Internal Resistance Interactive Calculator

Internal Resistance:

r = (ε - V) / I

Terminal Voltage:

V = ε - I·r

EMF (Open Circuit Voltage):

ε = V + I·r

Current through Circuit:

I = ε / (r + R)

Power Loss in Internal Resistance:

P_loss = I²·r

Circuit Efficiency:

η = (P_load / P_total) × 100% = (V / ε) × 100%

Variable Definitions:

• r = Internal resistance (Ω)
• ε = Electromotive force / EMF (V) - open circuit voltage
• V = Terminal voltage (V) - voltage under load
• I = Current (A)
• R = Load resistance (Ω)
• P_loss = Power dissipated in internal resistance (W)
• P_load = Power delivered to load (W)
• η = Efficiency (dimensionless, expressed as %)

▼ Why does terminal voltage decrease when current increases?

Terminal voltage decreases with increasing current due to the voltage drop across the source's internal resistance, which follows Ohm's law: V_drop = I·r. This internal resistance represents the cumulative resistance of all components within the source—electrode materials, electrolyte ionic conductivity in batteries, winding resistance in generators, or semiconductor junction resistance in power supplies. As current increases, the I·r product grows proportionally, subtracting a larger portion of the EMF before reaching the terminals. For example, a 12 V car battery with 0.1 Ω internal resistance shows 12 V at no load, but drops to 10 V when delivering 20 A to the starter motor. This phenomenon becomes critical in high-current applications where the internal voltage drop can exceed 10-20% of the EMF, significantly reducing available power and efficiency. The relationship is perfectly linear for resistive internal impedance, though reactive components in AC systems introduce frequency-dependent behavior that complicates analysis in power distribution networks.

▼ How does internal resistance affect battery life and performance?

Internal resistance directly governs battery power capability, thermal management requirements, and serves as the primary indicator of state of health throughout the battery's lifecycle. As batteries age through charge-discharge cycling, internal resistance increases due to electrode surface passivation (solid-electrolyte interphase growth in lithium-ion), electrolyte decomposition, active material loss, and current collector corrosion. This resistance increase manifests as reduced voltage under load, decreased available capacity (since the cutoff voltage is reached earlier), and elevated heat generation during operation. A battery with doubled internal resistance delivers only half the maximum power of a new cell, even if chemical capacity remains unchanged. The I²R power dissipation within the battery accelerates thermal degradation in a positive feedback loop—higher resistance causes more heating, which accelerates chemical degradation, further increasing resistance. Battery management systems monitor internal resistance through periodic impedance measurements, triggering replacement warnings when resistance exceeds 150-200% of the beginning-of-life value. In electric vehicles, pack-level resistance monitoring identifies weak cells requiring rebalancing or replacement before they compromise overall pack performance or create safety hazards through localized overheating.

▼ What causes the maximum power transfer to occur at 50% efficiency?

Maximum power transfer occurs when load resistance equals source internal resistance (R = r) because this condition optimizes the tradeoff between current magnitude and voltage across the load, even though it produces inherently low efficiency. When R << r, high current flows but most voltage is dropped internally, delivering little power to the load. When R >> r, high voltage appears across the load but current is minimal, again resulting in low power delivery. At R = r, the circuit current equals ε/(2r), terminal voltage equals ε/2, and load power reaches its maximum value of P_max = ε²/(4r). However, the source simultaneously dissipates identical power in its internal resistance, making total efficiency exactly 50%. This theorem finds application primarily in RF and telecommunications where signal power transfer matters more than efficiency—antenna impedance matching, audio amplifier output stages, and measurement instrumentation all operate near maximum power transfer conditions. In contrast, power delivery systems deliberately operate far from this point: switching power supplies maintain output impedance 50-100 times lower than load resistance to achieve 85-95% efficiency. The distinction highlights the fundamental difference between signal processing (maximize power transfer) and energy systems (maximize efficiency), representing different optimization objectives in electrical engineering design.

▼ How do you measure internal resistance accurately in practical circuits?

Accurate internal resistance measurement requires techniques that separate ohmic resistance from polarization, charge-transfer, and diffusion impedances that appear in electrochemical systems. The DC load-step method applies two different known currents (I₁ and I₂) while measuring corresponding terminal voltages (V₁ and V₂), calculating resistance from r = (V₁ - V₂)/(I₂ - I₁). This technique captures only the instantaneous resistive component, missing slower electrochemical processes. AC impedance spectroscopy provides superior characterization by injecting small sinusoidal currents across a frequency spectrum (0.01 Hz to 10 kHz) and analyzing the magnitude and phase of voltage response. The high-frequency asymptotic value represents pure ohmic resistance uncontaminated by diffusion processes, while the low-frequency impedance reveals charge-transfer kinetics and electrolyte transport limitations. Commercial battery analyzers typically use 1 kHz impedance as a standardized metric, providing repeatable measurements independent of state-of-charge variations. Practical considerations include temperature stabilization (resistance changes -0.5%/°C in lithium systems), rest period before measurement (allowing voltage relaxation after previous load), and contact resistance minimization through four-wire Kelvin connections. In production environments, automated testers perform simultaneous impedance measurements on hundreds of cells, sorting them into resistance-matched groups for series-parallel pack assembly—critical for preventing current imbalance and localized degradation in multi-cell systems.

▼ Why does cold temperature dramatically increase battery internal resistance?

Temperature reduction increases internal resistance through multiple parallel mechanisms affecting both ionic conductivity and charge-transfer kinetics in electrochemical systems. Electrolyte ionic conductivity decreases exponentially with temperature following an Arrhenius relationship with activation energy typically 0.15-0.25 eV for organic carbonate electrolytes in lithium-ion batteries. At -20°C, conductivity drops to 20-30% of room temperature values, directly increasing ohmic resistance. Additionally, charge-transfer resistance at electrode-electrolyte interfaces increases even more dramatically as the exchange current density (i₀) decreases exponentially with temperature, governed by the Butler-Volmer equation with activation energies of 0.4-0.6 eV. Lithium diffusion within electrode materials also slows, contributing diffusion-limited impedance that manifests as apparent resistance under DC load conditions. The cumulative effect produces 300-500% resistance increases at -20°C compared to 25°C, catastrophically reducing power capability—automotive starter batteries rated for 600 cold-cranking amps at -18°C may deliver less than 200 A, often insufficient for engine starting. Lead-acid batteries suffer additional degradation from sulfuric acid viscosity increases and reduced chemical reaction rates. Mitigation strategies include battery thermal management systems (heating elements maintaining temperature above 0°C), electrolyte formulation optimization with low-viscosity co-solvents, and battery management algorithms that limit current draw when low temperature is detected to prevent damage from lithium plating during charging or excessive voltage depression during discharge.

▼ How does internal resistance affect fast-charging capability?

Internal resistance fundamentally limits fast-charging rates through three interrelated constraints: maximum permissible voltage, thermal dissipation capability, and lithium plating prevention in lithium-ion systems. During charging, the charger must apply voltage V_charge = V_OCV + I·r, where V_OCV is the open-circuit voltage (state-of-charge dependent) and I·r is the internal resistance overpotential. As internal resistance increases with aging, higher charging voltages are required to maintain target current, eventually exceeding the cell's maximum voltage rating (typically 4.2 V for lithium-ion). The I²R power dissipation generates heat at rates proportional to the square of charging current—a cell with 50 mΩ internal resistance dissipates 12.5 W when fast-charging at 50 A (5C rate), requiring active cooling to prevent temperature rise above the safe operating limit of 45-60°C. Most critically, the internal voltage drop causes the anode potential to fall below lithium's plating potential during high-current charging, depositing metallic lithium dendrites that cause permanent capacity loss, increase internal resistance, and create internal short-circuit hazards. Fast-charging protocols therefore implement current tapering based on real-time internal resistance monitoring: initial constant-current charging at maximum safe rate transitions to constant-voltage mode as cells approach full charge, with current gradually reduced to prevent overvoltage. Advanced battery management systems measure impedance continuously, adapting charging current to maintain anode potential above the plating threshold—enabling 15-minute charging for low-resistance cells while protecting degraded cells through automatic current limitation.

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About the Author

Robbie Dickson — Chief Engineer & Founder, FIRGELLI Automations

Robbie Dickson brings over two decades of engineering expertise to FIRGELLI Automations. With a distinguished career at Rolls-Royce, BMW, and Ford, he has deep expertise in mechanical systems, actuator technology, and precision engineering.

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