Question 2.38: Calculate the self inductance between two parallel planes. ...
Calculate the self inductance between two parallel planes. This object is frequently called a “microstrip” or “microstrip line” transmission line and it is important for the understanding of integrated circuits. The separation distance d can be assumed to be much less than the width w.
Since the current in each of the each of the conductors is in the opposite direction, the magnetic field between the two conductors will be in the same direction. This magnetic field will pass through the rectangular area defined by the separation distance d and the length ∆z. An approximate value for the magnetic flux density between the two conducting strips can be obtained from Ampere’s law (2.108).
\mathbf{\oint B \bullet d l =\mu_{0} I _{ enc }} (2.108)
For one of the strips, we write 2\text{w}B \approx \mu_{0} I . Superposition applies and we make the assumption that the magnetic flux density is uniform in the entire region between the two strips. Therefore, the total magnetic flux \Psi_{m} is given by
\Psi_{m}=\int_{z=0}^{z=\Delta z} \int_{y=0}^{y=d} \frac{\mu_{0} I }{\text{w}} d y d z=\frac{\mu_{0} I}{\text{w}} d \Delta z
In this case, the total magnetic flux will be equal to the magnetic flux linkage. The inductance is computed from (2.164) to be
L_{j k} \equiv \frac{\Lambda_{j}}{I_{k}} (H) (2.164)
L =\mu_{ o } \frac{ d }{ \text{w} } \Delta z