A resistor bridge calculator solves for either the differential output voltage ($V_{out}$) of an unbalanced bridge or the unknown resistance ($R_x$) in a balanced bridge. For a standard unbalanced bridge with excitation voltage $V_{in}$, the output is calculated as: V_out = V_in * [R2 / (R1 + R2) - R4 / (R3 + R4)]. If the bridge is balanced ($V_{out} = 0$), the unknown resistor is calculated as R_x = R3 * (R2 / R1).
Whether you are designing a strain gauge load cell interface or debugging a platinum RTD temperature circuit, understanding the exact math behind the bridge prevents costly ADC saturation and noise floor issues. Below is the complete derivation, symbol mapping, and real-world magnitude data you need to size your components correctly.
The Core Bridge Equations and Symbol Definitions
A Wheatstone bridge consists of two parallel voltage dividers sharing a common excitation source. The output is the potential difference between the two midpoints. We use two primary formulas depending on the state of the bridge.
1. Unbalanced Bridge (Output Voltage):
V_out = V_in * ( R2 / (R1 + R2) - R4 / (R3 + R4) )
2. Balanced Bridge (Unknown Resistor):
R_x = R3 * (R2 / R1) (Assuming R4 is the unknown R_x and V_out = 0)
| Symbol | Parameter | Standard Unit | Practical Notes |
|---|---|---|---|
V_out |
Differential Output Voltage | Volts (V) | Measured between the midpoints of the two legs. Can be positive or negative. |
V_in |
Excitation / Input Voltage | Volts (V) | DC supply applied across the top and bottom nodes. Must be low-noise. |
R1, R2 |
Leg 1 Resistors | Ohms (Ω) | R1 is top, R2 is bottom. Midpoint is V_A. |
R3, R4 |
Leg 2 Resistors | Ohms (Ω) | R3 is top, R4 is bottom. Midpoint is V_B. |
R_x |
Unknown / Sensor Resistor | Ohms (Ω) | The variable element (e.g., strain gauge) replacing one fixed resistor. |
To ground these formulas in reality, here is a data-dense table showing what a resistor bridge calculator actually outputs for common physical sensors. Notice how small the differential voltages are compared to the excitation.
| Sensor Type | Nominal R (Ω) | Typical ΔR under load | Standard V_in (V) | Expected V_out Range |
|---|---|---|---|---|
| Foil Strain Gauge | 350 Ω | +0.1 Ω to +0.5 Ω | 5.0 V | 350 μV to 1.8 mV |
| Pt100 RTD (0°C to 100°C) | 100 Ω to 138.5 Ω | +38.5 Ω | 3.3 V | 0 mV to 285 mV |
| NTC Thermistor (10k) | 10,000 Ω | -4,000 Ω (at +25°C delta) | 5.0 V | 0 V to ~1.2 V (Highly non-linear) |
| CdS Photocell | 1,000 Ω (light) | +9,000 Ω (dark) | 5.0 V | 0 V to ~2.0 V |
Rearranged Forms for Component Selection
When designing a bridge, you rarely just 'calculate the output'. You usually know your desired output swing and your sensor's delta-R, and you need to back-calculate your fixed resistors or excitation voltage. Here are the algebraically rearranged forms of the unbalanced bridge equation (assuming R1=R3 and R2 is the sensor R_s):
- Solving for Excitation Voltage (V_in):
V_in = V_out / ( R2 / (R1 + R2) - R4 / (R3 + R4) ) - Solving for Fixed Resistor (R1, assuming R2 is known sensor):
R1 = R2 * ( (V_in - V_B) / V_B )where V_B is the voltage at the opposite midpoint. - Solving for Sensor Resistance (R4, given measured V_out):
R4 = R3 * ( V_A - V_out ) / ( V_in - V_A + V_out )where V_A is the voltage at Leg 1 midpoint.
V_in to get a larger V_out. Higher excitation voltages cause self-heating in the resistors (I²R losses), which introduces thermal drift that completely destroys your measurement accuracy. Stick to 3.3V or 5V and use gain in the instrumentation amplifier instead.
Worked Examples with Unit Tracking
Abstract formulas fail on the bench. Here are two step-by-step calculations tracking units from raw ohms to final ADC-readable voltages.
Example 1: Unbalanced Strain Gauge Bridge
Scenario: You have a 350 Ω strain gauge. Unloaded, all four bridge resistors are exactly 350.0 Ω. Under load, the gauge stretches, and its resistance increases to 350.2 Ω. Your excitation is a clean 5.000 V DC supply. What is the differential output?
- Identify Knowns:
V_in = 5.000 V
R1 = 350.0 Ω,R2 = 350.0 Ω
R3 = 350.0 Ω,R4 = 350.2 Ω(The stretched gauge) - Calculate Leg 1 Midpoint Voltage (V_A):
V_A = V_in * (R2 / (R1 + R2))
V_A = 5.000 V * (350.0 Ω / (350.0 Ω + 350.0 Ω))
V_A = 5.000 V * (350.0 / 700.0) = 5.000 V * 0.5 = 2.50000 V - Calculate Leg 2 Midpoint Voltage (V_B):
V_B = V_in * (R4 / (R3 + R4))
V_B = 5.000 V * (350.2 Ω / (350.0 Ω + 350.2 Ω))
V_B = 5.000 V * (350.2 / 700.2) = 5.000 V * 0.49985718... = 2.4992859 V - Calculate Differential Output (V_out):
V_out = V_A - V_B
V_out = 2.50000 V - 2.4992859 V = 0.0007141 V - Convert to Engineering Units:
0.0007141 V * 1000 = 0.7141 mV(or714.1 μV)
Result: The bridge outputs 0.714 mV. This tiny signal requires an instrumentation amplifier with a gain of at least 500 to map it to a standard 0-3.3V ADC range.
Example 2: Balanced Bridge for Unknown Precision Resistor
Scenario: You are using a manual bridge to find the exact resistance of a custom wirewound resistor ($R_x$). You adjust the precision potentiometer ($R_3$) until your center-zero galvanometer reads exactly 0 V.
- Identify Knowns at Balance:
V_out = 0 V
R1 = 1.000 kΩ = 1000 Ω
R2 = 2.500 kΩ = 2500 Ω
R3 (adjusted) = 4.750 kΩ = 4750 Ω - Apply Balanced Formula:
R_x = R3 * (R2 / R1) - Substitute and Track Units:
R_x = 4750 Ω * (2500 Ω / 1000 Ω)
Note: The units in the parenthesis cancel out (Ω/Ω = dimensionless ratio).
R_x = 4750 Ω * 2.5
R_x = 11875 Ω - Convert to Standard Prefix:
11875 Ω = 11.875 kΩ
Result: The unknown resistor is exactly 11.875 kΩ.
Assumptions, Unit Traps, and Realistic Magnitudes
A resistor bridge calculator is only as good as the assumptions you feed it. If your physical circuit violates these assumptions, your calculated numbers will not match your multimeter readings.
When the Formula Applies (and Core Assumptions)
- Ideal Excitation Source: The formula assumes $V_{in}$ has zero internal impedance. If you are powering the bridge from a high-impedance source (like a weak voltage divider or a nearly depleted battery), the bridge current will pull $V_{in}$ down, invalidating the calculation.
- Infinite Load Impedance: The unbalanced formula assumes no current flows out of the $V_{out}$ terminals. If you connect a standard DMM with 10 MΩ input impedance to a high-resistance bridge (e.g., 1 MΩ thermistors), the meter itself will load the bridge and skew the reading. You must buffer the output with an op-amp or use a high-impedance instrumentation amplifier like the TI INA125.
- Linear Components: The math assumes resistors are perfectly linear. If you use a thermistor or a varistor, the bridge will be highly non-linear, and this DC formula only gives you the static operating point, not the transfer function.
Unit Mistakes That Break the Math
The most common error when using a resistor bridge calculator is mixing units. If R1 is 10 kΩ and R2 is 350 Ω, you cannot type '10' and '350' into the ratio. You must convert both to base Ohms (10,000 and 350) or both to kilo-ohms (10 and 0.35). The ratio
R2/(R1+R2) is dimensionless, but only if the numerator and denominator share the exact same prefix.
Another fatal mistake is treating $V_{out}$ as a single-ended, ground-referenced voltage. $V_{out}$ is strictly differential. If you measure Leg 1 midpoint to Ground, you are only reading half the bridge. You must measure across the two midpoints, or your ADC must be configured for differential input (e.g., AIN0 - AIN1).
What a Realistic Answer Magnitude Looks Like
If your calculator spits out a $V_{out}$ of 2.5V for a strain gauge bridge, stop and check your inputs. You likely typed the delta-R wrong. As shown in the sensor table above, physical parameter sensors (strain, pressure, microscopic temperature shifts) yield differential outputs in the microvolt (μV) to low millivolt (mV) range.
A balanced bridge should yield exactly 0.000 V. In reality, due to resistor tolerance (even 0.1% precision resistors have a 0.1% mismatch), an 'unloaded' physical bridge will often show a residual offset voltage of 1 mV to 5 mV. This is why bridge circuits almost always require software or hardware offset nulling before taking actual measurements.
For deeper reading on bridge linearization and amplifier interfacing, refer to the foundational texts on Wheatstone Bridge circuits at All About Circuits and the Wheatstone Bridge tutorials at Electronics-Tutorials, which cover the AC impedance extensions of these DC formulas.
