Microstrip, Printed and Dielectric Antennas

Introduction

Microstrip, printed and dielectric antennas enable low-profile integration with radio-frequency (RF) circuits, enclosures and phased-array front ends. They are widely used in satellite terminals, radar, wireless access points, automotive sensors, wearable devices and Internet-of-Things equipment. Their apparent simplicity is deceptive: substrate dispersion, conductor and dielectric losses, surface waves, connector transitions, packaging and fabrication tolerances can materially change measured performance.

Design therefore requires a hierarchy of models. Closed-form transmission-line and cavity models provide physical insight and initial dimensions. Network and array models describe coupling and beam steering. Full-wave electromagnetic simulation then captures finite grounds, feeds, losses, housing interactions and radiation boundaries. Finally, prototype measurements close the model–manufacture–test loop.

Learning Objectives

After completing this chapter, the reader should be able to:

• design rectangular microstrip patches using transmission-line and cavity models;
• explain patch fields, resonant modes, losses, bandwidth and feeding methods;
• tune printed monopoles and slot antennas using physically correct scaling;
• formulate consistent array-factor, scan-phase and active-impedance equations;
• analyse dielectric resonator antenna modes through eigenvalue and driven full-wave models;
• design corporate and series feed networks;
• evaluate fabrication, packaging and tolerance effects; and
• establish a defensible simulation, optimization and measurement workflow.

1. Planar-Antenna Technologies

A microstrip patch antenna consists of a conducting radiator above a ground plane, separated by a dielectric substrate. Radiation occurs principally from fringing fields at discontinuities such as open patch edges.

A printed monopole is usually etched on one side of a substrate and excited against a finite printed ground. Its resonant current path, ground geometry and feed transition determine its impedance and radiation.

A slot antenna radiates through an aperture in a conducting surface. Slots may be excited by microstrip lines, coplanar waveguides, waveguides or coupling apertures.

A dielectric resonator antenna (DRA) uses a low-loss, high-permittivity dielectric body as the principal resonator. Unlike a metal cavity, its boundaries are open, so stored fields leak into free space and produce radiation.

Printed structures are easy to manufacture and integrate but can suffer conductor loss, dielectric loss and surface-wave excitation. DRAs can provide higher radiation efficiency at microwave and millimetre-wave frequencies, although their three-dimensional assembly and material tolerances require careful control.

2. Rectangular Microstrip Patch Design

2.1 Transmission-line interpretation

A rectangular patch of physical length \(L\), width \(W\), substrate thickness \(h\) and relative permittivity \(\varepsilon_r\) can be interpreted, near its fundamental resonance, as two radiating slots separated by a low-impedance transmission-line section. This model assumes a thin substrate, a sufficiently large ground plane and operation near the dominant mode.

A practical initial width for the dominant resonance at frequency \(f_r\) is

\[

W=\frac{c}{2f_r}\sqrt{\frac{2}{\varepsilon_r+1}},

\]

where \(c\) is the speed of light in free space. This width is a design choice that generally gives useful radiation conductance and efficiency; it is not itself the resonance condition.

For \(W/h\gtrsim 1\), an approximate effective relative permittivity is

\[

\varepsilon_{\mathrm{eff}}=\frac{\varepsilon_r+1}{2}

+\frac{\varepsilon_r-1}{2}\left(1+12\frac{h}{W}\right)^{-1/2}.

\]

A commonly used open-edge extension is

\[

\frac{\Delta L}{h}=0.412

\frac{(\varepsilon_{\mathrm{eff}}+0.3)(W/h+0.264)}

{(\varepsilon_{\mathrm{eff}}-0.258)(W/h+0.8)}.

\]

The effective resonant length of the dominant mode is approximately

\[

L_{\mathrm{eff}}=L+2\Delta L

=\frac{c}{2f_r\sqrt{\varepsilon_{\mathrm{eff}}}},

\]

so that

\[

L=\frac{c}{2f_r\sqrt{\varepsilon_{\mathrm{eff}}}}-2\Delta L.

\]

These formulas provide starting dimensions rather than final manufactured values. They neglect finite-ground effects, copper thickness, surface roughness, anisotropy, connector parasitics and detailed feed geometry.

2.2 Worked design: a 2.45 GHz patch

Design an initial patch on a substrate with \(\varepsilon_r=4.4\), \(h=1.6\) mm and target frequency \(f_r=2.45\) GHz.

First,

\[

W=\frac{3\times10^8}{2(2.45\times10^9)}

\sqrt{\frac{2}{5.4}}\approx37.3\text{ mm}.

\]

Thus \(W/h\approx23.3\), giving

\[

\varepsilon_{\mathrm{eff}}\approx4.08.

\]

The fringing extension is

\[

\Delta L\approx0.535h\approx0.856\text{ mm}.

\]

The effective length is

\[

L_{\mathrm{eff}}=\frac{3\times10^8}

{2(2.45\times10^9)\sqrt{4.08}}

\approx30.3\text{ mm}.

\]

Therefore,

\[

L\approx30.3-2(0.856)=28.6\text{ mm}.

\]

A full-wave model should subsequently include the actual board outline, feed, metal thickness and connector. The relatively lossy FR-4 substrate may produce lower gain and efficiency than a low-loss microwave laminate.

3. Cavity Model of the Patch

The cavity model treats the region between the patch and ground as a thin dielectric-filled cavity. The patch and ground-plane surfaces are perfect electric conductor walls, imposing

\[

\hat{n}\times\mathbf E=0.

\]

The open peripheral edges are approximated as perfect magnetic walls, imposing

\[

\hat{n}\times\mathbf H\approx0.

\]

This boundary assignment is essential: the top patch and bottom ground are electric walls, whereas the four side boundaries are approximate magnetic walls because of the strong open-edge fringing fields.

For a rectangular patch with dimensions \(L\) along \(x\) and \(W\) along \(y\), the approximate cavity resonance is

\[

f_{mn}\approx\frac{c}{2\sqrt{\varepsilon_{\mathrm{eff}}}}

\sqrt{\left(\frac{m}{L_{\mathrm{eff}}}\right)^2+

\left(\frac{n}{W_{\mathrm{eff}}}\right)^2},

\]

where \(m,n\) are nonnegative modal indices, not both zero. Effective dimensions account approximately for fringing. The usual dominant rectangular-patch mode varies primarily along the resonant length and is commonly denoted \(\mathrm{TM}_{10}\).

The cavity model explains modal field distributions, polarization and higher-order resonances more effectively than the two-slot transmission-line model. Neither approximation accurately represents a strongly thick substrate or electrically small ground plane; full-wave analysis is then preferable.

3.1 Input resistance and feed position

For the dominant mode, a simplified standing-wave approximation gives the edge-to-centre variation

\[

R_{\mathrm{in}}(x_0)\approx R_{\mathrm{edge}}

\cos^2\left(\frac{\pi x_0}{L}\right),

\]

where \(x_0\) is measured inward from a radiating edge. This approximation assumes a narrow probe or inset feed and weak perturbation of the modal field. It predicts maximum resistance near the edge and near-zero resistance at the centre. Mutual conductance between radiating edges and feed parasitics modify the exact value.

4. Feeding and Impedance-Matching Methods

4.1 Microstrip-line and inset feeds

An edge-connected microstrip feed is planar and inexpensive but introduces an impedance discontinuity and may radiate. An inset moves the feed toward a lower-resistance point. The inset slot itself changes current flow and should be included in the final model.

4.2 Coaxial-probe feed

A probe can be placed at a desired impedance point without an etched inset. Its inductance increases with substrate thickness, potentially requiring capacitive compensation. Connector-pin dimensions and the ground transition must be modelled.

4.3 Aperture coupling

A feed line beneath a ground plane couples through a slot to the patch. The ground isolates the feed network from the radiator, reducing spurious feed radiation. Slot length, slot width, feed-line stub and layer spacing control the coupling.

4.4 Proximity coupling

Electromagnetic coupling between an unconnected feed line and patch can provide broad bandwidth. It requires multilayer alignment and careful control of bonding-film thickness and permittivity.

4.5 Transformer sections

For a real load \(Z_L\) and source impedance \(Z_0\), a lossless quarter-wave transformer has characteristic impedance

\[

Z_t=\sqrt{Z_0Z_L}

\]

at its design frequency. Its physical length is approximately \(\lambda_g/4\), where \(\lambda_g\) is the guided wavelength. This result is narrowband and applies when the transformed load is real at the design plane.

5. Bandwidth, Efficiency and Loss

The unloaded quality factor can be represented by

\[

\frac{1}{Q_0}=\frac{1}{Q_{\mathrm{rad}}}+

\frac{1}{Q_c}+\frac{1}{Q_d}+\frac{1}{Q_{\mathrm{sw}}},

\]

where the terms represent radiation, conductor, dielectric and surface-wave losses. For a weakly dispersive dielectric,

\[

Q_d\approx\frac{1}{\tan\delta},

\]

provided dielectric loss dominates uniformly stored electric energy. Conductor loss rises because of skin effect and surface roughness. Surface waves become more important as substrate electrical thickness and permittivity increase.

For a simple isolated resonance, fractional impedance bandwidth is broadly inversely related to loaded quality factor. The exact bandwidth depends on coupling and the specified return-loss or voltage-standing-wave-ratio criterion; therefore, \(1/Q\) should not be used as a universal exact bandwidth formula.

Bandwidth can be increased using thicker low-permittivity substrates, stacked or parasitic patches, slots in the patch, coupled feeds, multiple resonances or matching networks. These measures involve trade-offs among profile, cross-polarization, surface waves, fabrication complexity and scan performance.

Radiation efficiency is

\[

\eta_{\mathrm{rad}}=\frac{P_{\mathrm{rad}}}{P_{\mathrm{accepted}}},

\]

where accepted power excludes reflected power. Realized gain additionally includes mismatch:

\[

G_{\mathrm{realized}}=(1-|\Gamma|^2)G,

\]

where \(G\) is gain based on accepted power and \(\Gamma\) is the input reflection coefficient.

6. Printed Monopoles and Slot Antennas

6.1 Printed monopoles

A printed monopole does not generally resonate at precisely one quarter of the free-space wavelength. Its effective current path includes the strip, feed transition, fringing field and part of the finite ground plane. Bevels, steps and multiple branches can create several nearby resonances for wideband operation.

A useful first-order tuning rule follows from approximate inverse scaling:

\[

f_rL_{\mathrm{eff}}\approx\text{constant}.

\]

Thus, if a design with effective dimension \(L_{\mathrm{old}}\) resonates at \(f_{\mathrm{sim}}\), a revised dimension targeting \(f_t\) is

\[

L_{\mathrm{new}}\approx L_{\mathrm{old}}

\frac{f_{\mathrm{sim}}}{f_t}.

\]

This rule is reliable only for a modest dimensional change when one geometric dimension controls the mode.

Example. A printed monopole of length 28.0 mm resonates at 4.45 GHz, but the target is 4.18 GHz. Because the simulated frequency is too high, its effective length must be increased:

\[

L_{\mathrm{new}}=28.0\frac{4.45}{4.18}

\approx29.8\text{ mm}.

\]

Subsequent simulation is necessary because the feed and ground also affect resonance.

6.2 Slot antennas

For a thin, perfectly conducting, infinite screen in a homogeneous medium, Babinet–Booker complementarity relates complementary dipole and slot impedances by

\[

Z_{\mathrm{slot}}Z_{\mathrm{dipole}}=\frac{\eta^2}{4},

\]

where \(\eta\) is the wave impedance of the surrounding homogeneous medium. The relation is not exact for finite grounds, dielectric interfaces, finite metal thickness or asymmetric feeding.

A resonant slot often has an effective length near one half of an appropriate guided wavelength, but end effects and substrate loading must be included numerically.

Example. A slot of length 15.82 mm resonates at 6.05 GHz instead of the desired 5.80 GHz. The required first correction is

\[

L_{\mathrm{new}}=15.82\frac{6.05}{5.80}

\approx16.50\text{ mm}.

\]

The slot must be lengthened because its current resonance is above the target.

7. Dielectric Resonator Antennas

7.1 Physical operation and modes

A DRA confines electromagnetic energy through the permittivity contrast between the resonator and its surroundings. Because the dielectric–air boundary is not a conducting wall, tangential electric and magnetic fields remain continuous and part of the field radiates.

Modes are commonly classified as transverse electric, transverse magnetic or hybrid relative to a selected axis. In finite three-dimensional resonators, both longitudinal field components may be present, making hybrid labels such as \(\mathrm{HEM}\) appropriate. Modal nomenclature must always be tied to geometry and axis convention.

7.2 Source-free modal formulation

For an isotropic, source-free structure with time dependence \(e^{-j\omega t}\), Maxwell’s equations lead to the electric-field eigenproblem

\[

\nabla\times\left(\mu^{-1}\nabla\times\mathbf E\right)

=\widetilde{\omega}^{\,2}\varepsilon\mathbf E.

\]

The boundary is open and must satisfy an outgoing-wave condition. In numerical models this is implemented using a radiation boundary, boundary-integral formulation or perfectly matched layer (PML). The complex eigenfrequency is written

\[

\widetilde{\omega}=\omega_r-j\gamma,

\]

where \(\gamma>0\) is the decay rate. The radiation quality factor of a lossless-material eigenmodel is

\[

Q_{\mathrm{rad}}=\frac{\omega_r}{2\gamma}

=\frac{\operatorname{Re}(\widetilde\omega)}

{-2\operatorname{Im}(\widetilde\omega)}.

\]

When material losses are included, the resulting complex frequency gives total modal \(Q\), not radiation \(Q\) alone.

The weak finite-element form is obtained by testing with \(\mathbf v\):

\[

\int_V(\nabla\times\mathbf v)^*\cdot

\mu^{-1}(\nabla\times\mathbf E)\,dV

-\widetilde\omega^{\,2}

\int_V\mathbf v^*\cdot\varepsilon\mathbf E\,dV=0,

\]

with PML or appropriate open-boundary terms included. Curl-conforming vector basis functions are required to suppress nonphysical scalar modes.

7.3 Cylindrical modal fields

Inside a uniform cylindrical dielectric region, a longitudinal field component has the separable form

\[

\Psi(\rho,\phi,z)=J_m(k_\rho\rho)

[A\cos(m\phi)+B\sin(m\phi)]Z(z),

\]

where \(J_m\) is a Bessel function and

\[

k_\rho^2+k_z^2=\varepsilon_r k_0^2,

\qquad k_0=\frac{\omega}{c}.

\]

The remaining field components follow from Maxwell’s curl equations. At every dielectric interface,

\[

\hat n\times(\mathbf E_2-\mathbf E_1)=0,

\qquad

\hat n\times(\mathbf H_2-\mathbf H_1)=0.

\]

Enforcing these conditions and the outgoing-wave condition gives a characteristic equation

\[

\det\mathbf M_m(\widetilde\omega)=0.

\]

The matrix contains Bessel functions inside and outward-radiating Hankel or spectral-domain fields outside. A finite cylinder is open in both radial and axial directions, so simple independent radial and axial resonance formulas are approximations rather than exact solutions. Accurate mode frequencies and radiation rates should therefore be obtained from an open-boundary eigenmode solver.

7.4 Coupling and driven response

A probe, microstrip line, coplanar line, slot or conformal strip excites a mode according to field overlap. In a reciprocal formulation, coupling strength is proportional to an overlap integral such as

\[

\kappa_n\propto\int_V\mathbf J_{\mathrm{feed}}\cdot

\mathbf E_n^*\,dV.

\]

A feed placed near an electric-field maximum couples strongly to that mode; symmetry can suppress unwanted modes. Eigenmode analysis identifies natural modes, whereas a driven simulation is required to predict port impedance, realized gain and feed parasitics.

Increasing \(\varepsilon_r\) generally reduces size but increases stored energy and may narrow bandwidth. Material anisotropy, air gaps and adhesive layers can shift resonance significantly.

8. Arrays and Feeding Networks

8.1 Consistent angle convention

Consider a linear array along the \(x\)-axis. Let \(\theta\) be measured from broadside, the \(+z\)-axis, in the \(x\)-\(z\) plane. For spacing \(d\), wavenumber \(k=2\pi/\lambda\), element currents \(I_n=I_0e^{jn\beta}\), and elements numbered \(n=0,\ldots,N-1\), the array factor is

\[

AF(\theta)=\sum_{n=0}^{N-1}

I_0e^{jn(kd\sin\theta+\beta)}.

\]

For uniform amplitudes,

\[

|AF|=\left|\frac{\sin(N\psi/2)}{\sin(\psi/2)}\right|,

\qquad

\psi=kd\sin\theta+\beta.

\]

A main beam at \(\theta_0\) requires

\[

\beta=-kd\sin\theta_0.

\]

For example, an array with \(d=0.5\lambda\) scanned to \(30^\circ\) from broadside requires

\[

\beta=-2\pi(0.5)\sin30^\circ=-\frac{\pi}{2}

\]

radians per element.

To avoid visible grating lobes throughout \(|\theta_0|\leq\theta_{\max}\), a useful one-dimensional condition is

\[

d\leq\frac{\lambda}{1+|\sin\theta_{\max}|}.

\]

This condition assumes scanning in the array plane and no reliance on element-pattern suppression.

8.2 Mutual coupling and active impedance

The terminal voltage of element \(p\) is

\[

V_p=\sum_q Z_{pq}I_q,

\]

where \(Z_{pp}\) is self-impedance and \(Z_{pq}\) is mutual impedance. The active impedance is

\[

Z_{\mathrm{act},p}=\frac{V_p}{I_p}

=\sum_qZ_{pq}\frac{I_q}{I_p}.

\]

For an ideal infinite periodic array with \(I_n=I_0e^{jn\beta}\), the reference-element active impedance becomes

\[

Z_{\mathrm{act}}(\beta)=\sum_mZ_{0m}e^{jm\beta}.

\]

Thus scan angle changes impedance through the progressive phase \(\beta=-kd\sin\theta_0\). A match obtained at broadside may deteriorate during scan. Embedded element patterns and active reflection coefficients should therefore be evaluated over the complete scan sector.

8.3 Corporate and series feeds

A corporate network uses repeated power division to deliver controlled amplitude and phase to each element. It supports broad bandwidth and flexible tapering but occupies area and accumulates line, bend and divider losses.

A series feed connects elements sequentially. It is compact and can be efficient, but phase and amplitude vary strongly with frequency, causing beam squint. At millimetre-wave frequencies, feed-network loss may dominate total array efficiency. Substrate-integrated waveguide, waveguide, suspended lines or antenna-in-package technologies may then be preferable to long microstrip routes.

9. Full-Wave Modelling and Verification

A defensible full-wave workflow includes the following steps.

1. Define measurable objectives: frequency band, accepted-power efficiency, realized gain, polarization, scan range, envelope-correlation requirement and physical limits.

2. Select the formulation: finite elements suit complex inhomogeneous volumes; finite-difference time-domain methods are useful for broadband transients; method-of-moments formulations efficiently model conducting surfaces; integral-equation and hybrid methods reduce open-region volume.

3. Enter dispersive materials: use frequency-dependent complex permittivity and conductivity where data exist. Include anisotropy when supported by the laminate specification.

4. Model the complete feed: include connector pin, launch, solder, via fence, reference plane and de-embedding structure.

5. Represent finite geometry: use the actual ground, board outline, housing, fasteners, battery, display or radome when they lie in significant near fields.

6. Set open boundaries: position radiation boundaries sufficiently far from strong reactive fields, or use a validated PML. For arrays, employ periodic boundaries and Floquet ports only when infinite periodicity is intended.

7. Mesh by physics: refine narrow gaps, slot ends, feed discontinuities, metal edges, high-permittivity resonators and thin bonding layers. Curved DRA surfaces require adequate geometric order.

8. Check convergence: refine the mesh until resonance, impedance, efficiency and pattern metrics change by less than an established tolerance. A visually smooth field is not proof of convergence.

9. Perform energy checks: confirm that accepted power is consistent with radiated power plus conductor, dielectric and other dissipative losses.

10. Cross-validate models: compare full-wave results with closed-form dimensions, eigenmodes, equivalent circuits or an independent solver where practical.

11. Run tolerance analysis: vary critical dimensions and material parameters using sweeps, Monte Carlo sampling or surrogate models.

12. Correlate with measurement: apply consistent reference planes, calibration and test fixtures; update uncertain model parameters only when physically justified.

AI-based surrogate models can accelerate optimization after being trained on verified electromagnetic data. They should not be extrapolated far outside their training domain, and final candidates must be validated by full-wave analysis and measurement.

10. Fabrication, Packaging and Tolerances

Frequency sensitivity to a parameter \(p\) can be quantified locally by

\[

S_p=\frac{p}{f_r}\frac{\partial f_r}{\partial p}.

\]

A first-order uncertainty estimate is

\[

\left(\frac{\sigma_f}{f_r}\right)^2

\approx\sum_iS_{p_i}^2

\left(\frac{\sigma_{p_i}}{p_i}\right)^2,

\]

provided parameter variations are small, approximately independent and locally linear. Correlated parameters require covariance terms.

Important uncertainties include etching bias, substrate thickness, dielectric constant, loss tangent, DRA dimensions, air gaps, adhesive thickness, connector location and assembly pressure. Copper roughness raises high-frequency loss, while plating and solder mask modify effective geometry and loading.

A protective radome or housing can detune an antenna by changing near-field permittivity. Metal brackets can block radiation or excite enclosure modes. Thermal expansion and moisture absorption may shift resonance in outdoor and aerospace systems.

Best practice is to design margin into bandwidth and efficiency rather than relying on nominal dimensions alone. Test coupons should be used to characterize transmission-line impedance, effective permittivity and process bias before committing to a production array.

11. Illustrative Application Case: Packaged 5.8 GHz Sensor Array

Background

This fictional case concerns a four-element printed-patch array for an industrial 5.8 GHz sensing unit. The array must scan to \(\pm30^\circ\), fit behind a polymer cover and maintain realized gain despite manufacturing variation.

Problem

An initial infinite-array simulation met the impedance target, but the first finite prototype resonated low and exhibited poor match near the scan limits. Measured gain was also 1.4 dB below prediction.

Analysis

Inspection identified four modelling omissions: the polymer cover, adhesive beneath the cover, connector launch and finite edge elements. Active-impedance analysis showed that mutual coupling changed the element match as \(\beta=-kd\sin\theta_0\) varied. Loss measurements also showed that the laminate loss tangent exceeded the nominal catalogue value.

For scanning to \(30^\circ\), the no-grating-lobe criterion gives

\[

d\leq\frac{\lambda}{1+sin30^\circ}

=0.667\lambda.

\]

The selected \(0.5\lambda\) spacing therefore remained appropriate. The required progressive phase at the scan limit was \(\mp\pi/2\) radians, with sign determined by scan direction.

Approach and solution

The engineering team rebuilt the model with the measured dielectric properties, complete connector, radome, adhesive and finite ground. Embedded element patterns and active reflection coefficients were calculated at broadside and scan extremes. The patch dimensions and feed inset were co-optimized rather than matched only at broadside. A Monte Carlo analysis included etching bias, laminate thickness and cover spacing. Test coupons were used to estimate effective permittivity and feed-line loss.

Outcome

The revised prototype met the specified return loss over the full scan sector and reduced the simulated-to-measured gain discrepancy to 0.4 dB. The final design used slightly broader impedance bandwidth than the nominal requirement to preserve yield.

Learning points

• Infinite periodic models do not capture finite-array edge elements.
• Scan-dependent active impedance must be included in matching.
• Packaging belongs in the electromagnetic model when it intersects the near field.
• Measured material data and tolerance analysis can be more valuable than aggressive nominal optimization.

12. Challenges, Regulation and Contemporary Developments

Planar and dielectric antennas face competing requirements: low profile versus bandwidth, miniaturization versus efficiency, high permittivity versus tolerance sensitivity, and dense spacing versus mutual coupling. Millimetre-wave systems add severe conductor loss, assembly error and connector uncertainty.

Contemporary developments include metasurface-supported radiators, substrate-integrated waveguide feeds, additive manufacturing of graded-permittivity resonators, antenna-in-package arrays, reconfigurable intelligent surfaces and digitally calibrated phased arrays. Characteristic-mode analysis is increasingly used to identify useful chassis and radiator currents before feed design.

Professional practice must address spectral masks, equivalent isotropically radiated power limits, human RF-exposure standards, electromagnetic compatibility and interference with safety-critical services. Designers should report gain, efficiency and bandwidth using traceable methods rather than selectively presenting favourable simulations. Environmental considerations include material toxicity, recyclability, repairability and the energy cost of large active arrays.

Chapter Summary

Microstrip patches can be initialized using transmission-line formulas and interpreted more deeply using the cavity model. In that model, the conducting patch and ground are electric walls, while the open peripheral edges are approximated as magnetic walls. Printed monopoles and slots are tuned by increasing effective length when simulated resonance is above the target. DRAs require open-boundary modal analysis because their dielectric interfaces support radiating hybrid fields.

Array analysis must employ a consistent angle convention. With angle measured from broadside, the phase term is \(kd\sin\theta+\beta\), and scanning to \(\theta_0\) requires \(\beta=-kd\sin\theta_0\). Mutual coupling makes active impedance scan dependent. Accurate engineering results ultimately require converged full-wave models, complete feed and package representations, measured material data, tolerance analysis and correlation with prototypes.

Key Terms

Active impedance: The voltage-to-current ratio at an array element when all elements are excited with specified relative phases and amplitudes.

Aperture coupling: Excitation of a radiator through a slot in an intervening ground plane.

Array factor: The pattern component determined by element locations and excitations.

Cavity model: Patch approximation using electric walls at conductors and magnetic walls at open peripheral edges.

Corporate feed: A branching network that distributes power through multiple divider stages.

Dielectric resonator antenna: An antenna whose principal resonant fields are supported by a dielectric body.

Effective permittivity: Equivalent permittivity accounting approximately for fields distributed between substrate and surrounding medium.

Embedded element pattern: Pattern of one array element with the remaining ports terminated in specified loads.

Fringing-field extension: Apparent increase in resonant length caused by fields beyond an open conductor edge.

Grating lobe: An additional visible array maximum caused by excessive element spacing or scan.

Inset feed: A microstrip feed inserted into a patch to access a lower-resistance point.

Perfectly matched layer: An artificial absorbing region used to truncate an open computational domain.

Printed monopole: A planar unbalanced radiator printed above or adjacent to a finite ground.

Radiation efficiency: Radiated power divided by accepted input power.

Realized gain: Gain including input mismatch loss.

Series feed: A feed architecture in which power reaches radiators sequentially.

Slot antenna: A radiating aperture formed in a conducting surface.

Surface wave: A substrate-guided wave that can reduce radiation efficiency and disturb patterns.

Tolerance analysis: Evaluation of performance variation caused by uncertain dimensions, materials and assembly parameters.

Review Questions

Short Answer Questions

1. State the electric- and magnetic-wall assignments in the cavity model of a microstrip patch.

2. Why is effective permittivity used in preliminary patch design?

3. What physical effects make a printed monopole differ from an ideal quarter-wave wire?

4. Distinguish radiation efficiency from realized gain.

5. Why does array active impedance vary with scan angle?

Long Answer Questions

1. Derive the preliminary length and width equations for a rectangular microstrip patch and discuss their limitations.

2. Compare microstrip-line, probe, aperture-coupled and proximity-coupled patch feeds.

3. Explain the source-free eigenvalue formulation of a dielectric resonator antenna, including open boundaries and complex eigenfrequency.

4. Discuss bandwidth and efficiency trade-offs in printed and dielectric antennas.

5. Develop a complete full-wave modelling and measurement-correlation workflow for a packaged antenna.

Analytical / Application Questions

1. Design the initial dimensions of a 3.5 GHz rectangular patch on a specified substrate, then identify at least five effects requiring full-wave correction.

2. A printed monopole resonates 8% above its target frequency. Estimate the required length correction and explain when this scaling may fail.

3. For an eight-element linear array with \(d=0.45\lambda\), calculate the progressive phase needed to scan to \(25^\circ\) from broadside and assess whether grating lobes can occur over a \(\pm40^\circ\) scan range.

Multiple Choice Questions with Answers

Question 1

In the cavity model of a rectangular microstrip patch, which boundary assignment is correct?

A. Patch and ground are magnetic walls; peripheral edges are electric walls

B. Patch and ground are electric walls; peripheral edges are approximated as magnetic walls

C. All six boundaries are electric walls

D. All six boundaries are magnetic walls

Correct Answer: B. Patch and ground are electric walls; peripheral edges are approximated as magnetic walls

Question 2

A 28.0 mm printed monopole resonates at 4.45 GHz instead of 4.18 GHz. Using inverse frequency scaling, what is the corrected length?

A. 26.3 mm

B. 28.0 mm

C. 29.8 mm

D. 31.9 mm

Correct Answer: C. 29.8 mm

Question 3

A 15.82 mm slot resonates at 6.05 GHz. Approximately what length targets 5.80 GHz?

A. 15.17 mm

B. 15.82 mm

C. 16.50 mm

D. 17.31 mm

Correct Answer: C. 16.50 mm

Question 4

For a linear array along x with angle measured from broadside, which phase progression scans the main beam to theta_0?

A. beta = -kd sin(theta_0)

B. beta = kd cos(theta_0)

C. beta = -kd cos(theta_0)

D. beta = kd/theta_0

Correct Answer: A. beta = -kd sin(theta_0)

Question 5

An array with d = 0.5 lambda is scanned to 30 degrees from broadside. What progressive phase is required?

A. -pi/4 radians

B. -pi/2 radians

C. pi radians

D. -2pi radians

Correct Answer: B. -pi/2 radians

Question 6

Which method most directly identifies the natural complex resonances of an open dielectric resonator antenna?

A. DC resistance analysis

B. Open-boundary eigenmode analysis

C. Quarter-wave transformer analysis

D. Geometrical optics alone

Correct Answer: B. Open-boundary eigenmode analysis

Question 7

For the e^(-j omega t) convention and complex eigenfrequency omega_tilde = omega_r - j gamma, what is the modal quality factor?

A. omega_r/gamma

B. gamma/(2 omega_r)

C. omega_r/(2 gamma)

D. 2 omega_r gamma

Correct Answer: C. omega_r/(2 gamma)

Question 8

Which antenna-feed method places a ground plane between the feed line and patch and transfers energy through a slot?

A. Inset feeding

B. Coaxial-probe feeding

C. Aperture coupling

D. Direct edge feeding

Correct Answer: C. Aperture coupling

Question 9

Which quantity includes both antenna gain and input mismatch loss?

A. Directivity

B. Radiation efficiency

C. Realized gain

D. Radiation resistance

Correct Answer: C. Realized gain

Question 10

For scanning over plus or minus 30 degrees, which spacing satisfies the stated no-grating-lobe criterion?

A. 0.80 lambda

B. 0.75 lambda

C. 0.70 lambda

D. 0.60 lambda

Correct Answer: D. 0.60 lambda