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Internal resistance

Internal resistance is the inherent opposition to the flow of electric current within the materials of a power source, such as a battery or electrochemical cell, typically modeled as a resistor connected in series with the ideal electromotive force (EMF) of the source.^[1] This resistance arises from the electrolyte, electrodes, and other internal components, converting some electrical energy into heat and reducing the effective output voltage under load.^[2] In practical terms, the terminal voltage V across a battery with EMF \mathcal{E} and internal resistance r when delivering current I is given by V = \mathcal{E} - Ir, demonstrating how internal resistance causes a drop in voltage proportional to the current drawn.^[1] For an ideal source with no internal resistance, the terminal voltage equals the EMF regardless of load; however, real sources exhibit this limitation, which becomes more pronounced at higher currents, potentially leading to zero or negative terminal voltage if the load resistance is too low.^[3] This effect is critical in applications like electric vehicles and portable electronics, where high current demands amplify the impact. The magnitude of internal resistance varies by battery chemistry, state of charge, temperature, and age; for instance, it differs between charging and discharging in lithium-ion batteries and increases over time due to degradation, signaling reduced state of health.^[4] High internal resistance lowers energy efficiency by dissipating power as heat (P = I^2 r), limits maximum power output, and can cause thermal runaway in extreme cases, making its measurement essential for battery management systems.^[5] Methods to determine it include DC load testing and AC impedance spectroscopy, with values typically ranging from milliohms in high-performance cells to ohms in older or low-quality units.^[6]

Fundamentals

Definition

Internal resistance, denoted as r, refers to the inherent opposition to current flow within a real voltage source, such as a battery or generator, which arises from the materials and construction of the source itself.^[7] This resistance causes the voltage measured across the source's terminals (terminal voltage) to be lower than the source's electromotive force (emf), denoted as \mathcal{E}, when current is drawn.^[1] In the basic circuit model, a real voltage source is represented as an ideal emf \mathcal{E} connected in series with the internal resistance r. The relationship between terminal voltage V, emf \mathcal{E}, current I, and internal resistance follows from Ohm's law applied to the equivalent circuit:

V = \mathcal{E} - Ir

Under open-circuit conditions (no current flow, I = 0), the terminal voltage equals the emf (V = \mathcal{E}); however, when a load is connected and current flows, the voltage drop across r (Ir) reduces the terminal voltage below the emf.^[1]^[7] The unit of internal resistance is the ohm (\Omega), consistent with standard resistance measurements. Typical values vary by source type but are often small; for example, a standard AA alkaline battery might have an internal resistance of 0.1–1 \Omega, depending on its chemistry and state of charge.^[8]^[9] This internal resistance plays a key role in the performance of electrochemical cells like batteries.^[7]

Physical origins

Internal resistance in voltage sources originates from microscopic physical mechanisms that impede the flow of charge carriers within the materials and interfaces. In electrochemical systems, the primary contributions include ionic resistance arising from the limited mobility of ions in the electrolyte, electronic resistance due to impeded electron transport in the electrodes, and contact resistance at the interfaces between components such as electrodes and current collectors.^[10] These ohmic losses collectively manifest as the internal resistance observed in circuit models. In solid conductors, such as the metallic components of electrodes, electrical resistance fundamentally stems from the scattering of conduction electrons by lattice imperfections and thermal vibrations. Key scattering mechanisms involve impurities that disrupt the periodic lattice potential, phonons representing quantized lattice vibrations that cause dynamic distortions, and structural defects like vacancies or dislocations that further localize electron paths.^[11] This scattering reduces the mean free path of electrons, directly increasing resistivity according to the Drude model of conduction.^[12] The dependence of resistance on temperature is a critical aspect, with most metallic materials exhibiting a positive temperature coefficient where resistivity rises with increasing temperature. This behavior results from intensified electron-phonon scattering at higher temperatures, as thermal energy amplifies lattice vibrations. The linear approximation for small temperature changes is given by

\rho = \rho_0 (1 + \alpha \Delta T),

where \rho is the resistivity at temperature T, \rho_0 is the reference resistivity, \alpha is the temperature coefficient of resistivity, and \Delta T = T - T_0./University_Physics_II_-Thermodynamics_Electricity_and_Magnetism(OpenStax)/09%3A_Current_and_Resistance/9.04%3A_Resistivity_and_Resistance) Material properties play a pivotal role in determining the magnitude of internal resistance. For electrodes, selecting high-conductivity metals like copper or aluminum minimizes electronic resistance by facilitating efficient electron transport.^[13] In electrolytes, the composition influences ionic resistance through factors such as viscosity, which inversely affects ion mobility; higher viscosity, often from solvent or salt choices, hinders ion diffusion and elevates resistance.^[14]

Electrochemical sources

In batteries

Internal resistance in batteries arises from the opposition to ion and electron flow within the electrochemical system, manifesting differently in primary and secondary types. Primary batteries, such as alkaline and zinc-carbon cells, are non-rechargeable and typically exhibit higher internal resistance due to their design for one-time use, with values around 150–300 mΩ for a fresh AA alkaline battery.^[15] In contrast, secondary or rechargeable batteries like lead-acid and lithium-ion cells are engineered for repeated cycling and feature lower internal resistance to support higher power delivery; for instance, a typical lead-acid cell has an internal resistance on the order of 1–5 mΩ, while lithium-ion cells, such as 18650 formats, often measure below 0.1 Ω, around 35 mΩ.^[16]^[17] These differences stem from the electrolyte composition and electrode materials, where primary cells prioritize energy density over rate capability, leading to greater ohmic losses during discharge.^[18] The internal resistance in batteries varies with the state of charge (SOC), generally increasing as the battery discharges due to ion depletion in the electrolyte and at electrode interfaces, which elevates ohmic and polarization losses.^[19] In lithium-ion batteries specifically, this rise is exacerbated by dendrite formation on the anode during low-SOC conditions or fast charging, where uneven lithium plating creates irregular metallic structures that increase local resistance and risk short circuits. Such variations can reduce effective voltage output by up to 20–30% at low SOC, limiting usable capacity and power delivery.^[20] Aging further amplifies internal resistance in batteries through electrode degradation, such as cracking and loss of active material in lithium-ion cathodes, which diminishes conductive pathways and raises ohmic resistance by 50–100% over hundreds of cycles.^[21] Electrolyte breakdown, including decomposition and solid electrolyte interphase (SEI) thickening, contributes to this by forming insulating layers that impede ion transport, ultimately leading to capacity fade of 20–30% and reduced overall efficiency.^[22] These effects are more pronounced in secondary batteries under repeated charge-discharge, shortening service life. Historically, early batteries like the Daniell cell of 1836 suffered from high internal resistance, around 3–5 Ω, due to porous separators and dilute electrolytes that limited ion mobility.^[23] Modern advancements have drastically reduced this through additives like conductive polymers and nanotechnology, such as graphene or carbon nanotubes, which enhance electrode conductivity and lower resistance to below 10 mΩ in lithium-ion systems, improving power density and cycle life.^[24]

In fuel cells

In fuel cells, internal resistance arises primarily from ohmic losses in the ionic conduction through the electrolyte and electronic conduction through electrodes and interconnects, differing from batteries due to the continuous supply of gaseous reactants that maintains fresh interfaces during operation.^[25] Key fuel cell types exhibiting distinct resistance characteristics include proton exchange membrane (PEM) fuel cells, which operate at low temperatures (around 80°C), and solid oxide fuel cells (SOFC), which function at high temperatures (600–1000°C).^[26] In PEM fuel cells, the primary contributor to internal resistance is the proton-conducting membrane, such as Nafion, with ohmic losses typically ranging from 0.1 to 0.5 Ω cm² under standard conditions, alongside contributions from bipolar plates (providing electronic pathways and gas distribution) and gas diffusion layers (facilitating reactant transport while adding electronic resistance).^[27] The ohmic overpotential is given by η_ohmic = i × (r_m + r_e), where i is the current density, r_m is the membrane ionic resistance, and r_e represents electronic resistances from other components.^[26] Operational factors significantly influence resistance; for instance, increasing humidity enhances proton conductivity in the membrane, thereby decreasing r_m and overall ohmic losses.^[28] In SOFCs, internal resistance stems from the solid ceramic electrolyte (often yttria-stabilized zirconia), which dominates ionic conduction losses, as well as from electrodes, interconnects acting as bipolar plates, and gas diffusion layers adapted for high-temperature gas flow.^[29] Elevated operating temperatures reduce electrolyte resistance by enhancing ionic mobility, though this can exacerbate other losses like electrode polarization.^[30] Compared to batteries, fuel cells exhibit lower internal resistance in steady-state operation because the continuous flow of reactants prevents buildup of reaction products at electrode interfaces, supporting higher power densities up to several hundred mW/cm².^[25] This steady-state advantage enables more consistent performance over extended durations without the degradation seen in static storage systems.^[31]

Measurement techniques

Direct methods

Direct methods for measuring internal resistance rely on applying a direct current load to the voltage source and observing the resulting voltage drop, based on the basic circuit model where the source has an electromotive force \epsilon in series with internal resistance r. The most common technique is the load method, which involves connecting a known external load resistor R_{\text{ext}} across the source terminals.^[32] To perform the measurement, first determine the open-circuit voltage \epsilon using a multimeter configured as a voltmeter connected directly across the source terminals, ensuring no load is attached. Next, connect the external load R_{\text{ext}} (typically chosen to draw a moderate current, such as 10-20% of the source's rated capacity) and an ammeter in series with the source. Measure the loaded terminal voltage V across the source terminals with the voltmeter and the current I through the circuit with the ammeter. The internal resistance is then calculated using the formula

r = \frac{\epsilon - V}{I},

where I = \frac{V}{R_{\text{ext}}} if the ammeter is not used and R_{\text{ext}} is precisely known (measured separately with an ohmmeter). To improve accuracy, multiple loads can be tested, and the results plotted as V versus I, where the slope is -r and the y-intercept is \epsilon.^[33]^[34] When using instruments, account for their internal resistances: the ammeter's low resistance (typically 0.1-1 \Omega) adds to the total circuit resistance and slightly reduces the measured current, while the voltmeter's high input resistance (often >10 M\Omega) draws negligible current and minimally affects the voltage reading. Lead and connection resistances should also be minimized by using short, thick wires.^[35] This method offers advantages in its simplicity, requiring only basic equipment like multimeters (functioning as voltmeters and ammeters) and resistors, making it ideal for educational demonstrations and quick assessments without specialized apparatus.^[32]^[34] However, it has limitations, particularly for sources with very low internal resistance (e.g., <0.01 \Omega), where lead and contact resistances can introduce errors exceeding 10-100% of the true value unless a four-wire Kelvin connection is employed to isolate them. It is also less suitable for high-power sources, as the required load currents may cause significant self-heating or safety risks during testing.^[36]^[5]

Indirect methods

Indirect methods for measuring internal resistance employ non-invasive techniques that perturb the electrochemical system minimally, often using alternating current (AC) signals or transient responses, enabling in-situ assessments without significant disruption to device operation. These approaches provide insights into both the total resistance and its individual components, such as ohmic and charge transfer resistances, by analyzing frequency-dependent or time-domain behaviors. Unlike direct current (DC) methods, indirect techniques account for capacitive and inductive effects, offering higher resolution for dynamic systems like batteries and fuel cells. Electrochemical impedance spectroscopy (EIS) is a widely adopted indirect method that applies a small-amplitude AC signal (typically 5-10 mV) over a broad frequency range, from millihertz to megahertz, to probe the impedance response of the system. The resulting data are plotted in a Nyquist diagram, where the imaginary impedance (Z'') is graphed against the real impedance (Z') for each frequency. The internal resistance, often referred to as the ohmic resistance R_\Omega, is determined from the high-frequency intercept of the Nyquist plot on the real axis, corresponding to Z' = R_\Omega as the angular frequency \omega \to \infty. This intercept represents the uncompensated resistance, including contributions from the electrolyte, electrodes, and contacts, while lower-frequency arcs reveal polarization effects like charge transfer. EIS is particularly valuable for resolving these components in operating devices, as demonstrated in lithium-ion battery diagnostics where it identifies degradation mechanisms through spectral fitting.^[37] The AC milliohm meter method involves injecting a sinusoidal AC current at a fixed frequency, commonly 1 kHz, and measuring the resulting AC voltage drop across the terminals. The internal resistance is calculated as r = \frac{V_{ac}}{I_{ac}}, where V_{ac} and I_{ac} are the amplitudes of the voltage and current signals, respectively. At this frequency, the method primarily captures the resistive component while minimizing influences from capacitance and inductance in most battery systems. This technique is rapid and suitable for production-line testing of lithium-ion cells, where low-amplitude signals (under 20 mV) ensure linearity and avoid heating effects.^[38] In the pulse method, a short current pulse (typically 1-10 seconds at 0.2C to 1C rate) is applied to or interrupted from the device, and the resulting voltage transient is analyzed. The ohmic internal resistance is derived from the initial instantaneous voltage change \Delta V immediately following the pulse onset or cessation, calculated as r = \frac{\Delta V}{I}, where I is the pulse current. This captures the rapid ohmic drop before slower polarization effects dominate, with standards like IEC 61960 specifying a 1-second measurement after a 10-second pulse at 0.2C for DC internal resistance (DCIR), which includes some polarization. The method is effective for in-situ measurements in batteries, providing a value that correlates well with DC methods but can be adjusted for reduced polarization interference when using very short pulses.^[38] These indirect methods excel in precision for operational environments, such as battery management systems, by enabling non-destructive, component-resolved analysis of internal resistance, including separation of charge transfer and diffusion contributions via EIS or transient fitting. Their ability to perform measurements under load or at varying states of charge supports real-time monitoring in electric vehicles and renewable energy storage, enhancing diagnostic accuracy over static approaches.^[37]^[38]

Effects and applications

Voltage regulation

In electrical circuits, the internal resistance of a voltage source causes a drop in the terminal voltage as current is drawn by the load. Consider an ideal electromotive force (EMF) \mathcal{E} in series with internal resistance r. When no current flows (I = 0), the terminal voltage V equals the EMF: V = \mathcal{E}. As load current I increases, a voltage drop Ir occurs across r, reducing the output voltage according to Kirchhoff's voltage law applied to the loop: \mathcal{E} = V + Ir, or rearranged, V = \mathcal{E} - Ir. This linear relationship shows that V decreases proportionally with I, with slope -r.^[7]^[39] Graphical representation of this behavior plots V versus I, yielding a straight line starting at \mathcal{E} on the y-axis (no-load condition) and declining with slope -r. For small r, the line remains nearly horizontal, indicating stable voltage; for larger r, the steep decline highlights poor stability under varying loads. This drop is inherent to real sources and limits their performance in applications requiring constant voltage.^[7] The internal resistance also influences maximum power transfer to the load. By the maximum power transfer theorem, for a Thevenin equivalent circuit (source \mathcal{E} with series r), maximum power P to load resistance R_L occurs when R_L = r. Here, current I = \mathcal{E} / (2r), so P = I^2 R_L = (\mathcal{E}^2 / (4r)). However, the terminal voltage at this point is V = \mathcal{E} / 2, half the no-load value, due to equal drops across r and R_L. This trade-off prioritizes power over voltage stability in design.^[40]^[41] Voltage regulation quantifies the stability impact, defined as the percentage change in terminal voltage from no-load to full-load conditions: \% \ regulation = \frac{V_{no-load} - V_{full-load}}{V_{full-load}} \times 100\%. For a source with full-load current I_{full}, V_{full-load} = \mathcal{E} - I_{full} r, so regulation depends on r relative to load. Good regulation typically means less than 5% drop, ensuring minimal variation for sensitive circuits; higher values indicate significant instability from internal resistance.^[42]^[43] To mitigate these effects and maintain stable output voltage V_{out}, voltage regulators such as linear low-dropout (LDO) or switching types are employed, which sense and adjust for drops caused by internal resistance. Additionally, selecting sources with low r (verified via measurement techniques) minimizes the inherent drop without additional circuitry. These approaches ensure consistent V_{out} across load variations in practical systems.^[44]^[45]

Power efficiency

Internal resistance in power sources, such as batteries and fuel cells, leads to significant energy losses primarily through heat dissipation, quantified by the power loss formula P_{\text{loss}} = I^2 r, where I is the current and r is the internal resistance.^[46] This loss reduces the useful output power delivered to the load, as the total power supplied by the source is divided between the load and the internal resistance. The overall power efficiency \eta is given by \eta = \frac{P_{\text{out}}}{P_{\text{out}} + P_{\text{loss}}} = \frac{R_{\text{load}}}{R_{\text{load}} + r}, where R_{\text{load}} is the external load resistance; this expression highlights that efficiency approaches 100% as R_{\text{load}} becomes much larger than r, but practical applications often balance efficiency against power delivery needs.^[46]^[47] In multi-cell configurations, the effective internal resistance varies with arrangement, impacting system efficiency. For n identical cells connected in parallel, the effective resistance decreases to r / n, allowing higher current delivery with proportionally lower losses and improved efficiency for high-power demands.^[47] In series connections, the effective resistance sums to n r, which can exacerbate losses unless compensated by parallel branches in hybrid setups.^[47] In electric vehicles (EVs) and renewable energy systems, high internal resistance limits discharge rates in batteries, often causing 10–20% efficiency losses during peak operation due to elevated I^2 r heating, particularly under cold conditions where resistance rises sharply.^[48] For renewables like solar or wind storage, similar losses reduce grid integration efficiency, as batteries struggle with fluctuating loads. Thermal management via cooling systems mitigates this by maintaining optimal temperatures (around 25–40°C), with advanced methods lowering effective resistance by up to 15% compared to conventional cooling under high-stress conditions and preserving efficiency.^[49] Optimization strategies focus on aligning load characteristics with source capabilities to maximize efficiency. Matching the load resistance to minimize deviation from ideal R_{\text{load}} \gg r conditions enhances energy conversion, avoiding excessive losses in variable-demand scenarios. Hybrid systems incorporating supercapacitors bypass high-resistance batteries for transient high-power needs, reducing overall I^2 r losses and boosting system efficiency by 10–20% in pulsed applications like EVs.^[50]^[51]

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