2.13 The Ground and Power Planes as a Tapered Transmission Line
A ground and power plane can be viewed as a parallel plate capacitor where a via and a trace is used to make a connection to one of the planes. When a connection is made in the middle of a board, the conductor geometry can be viewed as a tapered transmission line (TTL). In this geometry, the characteristic impedance varies with the radial distance from the point of entry. Energy is moved by waves that move radially into or out of the points of contact. When a switch connects a resistive load to the point of entry, the initial wave travels outward radially.
To understand how energy is transported, we will first consider the characteristic impedance of a thin annular ring at a distance r from the point of entry. This geometry is shown in Figure 2.10.
Figure 2.10 (a) A thin annular ring of a tapered transmission line. The H field is circular, the E field goes between the conducting planes, and the current I flows radially in the conducting planes. (b) The 
field pattern in the annulus.
The capacitance between two circles of radius r and r + dr and of thickness h is simply
2.13 
where A is the area, h is the spacing between planes, ε is the permittivity of free space, and ε R is the relative dielectric constant. The inductance of this annular ring can be calculated from the ratio of B flux per unit current. The B flux φ in this annulus is constant and is totally contained. The line integral of H around the annulus is simply 2πrH. This integral must equal the current I. The value of H is
2.14 
The induction flux φ is μH times the area enclosed or
The inductance L is the flux per unit current or
2.16 
The characteristic impedance of this annulus is (L/C)1/2. Combining Equations 2.12 and 2.15, the characteristic impedance at a radius r is
In a vacuum, the ratio (μ/ ε )1/2 is 377 ohm. Assume that the ratio of h to r is unity at the entry point. The value of Z is 377/2π = 60 ohm. If the relative dielectric constant ε R is 3.5, the impedance at the entry point is approximately 32 ohm. If the spacing between the planes is 10 mil, then at a distance of 0.1 inch from the point of entry, the characteristic impedance drops to 3.2 ohm. At a distance of 3 inches, the characteristic impedance is about 0.1 ohm. Equation 2.17 shows that the characteristic impedance of this type of transmission line is inversely proportional to the radius.