The automotive ignition system circuit generates a spark across the gap of a spark plug. This is accomplished by fluxing the primary coil of a transformer with a current, and then, suddenly interrupting the current, causing a large voltage to be induced across the secondary coil. Modern systems use a semiconductor switch to interrupt the current. The spark is generated when the large voltage appears across the gap.
The typical automotive ignition system circuit design is investigated. The motivation for this investigation is to show how a Laplace transform analysis can be used to understand how a common circuit works. An equivalent circuit model of the ignition system is derived using Laplace transforms. This model is tested in the laboratory and the system is simulated using PSpice.
1.1 MODELS
The first step in analyzing the system is to derive a model for the transformer. Figure 1 shows the equivalent circuit model for the transformer.
Equations (1) and (2) characterize the voltages across each coil of the transformer consisting of L1, the primary coil, and L2, the secondary coil, with mutual inductance M.
(1)
(2)
If the secondary is shorted, that is, v2 = 0, then the primary equivalent inductance can be measured. Using Equations (1) and (2) with the secondary shorted Equation (3) can be written.
(3)
This is an important result: Now, an expression for the mutual inductance in terms of L1, L2, and Leq (all values which can be measured) can be derived.
(4)
Figure 2 shows the schematic of the automotive ignition system. In this case, the secondary is connected to the spark plug, an open circuit. So, there is no current through the secondary and the voltage across the secondary as a function of time is simply the mutual inductance multiplied by the derivative of the current through the primary. This is the essence of the circuit: A sudden change in current through the primary produces a large voltage across the secondary.
The Laplace transform equivalent circuit model is shown in Figure 3. The secondary is replaced by a voltage source with a magnitude equal to the mutual inductance multiplied by the Laplace transform of the derivative of the current through the primary. The primary is replaced by its Laplace transform equivalent in parallel with a current source, because it is fluxed with a current Io. The capacitor and the ballast resistor are replaced with their Laplace transform equivalents.
Using the Laplace transform equivalent circuit model, the open circuit output voltage is derived in Appendix A and stated in Equation (5). For a given system, with known values, the inverse Laplace transform is performed to obtain a time domain solution.
(5)
When the circuit is connected so that the output voltage appears across the gap of a spark plug, however, Equation (5) is not valid for all time. In fact, when the voltage across the gap reaches a certain value, given in Equation (6), a spark is generated. The output voltage, then, returns to zero and the cycle is repeated.
(6)
1.2 SIMULATION
The automotive ignition circuit can be simulated using PSpice. Unlike the derivation of the mathematical model, values for the components must be picked in order to simulate the circuit. The same component values measured in the lab are used in the simulation and the sample calculations in the next section. This simplifies the comparisons.
Figure 4 shows the open circuit output voltage waveform of the system. The PSpice simulation is done without the sparkplug, since there is no available component that models its operation. With L2=55.8 H, L1=6.22 mH, C = 0.202 m F, Rsystem=1.28 W, and M=0.538, the peak output voltage is about -9 kV.
1.3 SAMPLE CALCULATIONS
Substituting the component values used in the simulation into Equation (5) and performing an inverse Laplace transform yields in an expression for the open circuit output voltage as a function of time.
Roots of D(s): 
N(+root): 
(7)
Equation (7) is the equation of an exponentially damped sine wave. The first peak will be the maximum value of the output voltage. The time that this peak occurs and the magnitude of the peak can be calculated by recognizing that this peak will occur when the angle is 90 degrees. The results of these calculations are derived below and stated in Equations (8) and (9). Figure 5 is a Mathcad plot of Equation (7).
(8)
(9)
2.1 Description of THE EXPERIMENT AND SETUP
The model derived for the output voltage can be tested in the laboratory if the proper equipment is available. To measure the large voltages induced, a high voltage probe (10,000:1) is needed. Using an ordinary probe would destroy the testing equipment.
The open circuit output voltage is measured by removing the spark-plug from the system. The semiconductor switch can be replaced by a manual switch. The output voltage is induced by opening the switch. The oscilloscope can be set up to trigger on the negative slope and capture the output wave form.
The circuit parameters, L1, L2, C, and R, can be measured using an LCR meter. By measuring the inductance of L1 with L2 shorted, Leq can be measured. The mutual inductance, M, can be calculated using Equation (4).
2.2 PRESENTATION OF MEASURED DATA
The voltage source used in the lab experiment is limited to an output current of 1.6 A. This means that Io is not equal to the input voltage divided by the ballast resister, as is the case in a real system. With L2=55.8 H, L1=6.22 mH, C = 0.202 m F, Rsystem=1.28 W, and M=0.538, the peak output voltage measured in the lab and the time that this peak occurs are given in Equations (10) and (11).
(11)
(12)
3.0 RESULTS AND COMPARISON
The experimental results for the output voltage waveform match the calculations and the simulation. The amplitude of the simulated waveform is about half of the calculated value. It is not clear what in the circuit is modeled differently by PSpice. The amplitude of the waveform obtained in the lab is about 30% less than the calculations. This discrepancy is due to experimental error. Specifically, it is impossible to instantaneously interrupt the current Io. While opening the switch, small sparks are observed at the points. This causes energy losses and reduces the amplitude of the output voltage waveform.
The experiment, simulation, and calculations for the output voltage waveform all agree on the phase and frequency. The experimental value for the time of the first peak is five microseconds more than the calculated value, a 10% discrepancy. The simulation value matches the calculations exactly.
4.0 CONCLUSIONS
From the results of the experiment and the derivation of the Laplace transform mathematical model, it is clear that the open circuit output voltage waveform for an automotive ignition system is an exponentially damped sine wave with a negative going zero axis crossing, The Laplace transform analysis worked well for this analyis. The circuit could have been solved by solving the governing differential equations, but the Laplace transform approach reduces the circuit to elements that can be analyzed using simple techniques. See Appendix A. This simplification is done without any approximations that sacrifice accuracy.