Solving the one-port calibration equations.

In a network analyzer one-port calibration, three standards are measured to determine the three error terms of the measurement system.  At first glance, it may appear that the set of three equations to be solved are nonlinear, but they can be manipulated into a set of three linear equations in three unknowns, as shown here.

Using the following notation (G is for gamma):

G_A = actual reflection coefficient
G_M = measured reflection coefficient
e₀₀ = directivity error
e₁₀e₀₁ = reflection tracking error
e₁₁ = source match error

The solution of the one-port measurement error model is:

G_M = e₀₀ + (e₁₀e₀₁) * G_A / (1 - e₁₁ * G_A)

Let:

a = e₁₀e₀₁ - e₀₀ * e₁₁
b = e₀₀
c = -e₁₁

Then:

G_M = (a * G_A + b) / (c * G_A + 1)
or,
G_A * a + b - G_A * G_M * c = G_M

Now measure the three known standards (G_A1, G_A2, G_A3), and with the resulting three measurements (G_M1, G_M2, G_M3), you have three linear equations to solve for a, b, and c:

G_A1 * a + b - G_A1 * G_M1 * c = G_M1
G_A2 * a + b - G_A2 * G_M2 * c = G_M2
G_A3 * a + b - G_A3 * G_M3 * c = G_M3

This works for any three standards, not just short, open, and load.