2.2 Battery Connections, Internal Resistance, and Alternative DC Sources

Battery Connections, Internal Resistance, and Alternative DC Sources

To supply the electrical demands of aircraft systems, individual cells are rarely used in isolation. Instead, they are combined in series, parallel, or series-parallel configurations to meet specific voltage and capacity requirements. Furthermore, practical DC sources exhibit internal losses that must be calculated, and alternative physical phenomena are harnessed to generate DC voltages for instrumentation and space applications.

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Series and Parallel Cell Connections

1. Series Connection

In a series connection, the positive terminal of one cell is connected to the negative terminal of the next.

• Total Voltage ($V_{\text{total}}$): The voltages of the individual cells add up. For $n$ cells in series: $V_{\text{total}} = V_1 + V_2 + \dots + V_n$ If the cells are identical: $V_{\text{total}} = n \times V_{\text{cell}}$
• Total Capacity (Ampere-hours, $Ah$): The capacity of the series combination is equal to the capacity of a single cell. The current flowing through each cell is the same: $C_{\text{total}} = C_{\text{cell}}$
• Total Internal Resistance ($r_{\text{total}}$): The internal resistances of the cells add together: $r_{\text{total}} = r_1 + r_2 + \dots + r_n$
• Application: Used when a higher voltage than that of a single cell is required (e.g., twenty $1.2\text{V}$ Ni-Cd cells connected in series to produce a $24\text{V}$ battery).

2. Parallel Connection

In a parallel connection, all positive terminals are connected together to one line, and all negative terminals are connected to another.

• Total Voltage ($V_{\text{total}}$): The voltage remains the same as that of a single cell (provided all cells have the same voltage): $V_{\text{total}} = V_{\text{cell}}$ Caution: Cells of different nominal voltages must never be connected in parallel, as the higher-voltage cell will discharge through the lower-voltage cell, causing overheating and potential damage.
• Total Capacity ($Ah$): The capacities of the individual cells add up. For $n$ identical cells in parallel: $C_{\text{total}} = n \times C_{\text{cell}}$
• Total Internal Resistance ($r_{\text{total}}$): The reciprocal of the total internal resistance is the sum of the reciprocals of the individual resistances. For $n$ identical cells: $\frac{1}{r_{\text{total}}} = \frac{1}{r_1} + \frac{1}{r_2} + \dots + \frac{1}{r_n} \implies r_{\text{total}} = \frac{r_{\text{cell}}}{n}$
• Application: Used when a higher current capability or longer operating time is required without increasing the system voltage.

3. Series-Parallel Connection

Cells can be arranged in a grid-like combination where groups of series-connected cells are connected in parallel. This configuration increases both the voltage and the capacity of the overall battery bank.

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Internal Resistance and Terminal Voltage

Every practical DC source (battery, generator, etc.) has a characteristic called internal resistance ($r$), which is the inherent opposition to current flow within the source itself.

Electromotive Force (EMF) vs. Terminal Voltage

• Electromotive Force (EMF, $E$): The open-circuit voltage of a cell when no current is flowing ($I = 0$). It is the maximum potential difference the cell can produce chemically.
• Terminal Voltage ($V_T$): The voltage measured across the battery terminals when a current is flowing.

Mathematical Relationships

1. Under Discharge (Battery Supplying Power): When the battery is discharging, current flows out of the positive terminal, creating an internal voltage drop ($I \cdot r$) across the internal resistance. The terminal voltage is less than the EMF: $V_T = E - (I \cdot r)$
2. Under Charge (Battery Receiving Power): When the battery is charging, current is forced into the positive terminal against the chemical EMF. The charging voltage must overcome both the EMF and the internal resistance: $V_T = E + (I \cdot r)$

Operational Significance in Aviation

• State of Health: As a battery discharges or ages, its internal resistance increases due to sulfation or active material degradation.
• High-Current Starting: When starting an aircraft engine, the starter motor draws a very high current ($I$). If the internal resistance ($r$) is high, the internal voltage drop ($I \cdot r$) becomes large, causing the terminal voltage ($V_T$) to drop. This can result in slow cranking speeds and may cause avionics to reset.
• Load Testing: To check a battery's condition, technicians perform a load test. By drawing a specified high current and measuring the terminal voltage drop, they can evaluate whether the internal resistance is within acceptable limits.

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Alternative DC Sources

In addition to chemical cells, electrical systems utilize other physical phenomena to generate DC power, primarily for instrumentation and aerospace power.

1. Thermocouples

A thermocouple is a device that converts heat energy directly into electrical energy based on the Seebeck effect.

• Construction: Formed by joining two dissimilar metals (such as Chromel and Alumel, or Copper and Constantan) at one end (the "hot" junction), while keeping the other ends (the "cold" or "reference" junction) at a known temperature.
• Operating Principle: When a temperature difference exists between the hot and cold junctions, a small DC voltage is generated. This voltage is directly proportional to the temperature difference.
• Aviation Applications:
  • Exhaust Gas Temperature (EGT): Thermocouple probes (often using Chromel-Alumel, Type K) are placed in the engine exhaust stream to monitor combustion efficiency.
  • Cylinder Head Temperature (CHT): Used on piston engines to monitor cylinder cooling.
  • Note: The output voltage of a single thermocouple is very small (in the millivolt range, e.g., $0\text{ to }50\text{ mV}$). To produce a usable voltage or drive an indicator directly, multiple thermocouples are connected in series to form a thermopile.

2. Photocells (Photovoltaic Cells)

Photocells convert light energy directly into DC electrical energy via the photovoltaic effect.

• Construction: Typically constructed from semiconductor materials, such as silicon. The semiconductor is doped to create a P-N junction.
• Operating Principle: When light photons strike the P-N junction, they transfer energy to electrons in the valence band, raising them to the conduction band. This creates free electron-hole pairs. The built-in electric field at the P-N junction pushes the free electrons toward the N-type material and holes toward the P-type material, establishing a potential difference (DC voltage) across the cell terminals.
• Aerospace Applications:
  • Solar panels on satellites and space probes.
  • Backup or primary power for solar-powered unmanned aerial vehicles (UAVs).
  • Individual silicon solar cells produce approximately $0.5\text{V}$ to $0.6\text{V}$ DC in bright sunlight and are connected in series-parallel arrays to achieve the required operating voltages and current capacities.

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Worked Exam Scenarios: Terminal Voltage Calculation

Scenario 1: A storage battery has an open-circuit voltage (EMF) of $24.0\text{ V}$ and an internal resistance of $0.05\ \Omega$. A load resistance of $1.95\ \Omega$ is connected across its terminals. Calculate the current in the circuit and the terminal voltage of the battery under this load.

Step-by-step Solution:

1. Calculate the total resistance in the circuit ($R_{\text{total}}$), which is the sum of the load resistance ($R_L$) and the battery's internal resistance ($r$): $R_{\text{total}} = R_L + r = 1.95\ \Omega + 0.05\ \Omega = 2.00\ \Omega$
2. Apply Ohm's law to find the total current ($I$) flowing in the circuit: $I = \frac{E}{R_{\text{total}}} = \frac{24.0\text{ V}}{2.00\ \Omega} = 12.0\text{ A}$
3. Calculate the terminal voltage ($V_T$) using the discharge formula: $V_T = E - (I \cdot r) = 24.0\text{ V} - (12.0\text{ A} \times 0.05\ \Omega)$ $V_T = 24.0\text{ V} - 0.6\text{ V} = 23.4\text{ V}$ Alternatively, calculate terminal voltage across the load: $V_T = I \times R_L = 12.0\text{ A} \times 1.95\ \Omega = 23.4\text{ V}$ Result: The terminal voltage drops to $23.4\text{ V}$ when supplying $12.0\text{ A}$ to the load.

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Worked Exam Scenarios: Series-Parallel Battery Pack

Scenario 2: An engineering design requires a power pack with a nominal voltage of $12\text{ V}$ and a total capacity of $30\text{ Ah}$. You are supplied with individual lead-acid cells, each rated at $2\text{ V}$ nominal and $10\text{ Ah}$ capacity. Describe the configuration required and calculate the total internal resistance if each cell has an internal resistance of $0.06\ \Omega$.

Step-by-step Solution:

1. Determine Series Cells per Branch: To obtain the required voltage of $12\text{ V}$ from $2\text{ V}$ cells, we must connect cells in series: $\text{Cells in series} = \frac{12\text{ V}}{2\text{ V/cell}} = 6\text{ cells}$ This single series branch will have a voltage of $12\text{ V}$ and a capacity equal to a single cell: $10\text{ Ah}$.
2. Determine Number of Parallel Branches: To scale the capacity from $10\text{ Ah}$ to the required $30\text{ Ah}$, we must connect multiple identical series branches in parallel: $\text{Parallel branches} = \frac{30\text{ Ah}}{10\text{ Ah/branch}} = 3\text{ branches}$
3. Calculate Total Cell Count: $\text{Total cells} = 6\text{ cells/branch} \times 3\text{ branches} = 18\text{ cells}$
4. Calculate Total Internal Resistance ($r_{\text{total}}$):
   • First, find the internal resistance of a single series branch ($r_{\text{branch}}$): $r_{\text{branch}} = 6 \times r_{\text{cell}} = 6 \times 0.06\ \Omega = 0.36\ \Omega$
   • Next, calculate the resistance of the 3 identical branches in parallel: $r_{\text{total}} = \frac{r_{\text{branch}}}{\text{number of branches}} = \frac{0.36\ \Omega}{3} = 0.12\ \Omega$ Result: The configuration is a series-parallel network of 3 parallel branches, with each branch containing 6 cells in series (total of 18 cells). The total internal resistance of the pack is $0.12\ \Omega$.