This problem presents the interesting physics of induction cookers. Induction cookers contain a coil which is driven by an alternating current to heat up a metal pan above it. Induction cookers are safe (no flammable gases), clean (no soot), heat quickly, and can be powered by renewable electricity. In this experiment, we will explore the fascinating physics of an induction cooker.
There are three parts in the experiment. First, we will measure the coil's inductance ($L$) and its internal resistance ($R_L$). Second, we will investigate the skin depth phenomenon in metals which is important for induction cooking. Finally, we will determine the specific heat capacity ($c$) of two different metal pans and their effective load resistance ($R_{\mathrm{LOAD}}$).
Parameter/Constant Symbol Value Stefan-Boltzmann $\sigma_S$ $5.670\times10^{-8}\:{\rm W\:m^{-2}K^{-4}}$ Magnetic permeability in vacuum $\mu_0$ $4\pi\times10^{-7}\: {\rm H/m}$ Mass density of Al $\rho_{\textrm{Al}}$ $2700 \, \mathrm{kg/m^3}$ Mass density of SS410 $\rho_{\textrm{SS410}}$ $7700 \, \mathrm{kg/m^3}$ Emissivity of Al $e_{\textrm{Al}}$ 0.65 Emissivity of SS410 $e_{\textrm{SS410}}$ 0.8
NOTE:
The first key component in the induction cooker is the coil. In this experiment we will measure the self inductance ($L$) of coil #1 (the top coil from Fig. 3b). This coil can be modelled as an ideal inductor $L$ in series with an internal coil resistor $R_L$.
We will use a series RLC circuit with the yellow metal resistor $R_1$, coil #1 and a capacitor. There are four different capacitors. Please note that the Function Generator output voltage might vary as you change the frequency because the load impedance may change.
NOTE:
A. Mutual inductance
In this experiment #2 we will use the two coils as shown in Fig. 4, but without any metal plates. First, we will measure the mutual inductance $M$ between both coils. Following Faraday’s law, the change in current in the first coil will induce a voltage in the second coil.
B. Skin depth experiment
The “skin-depth” concept plays an important role in the induction cooker. The “skin-depth” characterises the penetration depth of the alternating current (AC)-induced electromagnetic field into the metal. In this experiment we will investigate the skin depth of various metals that can be used as cooking pans. We will investigate the skin depth's frequency-dependence and measure the electrical conductivity ($\sigma$) of the metals.
We set coil #1 as the primary coil and coil #2 as the secondary coil. Since the total metal thickness ($\sim 3 \, \mathrm{mm}$) is small compared to coil-coil distance ($15 \, \mathrm{mm}$), we can assume that the magnetic field at the bottom, near the secondary coil, is approximately constant (if there is no metal).
Following Maxwell’s equations, when an oscillating electric or magnetic field penetrates a conductor, the field inside the conductor decreases exponentially with the penetration distance $z$:
$$B(z)= B_0\: e^{-z/\delta}\:\cos (\omega t - z/\delta + \phi)$$
where$B_0$ is the magnetic field amplitude before it enters the conductor, $\delta$ is the “skin depth” and $\phi$ is phase. Note: we ignore the phase factor $(-z/\delta+\phi)$ in this experiment.
The skin depth in a conductor is given as:
$$\delta = \sqrt{\frac{\sigma^m f^n}{\pi \mu}}$$
where $\sigma$ is the electrical conductivity, $f$ is frequency, $\mu = \mu_r \times \mu_0$ is the magnetic permeability, and $m$ and $n$ are power factors which are integers to be determined in this experiment.
We will perform experiments on four metals: (1) Aluminium, (2) Copper, (3) Stainless steel “SS304” and (4) Stainless steel “SS410”. By inserting the metals in between the coils, the voltage in the secondary coil will drop due to magnetic field “shielding” of the eddy current in the metal.
Note: First explore the appropriate range of frequencies that yield significant changes in secondary coil voltage.
Develop a model with equations and perform an experiment to determine $n$ for each metal (rounded to the nearest integer). Record your data. You may use linear regression to analyse the data as necessary to obtain data points to plot the final graphs for each metal to get $n$ and $\sigma$ (which will be asked in Q2.6).
Identify one metal that does not yield good data due to extreme values of skin depth. You can ignore it for Q2.5 and Q2.6.
NOTES:
In this experiment we will use the Aluminium and the SS410 metal squares as the “cooking pans”. First you will mount the Aluminum “pan” (item #11). Clamp it to the top platform (item #3) and then you flip it upside down as shown in Fig. 5. You will use coil #2, which is well separated from the “pan”, so that there is no heat transfer between them by conduction.
Place the setup inside the black box (item #8) so the convection loss is negligible. Since the metal "pan" sits on a plastic platform (a thermal insulator), we also assume no heat loss due to conduction. Thus the only heat loss is due to radiation to the surroundings. The radiation power of a body with temperature $T$ is given as:
$$P_{RAD}=e A \sigma_S T^4$$
where $e$ is the emissivity, $\sigma_S$ is the Stefan-Boltzmann constant and A is the radiating surface area.
We can measure the temperature of the metal “pan” by measuring the resistance of the NTC thermistor (attached to the "pan"), which is given as:
$$R_{NTC}=R_0\:\exp{[B(1/T-1/T_0)]}$$
where $R_0 = 10 \, \mathrm{k}\Omega$ is the nominal resistance at reference temperature $T_0 = 298 \, \mathrm{K}$, $B = 3950 \, \mathrm{K}$ is a constant, and $T$ is the thermistor temperature (in K).
Finally, we can model the heating of the metal “pan” as if it introduces a “load resistance” $R_{\mathrm{LOAD}}$ to the circuit as shown in Fig. 6. In other words, the coil and metal pan system can be modeled as coil inductance $L$, coil resistance $R_L$ and the “load resistance” $R_{\mathrm{LOAD}}$.
3.5
1.60
Develop a model and perform an experiment to determine $R_{\mathrm{LOAD}}$ for the Al “pan”. Plot the appropriate data.
Suggestion: Perform measurements after approximately 30 sec of applying power to ensure that the coil delivers steady power and so that the heat is distributed more uniformly.
D.1. FUNCTION GENERATOR BOX
Components:
D.2. DIGITAL OSCILLOSCOPE
1. PANEL KEY FUNCTIONS
These keys allow you to navigate through settings, select functions, and adjust measurements.
2. OSCILLOCOPE MEASUREMENT MODE:
In oscilloscope mode, the device only measures voltage and displays the waveform as a function of time. This mode can measure voltage signals with very high frequency up to $1 \, \mathrm{MHz}$.
3. MULTIMETER MEASUREMENT MODE:
In multimeter mode, the device is used to measure electrical parameters such as voltage and resistance. In AC voltmeter mode it gives numerical readings with up to 4 significant figures but the frequency is limited to between 40Hz and 1kHz.
4. ADDITIONAL FUNCTIONS
5. CHARGING THE DIGITAL OSCILLOSCOPE
To ensure the device is always ready to use, keep track of the battery levels.