Series/parallel combination of resistance/capacitance in an RC circuit (wrt LCR meter readings)

Question

I have this equivalent circuit I am working on. I have to find out individual values of all components in it. I can measure resistance, capacitance and impedance of the whole circuit. I am not sure resistance and capacitance values I measure with LCR meter are equivalent values of capacitance and resistance in this circuit? If so, how Ra, Rs and Rc are connected in terms of series and parallel to find equivalent resistance? Similarly, for capacitance?

Answer 1

In this particular circuit, if frequency goes towards zero, the measured resistance is equal to Ra + Rc + Rs. This is due to the impedance of the capacitors becoming infinite. On the opposite, if frequency goes towards infinity (i.e. is large enough), the measured resistance is equal to Rs.

It is harder to obtain CPE (and the exact value of Ra and Rc). What I would do is :

1. measure the impedance as a function of frequency, and save the results in a file ;
2. create a little program to calculate the same impedance for whatever values of the parameters (CPE, Ra and Rc). This can be done for example with Matlab (or its free clone Octave), or SciPy;
3. plot the measured impedance along with the calculated one, and adjust the parameters till there is a good correlation between the two curves

You will get a good set of parameters.

Note that you never will be able to distinguish Rc from Ra: these two resistance could be swapped, and you would get exactly the same impedance.

EDIT: typical LCR meter operation

Usually, you can choose the way an LCR meter gives you information about your circuit. This can be:

• Z (impedance) and \$\theta\$ (phase)
• Cs (equivalent series capacitance) and Rs (equivalent series resistance)
• Cp (equivalent parallel capacitance) and Rp (equivalent parallel resistance)
• ... (same for the inductance)

Note that, whatever the kind of data you choose, you will get a pair of values. Also: each of these two values is a function of frequency. For example, at 1kHz, a given circuit can be equivalent to a 1\$\Omega\$ resistor in series with a 1 µF capacitor, but at 10kHz it will be equivalent to a 2\$\Omega\$ resistor in series with a 0.2 µF capacitor.

In other words: LCR meters (except some high-end ones, and only for some basic circuits) won't give you an equivalent circuit, with FIXED values for a large range of frequencies. They will give you a very basic circuit with values (capacitance, resistance) accurate for ONE frequency only.

On the contrary, you want to have a (rather) complicated equivalent circuit with FIXED values for the different components. Provided the circuit is a good model for the actual component (it is not always true), you can try to fit for example the pair (Z(f), \$\theta\$(f)) calculated with a set of parameters (Ra, Rc, CPE...) with the experimental values. When the set of parameter gives you a good fit, you can consider that your equivalent circuit with your set of parameters is a good description of your actual circuit.