**Impedance Measurement**

The aim of impedance spectroscopy is to characterize the
electrical properties of devices or materials by the impedance function *Z**(ω). It is defined by the ratio of the voltage between two electrical ports of a
sample object and the resulting current through the ports.

*U/I*where

*U*and

*I*are constant in time. In an impedance measurement, however, an alternating ac voltage

*U*with a fixed frequency ω/2π is applied to the sample under test.

_{0}*U*causes a current

_{0}*I*at the same frequency in the sample. In addition, there will generally be a phase shift between current and voltage described by the phase angle φ.

_{0}The ratio of *U*_{0} and *I*_{0} and the phase
angle *j* are determined by the electrical properties of the sample. For simple
calculation and representation of the formulas, it is convenient to use
complex notation.

Voltage: | U(t)*(ω) := Ucos_{0}(ωt)
=Re[ expU*(ω)(jωt)] |

Current: | I(t)*(ω) := Icos(ωt+φ)
=Re[_{0}I*(ω) exp(jωt+φ)] |

with | |

U*=U'+jU'' =U_{0} | |

and | |

I*=I'+jI'';
I
_{0} = √[I'^{2}+I''^{2}];
tan(φ)=I''/I' |

where *U**, *U*', *U*'', *U*_{0}, *I**, *I*', *I*'', *I*_{0} and *j* all are
frequency dependent. For a sample with linear electrical response, the ratio
of *U**(ω) and *I**(ω) does not depend on the magnitude *U*_{0} of the
applied voltage and the measurement result can be reduced to one complex
function which can be defined by several basic representations. The most
common one is the impedance

*Z(ω)*=Z'(ω)+jZ''(ω)=U*(ω)/I*(ω)*

Other often used representations are

admittance: | Y*(ω) := 1/Z*(ω) |

capacity: | C*(ω) := -j/(ωZ*(ω)) |

inductance: | L*(ω) := -jZ*(ω)/ω |

These representations show the following properties.

For an ideal resistor or conductor with resistance *R* and
conductance *S=1/R*, *Z**(ω) = *R*, and *Y*(ω) = S*.

For an ideal capacitor with capacity *C*_{0}, *C**(ω) = *C*_{0.
}If a material is placed between two electrodes, *C**(ω) is directly
proportional to the complex material permittivity ε*(ω).

For an ideal inductor with inductance *L*0,
L*(ω) = *L*_{0. }If a material is placed into an inductive coil, *L**(ω)
is directly proportional to the complex material permeability µ*(ω).

The used representation depends on the sample type and the
personal preference of the researcher. In practice, samples are never ideal.
Nevertheless, one generally selects a representation close to the electrical
sample type. E.g. a mainly capacitive sample generally is represented in terms
of *C**(ω). Independent of the actual representation, one should keep in
mind that they all contain the same information, i.e., the sample response
current to an applied voltage.

loss factor | tan(δ) := -Z'(ω)/Z''(ω) = -C''(ω)/C'(ω) |

and | |

phase factor | tan(φ) := 1/tan(δ). |

The corresponding loss and phase angles are shown in the figure above.

Case | φ | tan(φ) | δ | tan(δ) |
---|---|---|---|---|

Ideal resistor | 0° | 0 | 90° | ∞ |

Ideal capacitor | -90° | -∞ | 0° | 0 |

Ideal inductor | 90° | ∞ | 180° | 0 |

In impedance spectroscopy *Z**(ω) is measured over a broad
frequency range e.g. from mHz to several GHz in order to get as most
information as possible.