For a homogeneous isotropic material, the dielectric constant can be expressed as a complex function
in which 
is a normalized
(relative) dielectric constant (real part), 
is the imaginary part, 
is the material conductivity, 
is the permittivity and 
is the real angular frequency. Changes in the resonance frequency
and loaded cavity quality
factor 
can be related to
the changes in the complex angular frequency
which can be related to the complex dielectric constant by the cavity perturbation formula
In these equations, the subscripts 1 and 2 refer to the "dark"
and "light" parameters, Vs and Vc are the sample and
cavity volumes, 
and 
are the magnetic and electric fields and 
is the complex magnetic permeability.
Perturbation theory connects the change in the resonance frequency and cavity quality factor with the real and the imaginary parts of dielectric constant. For a narrow thin strip of the sample placed in the maximum of electric field, the following expressions have been derived:
where: 
and 
are the changes in the real and the imaginary parts of dielectric
constant; and K is a numerical factor that depends on the
sample geometry. The inverse relations are also true:
Thus in these general expressions, the change for both real and imaginary parts of dielectric constant depend on the cavity quality factor change and the shift of the resonance frequency. As a result, one might expect to observe experimentally very complicated kinetics for the shift of resonance frequency and the change of the cavity quality factor. Therefore, because of Eqns. (4) and (5) it is not possible to relate these two parameters to either free or trapped electrons.
There are two cases when these expressions can be simplified depending on the material's parameters before illumination. The first case is for low conductivity materials. The real part is much greater than the imaginary part, and the change in the imaginary part is proportional to the change of the cavity quality factor while the change in the real part is proportional to the shift of the resonance frequency
in which 
and 
.
As a result, changes in the real and imaginary parts of the dielectric
constant can be determined separately from 
and 
, respectively. Using
Eq.(8) and Eq.(9), the loss tangent can be written as
in which 
represents the
change in the bandwidth at the half-maximum amplitude and is related
to the cavity quality factor by
.
This is a common case for dielectrics and low doped semiconductors.
Conduction band electrons cause an increase in a cavity quality
factor and trapped electrons cause a negative shift of the resonance
frequency (increasing the real part of dielectric constant). When
equilibration between trapped and free electrons occurs relatively
fast, the cavity quality factor and shift of a resonance frequency
decay with the same time dependence. Another situation occurs
in a resistive material with a low concentration of shallow traps
like intrinsic. In this case free electrons remain in the conduction
band for a relatively long time before trapping or recombination
occurred. As a result, the sign of the shift of the resonance
frequency is positive and originates from free electrons. The
change of the cavity quality factor is positive and caused by
conduction electrons also. This behavior resembles the metal-like
case where free electrons decrease the real part dielectric constant.
From these examples, one can see that contributions of free and
trapped electrons can be easily distinguished. The AMTMP method
can also determine the density of shallow traps in the 3 meV -
300 meV range. For materials with a continuous energy distribution
of traps such as CdSe, the integral density of traps was obtained
within the range of sensitivity and at the limit of trap saturation
only. The concentration of electrons 
in these shallow traps at any time t can be obtained using
the expression,
which results directly from the Clausius-Mosotti equation, with the polarizability estimated by the effective mass approximation.
A quite straight forward strategy relates components of the observed
transients to the trap depth for a material containing several
dominant discrete levels (single crystal GaAs, Si etc.). A simple
harmonic oscillator model was used to relate the oscillating frequency
of the electron to its
binding energy 
by the
equation
The resonance frequency change is related to the oscillating frequency
in such a manner that only small values of
(the shallow levels) can cause measurable changes in the frequency.
To obtain the density of excess free carriers at any time t,
the change in a cavity quality factor resulting from the light
pulse is related to the change in the conductivity 
of material by Eq.(1). The density of excess free electrons 
in the conduction band at any time t is obtained from the
conductivity change using
in which the drift mobility 
is equal to the Hall mobility.
In the second case involving highly conductive materials, the so called "cross-talk" condition occurs when the second terms in Eqns (4) and (5) dominate, so that the change of the cavity quality factor is proportional to the change of the real part instead of the imaginary part. Strictly speaking the term "photoconductivity" does not apply in this case. This behavior is hard to realize experimentally because the highly conductivity material significantly loads the cavity, causing a drastic drop in sensitivity.
In the intermediate case when the real and the imaginary parts of dielectric constant are comparable, each equation always contains both components. They can have the same or different sign (see above). As a result, the change of cavity quality factor and the shift of a resonance frequency are no longer directly proportional to free and trapped electrons because the transient decays have zero intercepts. For Si, this condition is avoided when the conductivity is much less than 0.1 (ohm cm)^1.
When only active losses are associated with the dissipation of
energy in the resonance circuit, the resonance profile has a Lorentzian
form. As a result, the difference signal 
can be expressed as the difference between the "dark"
and "light" Lorentz signals
In this equation, f is the microwave frequency, 
is the amplitude at the resonance frequency and 
is the bandwidth at half-maximum amplitude. Because the dark parameters
can be measured directly, a fit of Eqn. (14) to the difference
signal is made with three adjustable parameters, 
, which are used to calculate the changes in the real and imaginary
parts of the dielectric constant.