# Ideal and Real-World Circuit Elements

Source and linear circuit elements are **ideal** circuit elements. One central notion of circuit theory is combining the ideal elements to describe how physical elements operate in the real world. For example, the 1 kΩ resistor you can hold in your hand is not exactly an ideal 1 kΩ resistor. First of all, physical devices are manufactured to close tolerances (the tighter the tolerance, the more money you pay), but never have exactly their advertised values. The fourth band on resistors specifies their tolerance; 10% is common. More pertinent to the current discussion is another deviation from the ideal: If a sinusoidal voltage is placed across a physical resistor, the current will not be exactly proportional to it as frequency becomes high, say above 1 MHz. At very high frequencies, the way the resistor is constructed introduces inductance and capacitance effects. Thus, the smart engineer must be aware of the frequency ranges over which his ideal models match reality well.

On the other hand, physical circuit elements can be readily found that well approximate the ideal, but they will always deviate from the ideal in some way. For example, a flashlight battery, like a C-cell, roughly corresponds to a 1.5 V voltage source. However, it ceases to be modeled by a voltage source capable of supplying **any** current (that's what ideal ones can do!) when the resistance of the light bulb is too small.

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