3 Capacity Sensitivity

The sensitivity of capacity to a fundamental or physical parameter $x$ at operating point $\mathbf a$ is defined as the partial derivative of the logarithm of capacity with respect to the logarithm of the parameter:

Capacity sensitivity with respect to $$x \triangleq {\partial\log C({\mathbf a})\over\partial\log x}.$$

The logarithm is used to emphasize the sensitivity of the parameter without regard to the units in which the parameter is measured, and it allows us to effectively compare the relative sensitivities of various parameters. This is in contrast to the linear partial derivative $\partial C\over\partial x$, which has one value when, for example, $x=T_s$ is measured in nanoseconds, and a value one billion times smaller when $x=T_s$ is measured in seconds. If $x$ is a physical parameter, we may express the sensitivity with respect to $x$ at operating point $\mathbf a$ as

$${\partial \log C({\mathbf a})\over \partial \log x} = \left({1 \over C({\mathbf a})} \right) {\partial C({\mathbf a})\o... ...left({x \over C({\mathbf a})} \right) {\partial C({\mathbf a})\over \partial x}$$ (17)

$$= \left( {x \over C({\mathbf a})} \right) \left({\partial C({\mathb... ...}) \over \partial \beta_0} {\partial {\mathbf \beta_0} \over \partial x}\right)$$

$$= \left( {x \over C({\mathbf a})} \right) \nabla C({\mathbf a}) \cdot {\partial {\mathbf a} \over \partial x},$$ (18)

i.e., the normalized dot product of the gradient of $C({\mathbf a})$ and the vector $\partial{\mathbf a}\over \partial x$ which forms one of the columns of the Jacobian matrix of $\mathbf a$:

$$J({\mathbf a}) = \left[ \begin{array}{cccccccccc} {\partial\rho_0 \over\partia... ...tial R_L} & {\partial\beta_0\over\partial G} \end{array}\right]$$

$$= {\scriptsize\left[ \begin{array}{cccc} \frac{\eta e_{\text{-}... ..._L + 2 \kappa T \right) T_s \right) }^2} \end{array}\right]^T }$$

(Note that the expanded matrix has been written as a transpose.) To determine the sensitivity of capacity with respect to one of the physical parameters, we need only determine the gradient of the capacity expressed as a function of the four fundamental parameters and form the inner product with the appropriate column of $J({\mathbf a})$.