# Portable and Highly Versatile Impedance Meter for Very Low Frequency Measurements
## 2 Materials and Methods
### Principle of Operation
The voltage amplifier VA introduces a voltage gain \(A_{V}\) before the test signal, possibly superimposed to a DC component, is delivered to the DUT. For reasons that will be apparent when we will discuss the implementation detail of the block diagram in Figure 1, the voltage gain \(A_{V}\) has the form:  
\[A_{V}(f)=-\frac{A_{V0}}{1+j\frac{f}{f_{Pav}}};\ A_{V0}>0 \tag{5}\]  
In order to simplify the notation, we can make the following positions:  
\[ \begin{array}{c}A_{V}(0)=|A_{V}(0)|e^{j\angle A_{V}(0)}=A_{V0}e^{j\pi}\\ A_{V}(f_{LF})=|A_{V}(f_{LF})|e^{j\angle A_{V}(f_{LF})}=A_{VLF}e^{j\varphi_{VLF}} \end{array}  \tag{6}\]  
With these positions, the voltage \(v_{VD}\) can be written as:  
\[v_{VD}=-\frac{k_{V}A_{C}A_{DC}A_{V0}}{2}\cos(\varphi_{DC})+\frac{k_{V}A_{C}A_{ LF}A_{VLF}}{2}\cos(2\pi f_{LF}t+\varphi_{VLF}). \tag{7}\]  
The voltage \(v_{VD}\) is the voltage at one end of the DUT. The other end of the DUT is connected to the input of a Trans-Impedance Amplifier (TIA). Because of the virtual short at the input of the transimpedance amplifier, the voltage \(v_{VD}\) is indeed the voltage across the DUT. With reference to the admittance \(Y_{D}\)(\(f\)) of the DUT and with the following positions:  
\[ \begin{array}{c}Y_{D}(0)=|Y_{D}(0)|e^{jangle Y_{D}(0)}=Y_{D0}e^{j0}\\ Y_{D}(f_{LF})=|Y_{D}(f_{LF})|e^{jangle Y_{D}(f_{LF})}=Y_{DLF}e^{j\varphi_{DLF}} \end{array}  \tag{8}\]  
we obtain, for \(i_{DUT}\):  
\[ \begin{array}{c}i_{DUT}=-\frac{k_{V}A_{C}A_{DC}A_{V0}}{2}\cos(\varphi_{DC})Y _{D0}\\ +\frac{k_{V}A_{C}A_{LF}A_{VLF}}{2}Y_{DLF}\cos(2\pi f_{LF}t+\varphi_{VLF}+ \varphi_{DLF}).\end{array}  \tag{9}\]  
If we assume, for the transimpedance amplifier, a frequency response in the form:  
\[A_{R}(f)=-\frac{R_{R}}{1+j\frac{f}{f_{Pav}}} \tag{10}\]  
and if we make the following positions:  
\[ \begin{array}{c}A_{R}(0)=|A_{R}(0)|e^{jangle A_{R}(0)}=R_{R}e^{j\pi}\\ A_{R}(f_{LF})=|A_{R}(f_{LF})|e^{jangle A_{R}(f_{LF})}=A_{RLF}e^{j\varphi_{RLF}} \end{array}  \tag{11}\]  
We obtain, for the voltage \(v_{ID}\):  
\[ \begin{array}{c}v_{ID}=\frac{k_{V}A_{C}A_{DC}A_{V0}R_{R}}{2}\cos(\varphi_{DC })Y_{D0}\\ +\frac{k_{V}A_{C}A_{LF}A_{VLF}A_{BLF}}{2}Y_{DLF}\cos(2\pi f_{LF}t+\varphi_{VLF} +\varphi_{DLF}+\varphi_{RLF})\end{array}  \tag{12}\]  
The signals \(v_{VD}\) and \(v_{ID}\) are multiplied by the carrier \(v_{OR}\), thus obtaining the signals \(v_{IL}\) and \(v_{IR}\) that are within the bandwidth of the audio board and that can therefore be acquired and elaborated to extract the modulus and the phase of the unknown admittance \(Y_{D}\). The signals \(v_{IL}\) and \(v_{IR}\) are each the sum of three sinusoidal signals at frequencies \(f_{C}-f_{LF},f_{C}\), and \(f_{C}+f_{LF}\). In particular, we have for \(v_{IL}\):  
\[ \begin{array}{c}v_{IL}(t)=v_{IL1}(t)+v_{IL2}(t)+v_{IL3}(t)\\ v_{IL1}(t)=-\frac{k_{V}{{}^{2}A_{C}{{}^{2}A_{DC}}}A_{ND}}{2}\cos{(\varphi_{DC}) }\cos{(2\pi f_{c}t)}\\ v_{IL2}(t)=\frac{k_{V}{{}^{2}A_{C}{{}^{2}A_{LF}}}A_{VLF}}{4}\cos{[2\pi(f_{C}-f _{LF})t-\varphi_{ULF}]}\\ v_{IL3}(t)=\frac{k_{V}{{}^{2}A_{C}{{}^{2}A_{LF}}}A_{VLF}}{4}\cos{[2\pi(f_{C}+f _{LF})t-\varphi_{ULF}]}\end{array}  \tag{13}\]  
As far as \(v_{IR}\) is concerned, we have:  
\[ \begin{array}{c}v_{IR}(t)=v_{IR1}(t)+v_{IR2}(t)+v_{IR3}(t)\\ v_{IR1}(t)=-\frac{k_{V}{{}^{2}A_{C}{{}^{2}A_{DC}}}A_{ND}R_{R}}{2}\cos{(\varphi_{ DC})}Y_{D0}\cos{(2\pi f_{c}t)}\\ v_{IR2}(t)=\frac{k_{V}{{}^{2}A_{C}{{}^{2}A_{LF}}}A_{VLF}A_{RLF}}{4}Y_{DLF}\cos{ [2\pi(f_{C}-f_{LF})t-\varphi_{ULF}-\varphi_{DLF}-\varphi_{RLF}]}\\ v_{IR3}(t)=\frac{k_{V}{{}^{2}A_{C}{{}^{2}A_{LF}}}A_{VLF}A_{RLF}}{4}Y_{DLF}\cos{ [2\pi(f_{C}+f_{LF})t+\varphi_{ULF}+\varphi_{DLF}+\varphi_{RLF}]}\end{array}  \tag{14}\]  
From the amplitudes \(A_{IL1}\) and \(A_{IR1}\) of the signals \(v_{IL1}\) and \(v_{IR1}\) we obtain:  
\[\frac{A_{IR1}}{A_{IL1}}=R_{R}Y_{DO} \tag{15}\]  
That is, from the analysis of the sinusoidal signals at frequency \(f_{C}\) we obtain the admittance of the DUT in DC.