I'm studying semiconductor parameter analyzers. Those instruments are based on SMU (source and measuring unit) which can work either in the V-mode (they force a voltage on the device under test and measure the corresponding current) or in the I-mode (they force a current in the DUT and measure the corresponding voltage). Suddenly, it appears this slide:

It is just one slide and no further information is provided, but I cannot understand how this circuit works. On the web I found different configurations of this circuit.
I might surmise that the "force" and "sense" words in the picture make reference to the "source and measurement" properties of the SMU. Moreover, why do we have a variable battery? Why do we have two "force" wires (if I force a current through Rf1 like in the picture, I don't need another "forcing" action on the DUT through Rf2)? Why do we have two "sense" wires?

Thank you

The diagram is really confusing. In the 4-wire connection below on the right, for resistance measurement, the "force" wires are the ones (with red arrows) providing the current for the resistor under test. Through the "sense" wires only the current required by the voltmeter is circulating, causing a much lower drop.

In this example both current and voltage are being measured but the voltage source and current measurement could be replaced by a known current source. With the circuit on the left when you calculate the resistance from the current and voltage values you are actually calculating the resistance of the probe wires added to the DUT.

Regarding the "force voltage" "measure current" you mention, imagine if on the circuit on the left you didn't have the voltmeter. You would measure the current passing through the DUT, but the known voltage would be applied to the series association of the DUT, probe wires and the current meter. This would make the error even worse.

I might surmise that the "force" and "sense" words in the picture make reference to the "source and measurement" properties of the SMU.

Correct. Each SMU can output a "force" signal (the excitation signal), and it can measure ("sense") the effects of that signal upon the device under test (DUT).

Moreover, why do we have a variable battery?

The variable battery is called a "ground unit" (GNDU). It is an active circuit that produces a very precise reference potential for the parametric measurement. The circuit ground is usually too noisy to be useful for ultra-precise voltage and current measurements. For example, a Keysight Technologies B1505A Power Device Analyzer can measure currents down to "sub picoamp" levels (<1E-12 amps). This level of measurement resolution would be impossible without an ultra-clean, actively-driven "virtual ground" reference potential.

Why do we have two "force" wires (if I force a current through Rf1 like in the picture, I don't need another "forcing" action on the DUT through Rf2)? Why do we have two "sense" wires?

Consider the circuit shown in Figure 1. Voltmeter VM1 has very high input impedance; therefore, all (or nearly all) of I1's current flows through the two test leads and the DUT.

^{simulate this circuit – Schematic created using CircuitLab}

Figure 1. 2-wire resistance measurement.

The voltage measured by voltmeter VM1 is

$$
VM1 = I1,(R_TestLead + R_DUT + R_TestLead)
$$

Current source $I1$ outputs a known (calibrated) current level. Voltmeter VM1 measures the voltage across the two test leads and the DUT and then the device analyzer uses Ohm's law to calculate the measured value of the DUT's resistance $R_DUT(meas)$:

$$
R_DUT(meas) = frac VM1I1
\
= frac I1,(R_TestLead + R_DUT + R_TestLead)I1
\
= R_TestLead + R_DUT + R_TestLead
$$

EXAMPLE 1

The resistance in each test lead is $100,mOmega$, and the DUT's true resistance value is $R_DUT=1,kOmega$. In this case test the resistance in the two test leads introduces an error of about 0.02% to the measurement of the DUT's value.

$$
Error% = frac Measured-TrueTrue times 100
\
= frac R_DUT(meas)-R_DUTR_DUT times 100
\
= frac (0.1+1000+0.1),Omega - 1000,Omega1000,Omega times 100
\
= 0.02,%
$$

Therefore, if the resistance in the test leads is much less than the DUT's resistance, then we can ignore the test lead resistance terms and calculate $R_DUT$ as

$$
R_DUT(meas) approx frac VM1I1 biggrvert_R_TestLead <<< R_DUT
$$

If, however, the DUT's resistance is very small—e.g., a few ohms or less, then the test lead resistance cannot be ignored because it adds significant error into the measurement of the DUT's resistance.

EXAMPLE 2

The resistance in each test lead is $100,mOmega$, and the DUT's true resistance value is $R_DUT=1,Omega$. In this test case the resistance in the two test leads introduces an error of about 16.7% to the measurement of the DUT's value. And this error doesn't include the voltmeter's own measurement error, which further increases the overall uncertainty in the measurement.

$$
Error% = frac Measured-TrueTrue times 100
\
= frac R_DUT(meas)-R_DUTR_DUT times 100
\
= frac (0.1+1.0+0.1),Omega - 1.0,Omega1.0,Omega times 100
\
= 16.7,%
$$

To improve the device analyzer's measurement accuracy when measuring small resistances, a "4-wire" Kelvin measurement (Figure 2) must be used instead of the "2-wire" connection shown in Figure 1.

^{simulate this circuit}

Figure 2. Kelvin 4-wire resistance measurement.

Current source $I1$ outputs a known (calibrated) current level. Recall that voltmeter VM1 has very high input impedance, and therefore almost no current flows through VM1. Likewise, almost no current flows through the "sense" test leads (R_SENSE), and therefore there is no voltage change (voltage drop) across the sense test lead resistance R_SENSE, $V_R_SENSEapprox0,V$, which means the voltmeter is measuring the voltage at the DUT's input terminals:

$$
VM1 = frac I1,R_DUT,R_VM1R_DUT+2R_SENSE+R_VM1
$$

Note that if the voltmeter's input impedance $R_VM1$ is very high, then via L'Hôpital's rule:

$$
lim_R_VM1rightarrow infty VM1 = I1,R_DUT = V_DUT
$$

The measured voltage in a 4-wire Kelvin measurement, when used to calculate the DUT's resistance, yields a calculated resistance value that is much closer to the DUT's true resistance value when compared to a 2-wire measurement. If the device analyzer calculates the DUT's resistance value using

$$
R_DUT(calc) := frac VM1I1
= frac R_DUT,R_VM1R_DUT+2R_SENSE+R_VM1
$$

then the nominal percent error in the Kelvin measurement of the DUT's resistance due to circuit loading by the two sense test leads and the voltmeter's input impedance is

$$
Error% := frac R_DUT(calc)-R_DUTR_DUT times 100
\
= - frac R_DUT + 2 R_SENSE R_DUT + 2 R_SENSE + R_VM1 times 100
$$

EXAMPLE 3

The resistance in each sense test lead is $R_SENSE=0.1,Omega$, and the DUT's true resistance is $R_DUT=1,Omega$, and the voltmeter's input impedance is $R_VM1=1,GOmega$. The series impedance of the two sense test leads and voltmeter, when placed in parallel with the DUT, introduces a loading error of -0.00000012% into the calculated value for the DUT's resistance.