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9 years ago

The formula for the impedance of a capacitor is given by:

where f is the frequency of the AC voltage and current associated with the capacitor. It is often said that "DC is just AC with zero frequency". In a sense, this is true. Thus, it seems reasonable to believe that setting the frequency to zero in the equation above would give the impedance of the capacitor 'at DC'. Mathematically, this is problematic because division by zero is undefined, however it is clear that as the frequency approaches zero, the impedance increases without bound so one can properly say that the impedance of a capacitor 'goes to infinity' as the frequency goes to zero. What *exactly* does this mean?

To answer this, we need to understand how the formula for the impedance of a capacitor is derived. The first step in any deriviation of the impedance is to assume that the voltage across the capacitor is a sinusoidal function of time with *constant* amplitude, frequency, and phase. This step is *crucial* to the derivation of impedance and is equivalent to requiring that the circuit has settled into **AC steady state**. Now consider the following: if the frequency of this sinusoid is set to zero, *the voltage across the capacitor becomes constant*. In other words, setting the frequency to zero is equivalent to requiring that the circuit has settled into **DC steady state**. We already know that the DC steady state current through a capacitor is identically zero so it is reassuring to find that the impedance formula gives the same result. The question then becomes: do all circuits have a DC steady state solution?

Consider the case of a sinusoidal current source connected to a capacitor. Using the impedance form of Ohm's law:

we find that the voltage across the capacitor increases without bound as the frequency decreases to zero. How is this to be interpreted?

The voltage given by the formula above is a phasor voltage. This phasor gives the peak amplitude and phase of the sinusoidal voltage across the capacitor. However, the phasor representation assumes that the circuit is in AC steady state or, if the frequency is zero, in DC steady state.

For a circuit to be in DC steady state, the circuit must have a DC steady state solution in which to exist. The circuit composed of a non-zero constant current source and a capacitor *has no DC steady state solution*. This was made clear earlier when we found that if the current through a capacitor is constant and non-zero, the voltage must be changing at a constant rate. Thus, for a circuit that does not have a DC steady state solution, we should expect a meaningless answer from a method that requires that the circuit is in DC steady state. A infinite voltage amplitude is such a meaningless answer.

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