Birdcage Coil Designer (MRI)
The birdcage resonator is the standard volume coil for MRI. It produces a highly homogeneous circularly polarised B₁ field inside the bore. This tool designs low-pass (LP), high-pass (HP), and band-pass (BP) birdcage coils: given the bore dimensions and Larmor frequency, it computes the required element values and mode frequencies.
Equations & Parameters ▸
\(\omega_m^2 = \dfrac{2}{C}\!\left[L_{leg} + \dfrac{4L_{arc}}{N}\sin^2\!\left(\dfrac{m\pi}{N}\right)\right]^{-1}\qquad\text{(LP birdcage)}\)
| N | Number of rungs (legs). Typical values: 8, 12, 16. More rungs → better homogeneity. |
| r | Bore radius (mm) — radius of the cylindrical coil. |
| ℓ | Coil length (mm). Longer → more homogeneous central region. Entered as field ℓ (coil length). |
| d | Wire or strip width/diameter (mm). Sets wire inductance using the Rosa formula. |
| LP | Low-pass: capacitors in legs, inductors in end-ring arcs (or mesh). Common for high-field MRI. |
| HP | High-pass: inductors in legs, capacitors in end-ring arcs. Common for low-field. |
| f₀ | Target ¹H Larmor frequency. The m=1 mode must resonate here. |
Physical constants used
| µ₀ | 4π×10⁻⁷ H/m |
| ¹H γ/2π | 42.577 MHz/T (Larmor frequency per Tesla) |
| ¹H at 1.5 T | 63.87 MHz |
| ¹H at 3 T | 127.74 MHz |
| ¹H at 7 T | 297.7 MHz |
| σ_muscle @ 128 MHz | ≈ 0.77 S/m |
| ε_muscle @ 128 MHz | ≈ 58 |
| IEC SAR limit (WB normal) | 2 W/kg (over 6 min) |
| IEC SAR limit (head) | 3.2 W/kg (over 10 min) |
Geometry
8, 12, or 16 typical
mm
coil radiusmm
mm
conductor width or diameterMHz
¹H at 3T = 128 MHzResults
Element Values
Leg inductance, L_leg—
Ring arc inductance, L_arc—
Required capacitance, C—
Capacitor placement—
Resonance
Imaging mode (m=1) frequency—
Next mode (m=2) frequency—
Mode separation—
Q estimate (unloaded)—
Diagram