# Portable and Highly Versatile Impedance Meter for Very Low Frequency Measurements
## 2 Materials and Methods  
### Principle of Operation  
In approaching the design of an impedance meter to be used for the characterization of devices and materials in the frequency range from a few tens of mHz up to a few hundred Hz, we regarded low hardware complexity and high versatility as the most important goals to be obtained. By reducing the hardware complexity, the system can be more easily reproduced by researchers interested in its application. This is why we investigated in the first place the possibility of resorting to a standard audio board for signal generation and acquisition with a personal computer acting as a powerful platform for digital signal elaboration. While the modern audio board can have impressive performances in terms of resolution (24-bit resolution is common in many modern low to mid-range devices), due to AC coupling they cannot directly generate and acquire signals with frequencies below a few Hz. This problem, however, can be quite easily addressed and solved by resorting to simple modulation/demodulation operations as illustrated in Figure 1. Let us assume that the audio board is configured to work with a standard sampling frequency of 48 kHz and a resolution of 24 bits. One of the outputs of the audio board (\(OUT\)\(R\)) is used to generate a sinusoidal signal with constant amplitude \(A_{\mathrm{C}}\) and constant frequency \(f_{\mathrm{C}}\):  
\[v_{OR}=A_{\mathrm{C}}\cos(2\pi f_{\mathrm{C}}t) \tag{1}\]  
This signal \(v_{OR}\) is used as a carrier for synchronous demodulation/modulation and it reaches the inputs of all three analog multipliers \(M_{1}\), \(M_{2}\), \(M_{3}\) in Figure 1. Typically, \(A_{\mathrm{C}}\) is set at the maximum value compatible with the input dynamic of the multipliers. The frequency \(f_{\mathrm{C}}\) is chosen to be very large in comparison with the test frequencies to be delivered to the Device Under Test (DUT), but well within the passband of the audio board. Typically, \(f_{\mathrm{C}}\) is in the order of 10 kHz. The second output of the audio board (\(OUT\)\(L\)) is used to generate a signal that, when multiplied by \(v_{OR}\) and filtered by the Low Pass Filter (LPF) results in a low frequency sinusoidal signal, possibly superimposed to a DC level, at the output \(v_{OLP}\). In order to obtain this result, the signal \(v_{OL}\) has the form:  
\[v_{OL}=A_{DC}\cos(2\pi f_{\mathrm{C}}t+\varphi_{DC})+A_{LF}\cos[2\pi(f_{ \mathrm{C}}+f_{LF})t] \tag{2}\]  
where \(f_{LF}\) is the frequency at which we want to measure the impedance of the DUT and \(A_{DC}\), \(\varphi_{DC}\) and \(A_{LF}\) can be used to set the level of the DC bias and the amplitude of the sinusoidal signal at \(f_{LF}\), as will be apparent in the following. Starting from Equations (1) and (2), at the output \(v_{ILP}\) of the multiplier \(M_{1}\) we have:  
\[ \begin{array}{l}v_{ILP}=k_{V}v_{OR}v_{OL}=\frac{k_{V}A_{C}}{2}[A_{DC}\cos( \varphi_{DC})+A_{LF}\cos(2\pi f_{LF}t)]+\\ +\frac{k_{V}A_{C}}{2}\{A_{DC}\cos[2\pi(2f_{\mathrm{C}})t+\varphi_{DC}]+A_{LF} \cos[2\pi(2f_{\mathrm{C}}+f_{LF})t]\}\end{array}  \tag{3}\]  
where \(k_{V}\) is a dimensional constant introduced by the multiplier (\(k_{V}=1\) V\({}^{-1}\) in the case of the multiplier used in our prototype). All components at high frequencies (\(2f_{\mathrm{C}}\) and \(2f_{\mathrm{C}}+f_{LF}\)) are rejected by the low pass filter so that, at the output of the LPF we have:  
\[v_{OLP}=\frac{k_{V}A_{C}A_{DC}}{2}\cos(\varphi_{DC})+\frac{k_{V}A_{C}A_{LF}}{ 2}\cos(2\pi f_{LF}t) \tag{4}\]  
Figure 1: Block diagram of the proposed approach. where we assumed a unity gain for the LPF in the passband. From Equation (4) it is apparent that by setting the value of \(A_{LF}\) we set the amplitude of the test signal at the frequency \(f_{LF}\) while by setting the values of \(A_{DC}\) and \(\varphi_{DC}\) we can set the value of the DC bias, with \(\varphi_{DC}=0\) for positive biases and \(\varphi_{DC}=\pi\) for negative biases.