Waveguide E-Plane, H-Plane & Magic Tee Calculator

Q: Why can a waveguide Tee never be simultaneously matched at all ports without internal tuning posts?

According to fundamental network theorems, a lossless, reciprocal 3-port junction can never be matched at all three ports simultaneously. Unmatched tees have S11 = -6 dB. In a Magic Tee, an inductive cone/post in the H-arm and an inductive diaphragm in the E-arm cancel the junction reactances, achieving perfect matches at all 4 ports.

Q: How is a Magic Tee used in a radar monopulse antenna tracking receiver?

The two antenna feed horns connect to Ports 1 and 2. The H-arm (Port 3) outputs the sum signal (Σ = Port 1 + Port 2) for target range detection, while the E-arm (Port 4) outputs the difference signal (Δ = Port 1 - Port 2) for high-precision angular error tracking.

Analyze 3-port E-plane series and H-plane shunt junctions, evaluate 4-port Magic Hybrid Tees, compute TE10 guide wavelengths, and size inductive matching posts.

Standard Waveguide Band

Operating Freq f (GHz)

Junction Configuration

Internal Waveguide Dimensions

Broad Wall a (mm):

Narrow Wall b (mm):

Input Power P_in (Watts)

Input Port Selected

Guide Wavelength (λ_g)

39.71 mm

TE10 Cutoff: 6.56 GHz

Side Arms Output Power

50 W each (-3 dB)

Phase: 180° Out of Phase

Magic Tee Ideal Scattering Matrix

[S] = (1/√2) ×
  [ 0, 0, 1, 1 ]
  [ 0, 0, 1, -1 ]
  [ 1, 1, 0, 0 ]
  [ 1, -1, 0, 0 ]

Waveguide Wave Impedance (Z_TE): 499.2 Ω

Isolation Between E-Arm & H-Arm: ∞ (Theoretical Zero Coupling)

Internal Post & Iris Matching

H-Plane Inductive Post Diameter: ~1.85 mm at junction center

E-Plane Capacitive Iris Gap: ~4.20 mm aperture

Matched VSWR at f₀: < 1.15 : 1 (Over 10% band)

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Waveguide E-Plane, H-Plane & Magic Tee Junctions

Rectangular waveguides propagate the fundamental dominant (TE_{10}) mode. Combining branches in the electric plane (E-plane) and magnetic plane (H-plane) creates iconic 3-port and 4-port microwave power-splitting networks.

1. E-Plane Tee (Series / Difference Junction)

The side arm branches out perpendicular to the broad wall (parallel to the electric field (E)). When power enters the E-arm (Port 4), boundary conditions enforce an antisymmetric electric field at the collinear arms (Ports 1 and 2):

$$S_{14} = -S_{24} = \frac{1}{\sqrt{2}} \iff \text{Equal amplitude, } 180^\circ \text{ out-of-phase}$$

2. H-Plane Tee (Shunt / Sum Junction)

The side arm branches out parallel to the narrow wall (in the plane of the magnetic field (H)). When power enters the H-arm (Port 3), the electric field enters both collinear arms with identical orientation:

$$S_{13} = S_{23} = \frac{1}{\sqrt{2}} \iff \text{Equal amplitude, } 0^\circ \text{ in-phase}$$

3. The Magic Tee (Hybrid 4-Port Coupler)

By combining an E-plane branch and an H-plane branch at the exact same longitudinal junction, a 4-port Magic Tee is created. Due to field orthogonal symmetry, the E-arm and H-arm are completely isolated from each other ((S_{34} = S_{43} = 0)).

Frequently Asked Questions

Why can a waveguide Tee never be simultaneously matched at all ports without internal tuning posts?

According to fundamental network theorems, a lossless, reciprocal 3-port junction can never be matched at all three ports simultaneously. Unmatched tees have S11 = -6 dB. In a Magic Tee, an inductive cone/post in the H-arm and an inductive diaphragm in the E-arm cancel the junction reactances, achieving perfect matches at all 4 ports.

How is a Magic Tee used in a radar monopulse antenna tracking receiver?

The two antenna feed horns connect to Ports 1 and 2. The H-arm (Port 3) outputs the sum signal (Σ = Port 1 + Port 2) for target range detection, while the E-arm (Port 4) outputs the difference signal (Δ = Port 1 - Port 2) for high-precision angular error tracking.