Characteristic Impedance Odd
Understanding Characteristic Impedance Odd
Odd-Mode Excitation and Mutual Capacitance
Coupled transmission lines are commonly used in high-speed digital and RF designs to transmit differential signals or construct passive components. The electromagnetic coupling between the lines alters their wave propagation. To analyze this coupled system, engineers divide the signals into even (in-phase) and odd (out-of-phase) modes. The Characteristic Impedance Odd ($Z_{0o}$) is the characteristic impedance of one line under odd-mode excitation.
Odd-mode excitation occurs when the two lines are driven with signals of equal amplitude but opposite phases ($V_1 = -V_2$). Under these conditions, a virtual ground plane (zero potential boundary) is formed exactly halfway between the two traces. This creates a high potential gradient between the lines, activating the mutual capacitance ($C_m$) between them. The effective capacitance of each line increases to $C_p + 2C_m$, which lowers the characteristic impedance ($Z_{0o} < Z_0$).
Application in Differential Impedance Matching
Odd-mode impedance is a critical parameter for differential signaling, which is used in high-speed digital interfaces (such as HDMI, PCIe, and USB) to reject common-mode noise. A differential transmission line consists of two coupled traces. The differential impedance ($Z_{\text{diff}}$) is the impedance measured between the two lines under differential (odd-mode) excitation.
The differential impedance is exactly twice the odd-mode characteristic impedance ($Z_{\text{diff}} = 2 Z_{0o}$). To prevent reflections in differential links, the line geometry must be designed so that $Z_{\text{diff}}$ matches the system specification (typically 90 or 100 ohms). This requires calculating the trace widths and coupling gaps carefully to achieve a $Z_{0o}$ value of 45 or 50 ohms respectively, maintaining signal integrity over long transmission paths.
Key Mathematical Relations
Technical Specifications Comparison
| Differential Interface Standard | Target Differential Impedance (\$Z_{\text{diff}}\$) | Required Odd Impedance (\$Z_{0o}\$) | Typical Substrate Stackup Height | Typical Trace Gap Spacing |
|---|---|---|---|---|
| PCI Express (Gen 3 - 5) | 85 \$\Omega\$ | 42.5 \$\Omega\$ | 100 \mum (FR4/Megtron 6) | Narrow (tight coupling) |
| USB 3.0 / HDMI | 90 \$\Omega\$ | 45.0 \$\Omega\$ | 120 \mum (High-speed FR4) | Narrow - Medium |
| Ethernet / LVDS | 100 \$\Omega\$ | 50.0 \$\Omega\$ | 150 \mum (FR4) | Medium gap spacing |
| Serial ATA (SATA) | 100 \$\Omega\$ | 50.0 \$\Omega\$ | 120 \mum (Low-loss dielectric) | Medium gap spacing |
Frequently Asked Questions
Why is the odd-mode impedance lower than the system impedance of a single line?
In odd-mode excitation, the two lines carry opposite-phase voltages, creating a voltage difference between them. This activates the mutual capacitance ($C_m$) between the traces. The effective capacitance of each line increases because it now includes both the capacitance to ground and the mutual capacitance path, which decreases the impedance ($Z_0 = \sqrt{L/C}$).
What is the relationship between odd-mode impedance and differential impedance?
For a coupled transmission line system, the differential impedance ($Z_{\text{diff}}$) is defined as the impedance measured between the two active conductors when driven differentially (which is odd-mode excitation). Because the signals are opposite in phase, the differential impedance is exactly twice the odd-mode characteristic impedance: $Z_{\text{diff}} = 2 Z_{0o}$.
How does the gap spacing between coupled traces affect odd-mode impedance?
Reducing the gap spacing between the two traces increases the electromagnetic coupling, which increases the mutual capacitance ($C_m$) between them. Since $Z_{0o} = \sqrt{L/(C_p + 2C_m)}$, a larger mutual capacitance increases the denominator, which causes the odd-mode impedance to drop.