# Portable and Highly Versatile Impedance Meter for Very Low Frequency Measurements
## 2 Materials and Methods
### Hardware Configuration  
As we have noted above, from Equation (16), it would appear that the only relevant parameter for the estimation of the admittance (or impedance) of the DUT is the frequency response of the transimpedance amplifier. While this is true in principle, all other circuit parameters must be carefully selected r to take full advantage of the available dynamic ranges and to reduce the influence of disturbances on the estimation process. The actual circuit developed in order to demonstrate the approach we propose is reported in Figure 2. It is important to stress the fact that while the modulator/demodulator section is intended to remain the same regardless of the specific measurement applications, the front-end section can be adapted depending on the nature of the DUT and on the desired measurement conditions. The multipliers \(M_{1}\), \(M_{2}\), and \(M_{3}\) in Figure 1 are implemented by resorting to the three identical monolithic four-quadrant multipliers AD 835 [37] that require essentially no external component for their operation (save for the power supply decoupling capacitors not shown in the figure). The Low Pass Filter (LPF) is implemented by resorting to a switched capacitor fifth-order Butterworth filter (LT1063 [38]) configured in self-clocking mode (\(R_{\text{CK}}=50\) k\(\Omega\) and \(C_{\text{CK}}=200\) pF) to obtain a corner frequency of 1 kHz. The passive low pass filter \(R_{AL}\)\(C_{AL}\) (\(R_{AL}=10\) k\(\Omega\), \(C_{AL}=47\) nF) acts as an anti-aliasing filter with almost no influence on the response of the system. The Demodulator/Modulator section is supplied by a miniaturized DC/DC converter supplied by the USB port of the personal computer. Depending on the required voltage range and current capability required by the DUT to be characterized, the same DC/DC converter can be used for the front-end section. Since the input and output dynamic ranges for the AD835 are limited to \(\pm 1\) V and \(k_{\text{V}}=1\) V\({}^{-1}\), this means that the instantaneous value of \(v_{\mathit{ULP}}\) in Equation (3) must remain within \(\pm 1\) V. Since the passband gain of the LPF is 1, this means that the instantaneous value of \(v_{\mathit{OLP}}\) cannot exceed \(\pm 500\) mV. The amplifier VA can be used to extend the range of \(v_{\mathit{VD}}\) with respect to \(v_{\mathit{OLP}}\). However, when the circuit configuration corresponding to \"opt 1\" (Figure 2) is employed, since \(v_{\mathit{VD}}=v_{\mathit{DUT}}\) is also at the input of the multiplier \(M_{2}\), \(v_{\mathit{DUT}}\) cannot exceed \(\pm 1\) V. If larger voltages are required across the DUT, as it is often the case, the configuration corresponding to \"opt 2\" (Figure 2) must be used, where the voltage divider \(R_{P1}-R_{P2}\), if properly designed, allows the voltage \(v_{\mathit{VD}}\) to remain within the input dynamic range of the multiplier \(M_{2}\). Note that the block diagram in Figure 1 corresponds to the case in which \"opt 1\" in Figure 2 is used. When \"opt 2\" is used, the presence of the voltage divider can be easily incorporated in Equations (6)-(16). The frequency response \(A_{V}\) of the amplifier (Equation (5)) is characterized by the following parameters:  
\[A_{V0}=\frac{R_{2}}{R_{1}}\ ;\ f_{\mathit{PAV}}=\frac{1}{2\pi R_{2}C_{2}} \tag{17}\]  
Figure 2: Complete schematic for the implementation of the proposed approach. Depending on applications, different Front Ends can be used. The capacitor \(C_{2}\) in parallel to the resistance \(R_{2}\) is used to limit the bandwidth of the amplifier close to maximum measurement frequency. This bandwidth limitation serves the purpose of reducing the noise at the output of AV at higher frequencies that, because of the likely capacitive nature of the DUT would result in large noise at the output of the TIA that could cause the voltage \(v_{ID}\) to exceed the maximum allowed dynamic range (\(\pm 1\) V) even with \(v_{OLD}=0\).