Understanding Resistance R vs Internal Resistance r in DC Circuits

In circuit theory, internal resistance r (lowercase) is the inherent opposition to current flow within a power source itself, while uppercase R represents the external load resistance connected to that source. This single lowercase letter is the reason a freshly charged 12V battery might output 13.2V on your multimeter, but immediately sag to 11.5V the moment you connect a high-draw DC motor. Understanding what r changes in a real installation is critical: it dictates your maximum available short-circuit current, causes internal heat dissipation, and determines whether your Battery Management System (BMS) will prematurely trip a low-voltage cutoff under heavy loads.

Beginners commonly confuse internal resistance r with load resistance R, assuming a power supply will deliver its rated voltage regardless of the current drawn. Others mistakenly conflate DC internal resistance with AC impedance (Z), which includes frequency-dependent reactance. To visualize the concept, think of a water pump (the battery) with a narrow, kinked output hose (internal resistance r) feeding a large sprinkler system (load resistance R). The kink limits the maximum flow rate and drops the pressure before the water even reaches the sprinkler heads.

The Core Difference: Load Resistance (R) vs. Internal Resistance (r)

When you read a schematic, uppercase R designates the components doing the useful work—resistors, heating elements, motor windings, or LED strings. Lowercase r designates the parasitic, unavoidable resistance inside the voltage source and its immediate conductors.

Every real-world power source has an electromotive force (EMF, denoted as E), which is the ideal, unloaded voltage. The actual terminal voltage (V_term) you can measure with a multimeter while the circuit is active is governed by the equation:

V_term = E - (I × r)

Where I is the current flowing through the circuit. Because r is in series with R, any current drawn by the load must also pass through the source's internal resistance, generating a voltage drop that subtracts directly from the available terminal voltage.

Worked Example: Calculating Voltage Sag Under Load

Let’s look at a real-world bench scenario. You are powering a 12V DC water pump from a 12V LiFePO4 battery.

• Source EMF (E): 13.2V (measured open-circuit)
• Internal Resistance (r): 0.05 Ω (typical for a mid-size LiFePO4 cell)
• Load Resistance (R): 2.15 Ω (the effective operating resistance of the pump)

First, calculate the total circuit resistance and the actual current draw:

1. Total Resistance: R_total = R + r = 2.15 Ω + 0.05 Ω = 2.20 Ω
2. Circuit Current (I): I = E / R_total = 13.2V / 2.20 Ω = 6.0 A

Now, calculate the voltage drop caused strictly by the internal resistance:

• Internal Voltage Drop: V_drop = I × r = 6.0 A × 0.05 Ω = 0.30 V
• Terminal Voltage: V_term = 13.2V - 0.30V = 12.9 V

Finally, consider the thermal impact. The power dissipated as heat inside the battery is P = I² × r. In this case, 6.0² × 0.05 = 1.8 Watts. While 1.8W won't melt a 100Ah battery, if you were pulling 100A from a cell with a degraded r of 0.15 Ω, you'd be dissipating 1,500W internally—leading to rapid thermal runaway.

Where You Meet Internal Resistance (r) in Practice

You won't see a discrete component labeled "r" on a PCB, but you will fight its effects constantly in physical installations.

Battery Health and BMS Cutoffs

As lithium-ion and LiFePO4 cells age, their internal resistance increases due to Solid Electrolyte Interphase (SEI) layer growth and electrolyte depletion. I frequently see DIY solar builders complain that their 2000W inverter trips the BMS low-voltage disconnect (LVD) when the microwave turns on, even though the battery monitor says 80% State of Charge. The culprit is high r. The massive current spike causes a severe internal voltage sag, dropping the terminal voltage below the BMS cutoff threshold (usually around 10.5V for 12V systems) for a fraction of a second.

Solar Panel I-V Curves

In photovoltaics, series resistance (R_s) inside the silicon cell and the silver grid fingers limits the "fill factor" of the panel. A panel with high internal resistance will show a rounded, sloping shoulder on its Current-Voltage (I-V) curve, drastically reducing the Maximum Power Point (MPPT) harvest compared to a panel with low R_s.

PCB Traces and Wire Sizing

When designing a custom PCB for an ESP32 or Arduino, the copper traces themselves possess internal resistance. A standard 1-ounce copper trace that is 10 mils wide has an r of roughly 0.05 Ω per inch. If you route a 2A motor supply through 4 inches of this trace, you lose 0.4V before the power even reaches the motor driver IC, potentially causing logic brownouts.

Typical Internal Resistance (r) Values by Chemistry

Knowing the baseline r for different chemistries helps you select the right power source for high-drain applications. The data below reflects typical values for fresh, room-temperature (25°C) cells.

For deeper technical analysis on how degradation impacts these numbers over time, the Cadex Battery University guide on internal resistance provides excellent baseline metrics for aging cells. Furthermore, Electronics Tutorials on Internal Resistance offers a solid mathematical breakdown of how these values interact in complex mesh circuits.