Impedance/capacitance measurements

Question

I'm trying to do some impedance measurements on passive components with the simple circuit shown below. I find it difficult to make sense of the results and hope somebody can help me understand what I'm measuring and identify possible sources of error.

^{simulate this circuit – Schematic created using CircuitLab}

V_in is a function generator producing sine waves of various frequencies from 10 Hz to 10 MHz. R1 is a resistor. The device under test at Z is a resistor, capacitor or inductor.

I use the built-in measurement functions of my scope to measure peak-to-peak voltage at V1 and V2 as well as the phase of V2 relative to V1. All measurements are averaged over at least 128 samples. After representing the voltages as phasors, I compute the impedance Z as $$\mathbf{Z}=\dfrac{R_1\mathbf{V}_2}{\mathbf{V}_1-\mathbf{V}_2}$$ Then I obtain the resistive part of the impedance as the real part of Z and the capacitive or inductive part as $$L=\dfrac{\operatorname{Im}(\mathbf{Z})}{2\pi f}$$ and $$C=\dfrac{-1}{2\pi f\operatorname{Im}(\mathbf{Z})}$$ depending on whether the imaginary part is positive or negative, respectively.

I have R1 and the DUT on a breadboard. The voltages at V1 and V2 are measured using freshly compensated 10X probes. The output of the function generator is fed into the circuit using another scope probe in 1X mode.

The table below shows measured V1, V2 and phase angle as well as computed R, L and C values for a resistor and a few capacitors at different frequencies.

Observations:

• The resistor behaves resistively as expected, but the estimated value is about 5% off from what my ohmmeter says.
• The capacitance of the 10 pF capacitor is estimated almost 2 orders of magnitude too large.
• The 470 pF capacitor is better, but still a bit off.
• The 220 nF electrolytic has an almost purely resistive impedance.
• In a few cases, the resistance comes out negative. What does that mean?

I don't expect my experiment to be very accurate, but this is quite far off the mark. I'm not sure if the problem is with my experimental setup, or with the maths, or with both. Any suggestions to improve my measurements are appreciated!

                        Frequency  V1 (V)   V2 (V)  Theta (deg)    R (ohm)    L (nH)       C
Resistor 17.2k          10 Hz        4.14     3.28        0         18116          0       0
Resistor 17.2k          100 Hz       4.14     3.28       -0.02      18116          0      52.3  uF
Resistor 17.2k          1 kHz        4.13     3.28       -1.74      17775          0      61.1  nF
Resistor 17.2k          10 kHz       4.12     3.28       -0.9       18023          0      11.7  nF
Resistor 17.2k          100 kHz      4.12     3.28       -1         18426          0       1.0  nF
Ceramic 10 pF           1 kHz        4.17     4.16       -2.5        3598          0       1.47 nF
Ceramic 10 pF           10 kHz       4.12     3.97      -16.1        -167          0     964    pF
Ceramic 10 pF           100 kHz      4.04     1.38      -68.4          50          0     912    pF
Ceramic 10 pF           1 MHz        4.04     0.185     -78            35.8        0     735    pF  (noisy V2)
Ceramic 10 pF           10 MHz       2.8      0.04      136           -48        734       0        (very noisy V2)
Ceramic 470 pF          10 kHz       4.12     4.12       -6         -2375          0     351    pF
Ceramic 470 pF          100 kHz      4.08     2         -49.6         607          0     542    pF
Ceramic 470 pF          1 MHz        4.04     0.4       -62           190          0     351    pF
Ceramic 470 pF          10 MHz       2.8      0.07      103           -29       1830       0        (very noisy V2)
Electrolytic 220 nF     10 Hz        4.18     4.12       -2         45739          0     127    nF
Electrolytic 220 nF     100 Hz       4.17     4.12       -0.1      383325          0    28.5    nF
Electrolytic 220 nF     1 kHz        4.19     4.12        0        279571          0       0

Update:

Based on the answer I received, I made the following improvements to my setup:

1. I got rid of the breadboard and replaced it with a "free-floating" setup held together by the scope probes.
2. I varied the value of R1 across experiments to make sure V2 isn't too close to V1.
3. For the electrolytic capacitor, I added a DC offset to the input signal to make sure it doesn't get reverse-biased.

I also added measurements of another small ceramic capacitor of 47 pF.

After making these changes, I get much more reasonable results:

   Device under test         R1          f        V1          V2     Theta          R            L           C
      Resistor 17.2k   4.75 kOhm    10.00 Hz    4.18 V      3.28 V    1.2°    17.16 kOhm     27.26 H         0 F
      Resistor 17.2k   4.75 kOhm   100.00 Hz    4.20 V      3.28 V    1.2°    16.81 kOhm      2.47 H         0 F
      Resistor 17.2k   4.75 kOhm    1.00 kHz    4.12 V      3.28 V    0.3°    18.54 kOhm    80.81 mH         0 F
      Resistor 17.2k   4.75 kOhm   10.00 kHz    4.12 V      3.28 V    0.1°    18.55 kOhm     3.54 mH         0 F
      Resistor 17.2k   4.75 kOhm  100.00 kHz    4.12 V      3.29 V   -3.2°    17.60 kOhm         0 H   323.78 pF
       Ceramic 10 pF  47.40 kOhm    1.00 kHz    4.13 V      3.96 V  -16.2°    850.09 Ohm         0 H   977.01 pF
       Ceramic 10 pF   4.75 kOhm   10.00 kHz    4.12 V      3.96 V  -15.5°    157.60 Ohm         0 H   931.67 pF
       Ceramic 10 pF   4.75 kOhm  100.00 kHz    4.04 V      1.40 V  -66.9°     88.89 Ohm         0 H   891.58 pF
       Ceramic 10 pF  331.00 Ohm    1.00 MHz    2.84 V      1.36 V  -60.0°      4.46 Ohm         0 H   870.08 pF
       Ceramic 10 pF  331.00 Ohm   10.00 MHz    2.20 V   160.00 mV  -90.0°     -1.74 Ohm         0 H   664.64 pF
      Ceramic 470 pF  47.50 kOhm   10.00 kHz    4.12 V      2.44 V  -54.3°   -370.72 Ohm         0 H   459.50 pF
      Ceramic 470 pF   4.75 kOhm  100.00 kHz    4.04 V      2.41 V  -52.5°     55.03 Ohm         0 H   445.72 pF
      Ceramic 470 pF   4.75 kOhm    1.00 MHz    4.03 V   363.00 mV  -75.7°     69.68 Ohm         0 H   369.91 pF
      Ceramic 470 pF  331.00 Ohm    1.00 MHz    3.22 V      2.49 V  -37.8°     11.48 Ohm         0 H   381.39 pF
      Ceramic 470 pF  331.00 Ohm   10.00 MHz    1.91 V   268.00 mV  -74.0°      6.67 Ohm         0 H   335.93 pF
 Electrolytic 220 nF  47.50 kOhm    10.00 Hz    4.13 V      3.49 V  -32.3°     31.70 Ohm         0 H   211.87 nF
 Electrolytic 220 nF   4.75 kOhm   100.00 Hz    4.16 V      3.51 V  -34.3°   -222.56 Ohm         0 H   224.00 nF
 Electrolytic 220 nF  331.00 Ohm    1.00 kHz    3.55 V      3.24 V  -24.0°      1.59 Ohm         0 H   214.28 nF
 Electrolytic 220 nF  331.00 Ohm   10.00 kHz    2.54 V   592.00 mV  -68.8°     11.20 Ohm         0 H   196.00 nF
       Ceramic 47 pF  331.00 Ohm   10.00 MHz    2.30 V      1.30 V  -54.6°      3.96 Ohm         0 H    69.36 pF
       Ceramic 47 pF  331.00 Ohm   20.00 MHz    1.88 V   680.00 mV  -68.2°      1.34 Ohm         0 H    61.72 pF
       Ceramic 47 pF  331.00 Ohm   40.00 MHz    2.56 V   480.00 mV  -76.3°      3.24 Ohm         0 H    62.45 pF

I'm still puzzled by two things:

1. The measurements of the 470 pF and the 47 pF ceramic caps are quite good now. Why is the 10 pF cap so far off (higher than the 470pF)? The setup and the sizes of the caps are exactly the same, I can't think of a reason why there should be much greater parasitics.
2. In a few cases, I still get negative resistances. How should I interpret those?

Answer 1

This looks like reasonable experimental error to me, given the nature of your measurements.

Trying to measure 10pF with a breadboard circuit is not going to work. The parasitic capacitances will overwhelm your measurements.

What is the tolerance on the 470pF capacitor? It's not unusual for capacitors to be +/- 20%. Your data for it looks pretty good to me considering the crude setup you have. Electrolytic capacitors can be -20%/+80%...and you are measuring a polarized capacitor with an ac waveform.

Also, any measurement where the phase difference between V1 and V2 is small, or where V2 is very small compared to V1, will be prone to having a large error.