Question

The heat generated in the circuitry on the surface of a silicon chip \((k=130 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is conducted to the ceramic substrate to which it is attached. The chip is \(6 \mathrm{~mm} \times 6 \mathrm{~mm}\) in size and \(0.5 \mathrm{~mm}\) thick and dissipates \(5 \mathrm{~W}\) of power. Disregarding any heat transfer through the \(0.5-\mathrm{mm}\) high side surfaces, determine the temperature difference between the front and back surfaces of the chip in steady operation.

Short answer

Answer: The temperature difference between the front and back surfaces of the silicon chip in steady operation is approximately 0.534 K.

Step-by-step solution

List the known parameters

The following parameters are given in the problem: - Chip's thermal conductivity, k = 130 W/m·K - Chip's width, w = 6 mm - Chip's length, l = 6 mm - Chip's thickness, t = 0.5 mm - Power dissipated by the chip, P = 5 W Convert the given dimensions to meters for consistency: - w = 6×10⁻³ m - l = 6×10⁻³ m - t = 0.5×10⁻³ m

Use the heat conduction equation

Considering the heat conduction through the thickness of the silicon chip, we can write the heat conduction equation: P = k × A × (ΔT)t Where: - A is the cross-sectional area of the chip - ΔT is the temperature difference between the front and back surfaces of the chip - t is the thickness of the chip Now, we can calculate the chip's area: A = w × l = (6 × 10⁻³) × (6 × 10⁻³) = 36 × 10⁻⁶ m²

Find the temperature difference ΔT

We can now rearrange the heat conduction equation to solve for ΔT: (ΔT) = P × t / (k × A) Plug in the known values: (ΔT) = (5) × (0.5 × 10⁻³) / (130 × (36 × 10⁻⁶)) (ΔT) = 2.5 × 10⁻³ / 4.68 × 10⁻³ (ΔT) = 0.534 So the temperature difference between the front and back surfaces of the chip in steady operation is approximately 0.534 K.

Key concepts

Thermal Conductivity

Thermal conductivity is a measure of how easily heat can pass through a material. It is denoted by the symbol \(k\) and is calculated in watts per meter per kelvin (\(W/m\cdot K\)). Materials with high thermal conductivity, like copper and aluminium, are excellent at transferring heat, while those with low thermal conductivity, like rubber and wood, are better insulators.In the context of electronics, like a silicon chip, high thermal conductivity is crucial because it allows the generated heat to be effectively conducted away from the heat-sensitive parts. This helps in avoiding overheating and contributes to the chip's efficiency and longevity. The provided exercise illustrates the importance of thermal conductivity in determining how the heat generated by the silicon chip goes through it and onto the substrate it is attached to.

Heat Transfer Equation

The heat transfer equation \(P = k \times A \times (\Delta T) / t\) is integral to understanding how heat conduction works. In this formula, \(P\) represents the power, or the amount of energy per unit time, being transferred. The area through which the heat is being conducted is represented by \(A\), and \(\Delta T\) illustrates the temperature difference across the material. \(t\) stands for the thickness of the material that the heat is being conducted through.Understanding the Equation:

\t
• \(P\): The electrical power generated by the device, indicative of how much heat must be transferred or dissipated.
\t
• \(k\): The thermal conductivity which tells about the material's ability to conduct heat.
\t
• \(A\): The cross-sectional area perpendicular to the direction of the heat flow.
\t
• \(\Delta T\): The temperature difference driving the heat transfer.
\t
• \(t\): The distance between the areas at different temperatures, or the thickness of the material.

When applied to the exercise scenario, by inputting the known values into the equation, we determine the temperature difference across the chip. This equation underlines how power dissipation, material properties, and dimensions all play a part in the thermal management of electronic components.

Steady State Temperature Difference

The steady state temperature difference refers to the constant temperature difference established across a material when it is subject to continuous heat flow. In a steady state, the heat entering any section of the material is equal to the heat leaving it, signifying that the internal energy of the material remains unchanged over time.In electronic systems, reaching a steady state temperature is vital as it means a balance has been achieved between heat being generated by electronic components and the heat being dissipated or conducted away. It is this balance that dictates the effectiveness of the system's thermal design. In our textbook problem, the temperature difference represents the equilibrium point where heat produced by the silicon chip is effectively transferred to the ceramic substrate without causing a continued increase in temperature, ensuring the device's reliability and performance are maintained.