking the limit of the result as r approaches infinity. The other involves 1 writing H = Α in terms of the components of the vector potential and eliminating µ various components on the basis of their limiting ( r ) values. The latter requires a little more thought than the former but results in far less wear and tear on the pencil and fingers.)

22 8- For a very thin center-fed half-wave dipole lying centered at the origin along the z-axis: a) Find the charge distribution on the dipole if current distribution is I ( z) = I0 cos( β z). b) Repeat part (a) for I ( z) = I0(1 4 z λ), z λ 4.

23 3- A 15 MHz uniform plane wave having a peak electric field intensity E 0 = 0.05 V/m is incident on a.5 λ long dipole at an angle θ. Assume lossless dipole. a) Plot the current on the dipole. b) Find the angles of incidence θ that will give a zero open circuit voltage Voc at the terminals of the dipole. c) Plot the far-field E plane pattern of the dipole (Polar plot).

24 d) Find the maximum effective length of the dipole. e) If the dipole is connected to a matched load, what is the maximum power P L that can be delivered to the load? Or

25 3- Two half wave dipoles are arranged as shown in Fig. The first antenna is transmitting 300W at 300MHz. Find the open circuit voltage induced at the terminals of the receiving second antenna and its effective area. Find available power at antenna two. The open circuit voltage at antenna will be Vo = he E1. The antenna effective height is L sinθ L jβ z cosθ Im h ( θ ) = Im sin β ( z) e dz F( θ ) I = β I F( θ ) o L L L cos( β cos θ ) cos( β ) sinθ =, or after substitution, for a half wave dipole L=λ/ and where θ θ 90o o = =, h π I cos( cos θ) m Im c = = =. β I sinθ β I π f o o The E1 in place of antenna will be L L jβ z cosθ j Io m. L r µ o sinθ 60 E1( θ ) = j I sin β ( z) e dz = F( θ ) ε 4π r o For a half wave dipole, π cos( cos 60) F( θ 1) =. So the electric field will be sin 60

26 µ o π ε cos( cos60) o E( θ 1) =, or in the terms of effective antenna 1 height, π r sin 60 µ o β π ε cos( cos 60) o E( θ 1) = he1, where he1 4π r = β sin 60. We can calculate this to be π 1 60I cos( ) 1 E( θ 1) = = 0.6I1 = 0.49I The current I1 can be found from the given power P1, 1 P 300 P = R I I = = =.87A 73Ω 73 1 λ Finally, Vo = he E1 = = V. π The available power comes from the assumption that the antenna has a matched resistance for the rest of the circuit, 1 1 V ( he E1) Pa = Rant I = Rant ( ) = = = 344µ W. R 8R 8 73 ant ant

27 3- A half-wave dipole at GHz is connected to a generator with Z g =50 Ω. The generator can deliver a maximum of 1W of power to a load under impedance matched conditions: a) Calculate the power delivered to the λ dipole. Z Z L 0 Γ = = = 0.186, Pincident = 1W Pdelivered = (1 Γ ) Pincident = 965mW ZL + Z b) Calculate the current at the antenna terminal and the far-field electric field at a distance of 00m from the antenna (observation point) in the direction of maximum radiation. 1 Prad Prad = I0 Rrad I0 = = = 16.6 ma R 73 rad π f = GHz, λ= = 0.15m, k = = 41.89rad/m 9 10 λ cos( πcos θ) jkr e o Eθ = j60i0 max radiation at θ = 90 sinθ R j(41.89)(00) e 3 j8378 Eθ = j60(0.166) = j e 00 c) If a small solenoid with N = 10 turns is placed at the observation point: i) How would you orient the solenoid to pick up the antenna radiated field? ii) If the radius of the solenoid loops is 1 cm, calculate the induced rms voltage across the terminals of the solenoid.

28 5- A half-wave dipole along the z-axis is described by the constant current density Jr ( ) = zi ˆ cos( kz) for λ 4 z λ 4. 0 a) Find the magnetic vector potential in the far field. jkr 4 jkr µ 4 0I0 e λ jkz cos 0I0 e λ θ µ 1 jkz jkz jkz cosθ A zˆ cos( kz) e dz zˆ ( e e ) e dz 4π r = λ 4 4π r + λ 4 jkr π µ 0I0 e cos( cos θ) A z ˆ kπ r sin θ (b) Find the electric far field. (c) Find the radiated power. Use the fact that πcos[( π)cos θ] dθ sinθ (d) Find the antenna gain pattern.

Thus the current density J is described as Recall from lecture that, in the far field, the effect of moving an antenna is

29 - An antenna operating at 100 MHz is composed of a system of four elementary dipoles all located on and parallel to the z axis, 50 cm apart. Thus the current density J is described as Recall from lecture that, in the far field, the effect of moving an antenna is calculated quite ( 1) simply, namely if Jr ( ) Er ( ) then Jr ( r1) e jβrr Er ( ). Using this fact, address the following questions: a) compute the vector potential A and electric field E. b) compute the Poynting vector S, and intensity U.

30 c) compute the normalized power pattern, plot it. d) compute the directivity. In what direction(s) does the peak occur?

31 3- In the previous problem, move all the antenna elements in the positive direction by an amount 75 cm, and repeat the calculations. Which answers change? Which don t? Why?

32 - What antenna current is required so that the antenna will radiate 0 W if a) the antenna is a m long dipole antenna operating in the AM broadcast band at 1 MHz. f = 1 MHz, λ = c f = 300 m, l = m < λ 50 = 6 m short dipole: P = Ppoint dipole 4 z P = 40 π I I = A 4 λ b) the antenna is a 1 m long dipole antenna operating in the FM/TV broadcast band at 150 MHz. f = 150 MHz, λ = c f = m, l = 1 m = λ half-wave dipole: P = 36.6 I I = 0.74 A 0 0 c) For the above two antennas, find the maximum distance from the antenna in the horizontal plane at which the FCC standard of 5 mv/m field strength is maintained. Assume antennas are vertical. η0k( I0 ) z η0ki0 z Short dipole E = sinθ 5 mv/m, r sinθ 4π r 8π 0.05 η0ki0 z Horizontal plane θ = π, sinθ = 1, r = 1.7 Km 8π 0.05 Half-wave dipole π ( θ ) 0I cos 0 cos η E = 5 mv/m π r sinθ, θ = π, sinθ = 1, cos 0 π cos cosθ = 1 θ =, ( ) η0i0 r = 1.8 Km π 0.05

33 - Consider an elementary, z-directed electric dipole of current moment II on the x-axis at x=d. Assume that d is very large. a) What is the direction and magnitude of the electric field at the origin due to this dipole?

34 b) Let the dipole continue to be z-directed but move it to the point (d,d,0) in the xy plane. What is the direction of the dipole s electric field now? Alternate solution method by reciprocity

35 3- Assume a cross dipole antenna as shown below figure. It is made of two half-wavelength dipoles along z and y directions. Find the radiated fields assuming that the antennas are excited by current sources with equal amplitude and phase. What is the polarization of electric field in the far-zone region in the xz plane?

37 3- The transmitting antenna of a radio navigation system is a vertical metal mast of height h=40 m insulated from the earth. A source operating at f=180 KHz sends a current with amplitude I m =100 ma into the base of the mast. Assuming the current amplitude in the antenna to decrease linearly toward zero at the top of the mast and the earth to be a perfectly conducting plane, determine (a) the effective length of the antenna (The effective length of a linear antenna of length L is defined as the length l e of an equivalent linear antenna with uniform current distribution I(0) such that it radiates the same far-zone fields in the H-plane, i.e. l e 1 I (0) L = I( z ) dz.) L (b) the maximum field intensity at a distance of R=160 km from the antenna, (c) the time-average radiated power, (d) the radiation resistance.

38 4- A thin linear dipole of length L is placed symmetrically about the z-axis. Find the far-zone electric and magnetic fields assuming the current distribution is given by

40 7- A dipole antenna, with a triangular current distribution, is used for communicating with a submarine at f =110 khz. The overall length of the dipole is L = 30 m and the wire radius is a = 1.5 m. Assume a loss resistance of 3 Ω. (a) Evaluate the input impedance of the antenna, including the loss resistance. The input reactance can be approximated by

41 (b) (3 points) Evaluate the radiation efficiency of the antenna Hint: The far-field pattern of the above dipole has a term sin (0.5βLcosθ). Use a one term Maclaurin (Taylor) series expansion of this function before you use it in part (a).

42 9- Consider the double-dipole antenna. The dipoles are oriented along the x-axis and are spaced a distance d from each other on the y-axis. The dipoles are fed in phase and with the same amplitude (for cases 1-4). The dipole length is 0.65 lambda. a) Calculate the antenna pattern (normalized). Define the E-plane and the H-plane, and write their patterns mathematically.

43 b) For d=0.5 lambda, plot the E-plane and H-plane co-pol and x-pol patterns (linear in power, radial in angle). Plot also the 45 degree co-pol and x-pol patterns. Clearly label the polarization in the E-plane and H-plane patterns and on the z-axis.

44 c) The double-dipole is now placed a height h (h=0.5 lambda) above the ground plane (x-y plane). Calculate the new pattern, determine the 3-dB beamwidth, and estimate the directivity of the antenna (do not use the equation in Balanis). What do you think happens to the crosspolarization level in the presence of the ground plane (assume the ground plane is infinite in extent)?

45 d) Calculate the driving point impedance for one dipole antenna (in the double-dipole antenna) in the presence of the ground plane. Use the graphs in the notes. e) Which f/d reflector would you choose to match the antenna of case (3) above (with a finite but small ground plane). Clearly state your reasons. Also, draw a realistic feeding mechanism for the double-dipole antenna.

Linear Wire Antennas. EE-4382/ Antenna Engineering

Linear Wire Antennas. EE-4382/ Antenna Engineering Linear Wire Antennas EE-438/5306 - Antenna Engineering Outline Introduction Infinitesimal Dipole Small Dipole Finite Length Dipole Half-Wave Dipole Ground Effect Constantine A. Balanis, Antenna Theory: