What are even and odd mode impedances?

When two transmission lines are coupled (close together), the electromagnetic fields interact. The coupled system supports two fundamental modes: Even mode: both lines carry signals of equal amplitude and equal phase. The symmetry plane between the lines acts as a magnetic wall (open circuit). The electric field between the conductors is reduced, so the capacitance per unit length decreases and the impedance increases: Z0e is greater than Z0 (uncoupled). Odd mode: both lines carry signals of equal amplitude but opposite phase (180 degrees apart). The symmetry plane acts as an electric wall (ground plane). The electric field between the conductors increases, so the capacitance per unit length increases and the impedance decreases: Z0o is less than Z0 (uncoupled). The relationship to the uncoupled impedance: Z0 = sqrt(Z0e times Z0o). This means the geometric mean of the even and odd mode impedances equals the uncoupled impedance. For a 50 ohm system with 10 dB coupling: Z0e approximately equals 69 ohms, Z0o approximately equals 36 ohms. For 3 dB coupling: Z0e approximately equals 120 ohms, Z0o approximately equals 21 ohms (very tight coupling, difficult to achieve in microstrip).

How are even/odd modes used in coupler design?

Quarter-wave coupled-line couplers use the difference between Z0e and Z0o to create directional coupling. The coupling factor is: C = (Z0e minus Z0o) / (Z0e plus Z0o). For a 10 dB coupler (C = 0.316): Z0e = 69.3 ohms, Z0o = 36.1 ohms. For a 3 dB coupler (C = 0.707): Z0e = 120.7 ohms, Z0o = 20.7 ohms. The coupled section is a quarter wavelength long at the center frequency. At this frequency, the input signal splits between the through port (most power) and the coupled port (coupled power), with the isolated port receiving ideally zero power. Directivity = isolation minus coupling, and depends on the equality of even and odd mode phase velocities. In stripline (homogeneous medium): v_e = v_o, so directivity is theoretically infinite. In microstrip (inhomogeneous: air plus dielectric): v_e is not equal to v_o because the even mode has more field in the dielectric. This velocity mismatch limits directivity to 10 to 20 dB in microstrip. Solutions: Lange coupler (interdigitated fingers equalize velocities), wiggle-line compensation, overlay coupled lines.

How do even/odd modes relate to differential signaling?

Even mode in coupled lines is equivalent to common mode in differential signaling. Odd mode is equivalent to differential mode. The impedances are related: Z_differential = 2 times Z0o (each line sees Z0o, in series for the differential signal). Z_common = Z0e / 2 (each line sees Z0e, in parallel for the common-mode signal). For a 100 ohm differential pair: Z0o = 50 ohms, so Z0e depends on coupling. Loosely coupled: Z0e approximately equals 55 to 60 ohms. Tightly coupled: Z0e approximately equals 70 to 80 ohms. High-speed digital standards specify differential impedance: USB 3.x: 90 ohms differential (Z0o = 45 ohms). PCIe: 85 ohms differential (Z0o = 42.5 ohms). HDMI: 100 ohms differential (Z0o = 50 ohms). The coupling between the lines (determined by spacing) affects both Z0e and Z0o. Tighter coupling (closer spacing) increases Z0e and decreases Z0o, increasing the difference between common-mode and differential-mode impedances. This is desirable because it increases the common-mode rejection of the differential pair.

Coupled Lines

Even Mode

/ee-ven mohd/

Equal-phase signals on both conductors. Z_0e > Z₀ (reduced inter-conductor E-field). Odd mode: opposite phase, Z_0o < Z₀. Z₀ = √(Z_0e×Z_0o). Coupling: C=(Z_0e−Z_0o)/(Z_0e+Z_0o). 10dB: Z_0e=69Ω, Z_0o=36Ω. Z_diff=2Z_0o. Z_cm=Z_0e/2. Stripline: v_e=v_o (high directivity). Microstrip: v_e≠v_o (limited directivity).

Z_0e: >Z₀

Z_0o: <Z₀

C: (Z_0e−Z_0o)/(Z_0e+Z_0o)

Understanding Even Mode

Even and odd mode analysis is the foundation of coupled-line circuit design. Every directional coupler, coupled-line filter, and differential transmission line is designed using these two fundamental modes. By decomposing any excitation into even and odd mode components, the behavior of coupled structures can be analyzed using simple single-line theory applied twice.

The key insight is that coupling between lines is determined entirely by the difference between Z_0e and Z_0o. Tight coupling requires a large Z_0e/Z_0o ratio, which demands close spacing and is limited by fabrication tolerances.

Even/Odd Mode Equations

Mode impedances:
Z₀ = √(Z_0e×Z_0o)
Z_0e > Z₀ > Z_0o

Coupling coefficient:
C = (Z_0e−Z_0o)/(Z_0e+Z_0o)
10dB: C=0.316, Z_0e=69, Z_0o=36
3dB: C=0.707, Z_0e=121, Z_0o=21

Differential relationship:
Z_diff = 2×Z_0o
Z_cm = Z_0e/2

Coupled-Line Applications

Structure	Coupling	Z_0e/Z_0o	Medium	Use
10dB coupler	Loose	1.9	Microstrip	Power monitor
3dB Lange	Tight	5.8	Microstrip	Balanced amp
Coupled filter	Variable	1.2-3	Strip/micro	BPF
Diff pair	Moderate	1.1-1.6	PCB	USB/PCIe
Broadside	Tight	3-10	Stripline	3dB coupler

Common Questions

Frequently Asked Questions

Z_0e/Z_0o?

Even: same phase, symmetry plane = magnetic wall, less C between lines, Z_0e>Z₀. Odd: opposite phase, electric wall (ground), more C, Z_0o<Z₀. Z₀=√(Z_0e×Z_0o). Tighter coupling: larger Z_0e/Z_0o ratio. 3dB: ratio=5.8 (hard in microstrip).

Couplers?

C=(Z_0e−Z_0o)/(Z_0e+Z_0o). λ/4 coupled section: directional coupler. Stripline: v_e=v_o, infinite directivity (ideal). Microstrip: v_e≠v_o, 10-20dB directivity. Lange coupler: interdigitated fingers equalize velocities. Overlay: broadside coupling for tight C.

Differential?

Even mode = common mode. Odd mode = differential mode. Z_diff=2Z_0o. Z_cm=Z_0e/2. USB 3.x: 90Ω diff (Z_0o=45). PCIe: 85Ω (Z_0o=42.5). Tighter coupling: more CM rejection. Spacing controls Z_0e/Z_0o split.

Related Terms

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Coupled Structures

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