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Capacitive impedance is the opposition that a capacitor presents to alternating current (AC).
For an ideal capacitor, its impedance is:
\[ Z_C = \frac{1}{j\omega C} \]
or equivalently:
\[ Z_C = -\frac{j}{\omega C} \]
where:
The magnitude of capacitive impedance is called capacitive reactance:
\[ X_C = \frac{1}{2\pi f C} \]
So, a capacitor has high impedance at low frequency and low impedance at high frequency.
A capacitor stores energy in an electric field. Because of this, it does not behave like a resistor. A resistor opposes current in the same way at all frequencies, but a capacitor’s opposition depends strongly on frequency.
The basic capacitor current-voltage relationship is:
\[ i(t) = C \frac{dv(t)}{dt} \]
This means the current through a capacitor depends on how quickly the voltage across it changes.
For DC, after the capacitor charges, the voltage no longer changes:
\[ \frac{dv}{dt} = 0 \]
so the current becomes zero. Therefore, at DC:
\[ f = 0 \]
\[ X_C = \frac{1}{2\pi f C} \rightarrow \infty \]
So an ideal capacitor behaves like an open circuit to DC.
For high-frequency AC, the voltage changes rapidly, so \(\frac{dv}{dt}\) is large. The capacitor can then conduct more AC current. As frequency increases:
\[ X_C = \frac{1}{2\pi f C} \]
decreases. Therefore, a capacitor behaves more like a short circuit at high frequencies.
In a purely capacitive circuit, the current leads the voltage by:
\[ 90^\circ \]
This is an important distinction from a resistor, where voltage and current are in phase.
For a capacitor:
\[ Z_C = -jX_C \]
The negative imaginary term \(-j\) represents the fact that the capacitor causes a phase shift: voltage lags current by \(90^\circ\), or equivalently, current leads voltage by \(90^\circ\).
A common memory aid is:
ICE: In a Capacitor, current \(I\) leads voltage \(E\).
Suppose you have a capacitor:
\[ C = 1 \ \mu F \]
at a frequency:
\[ f = 1 \text{ kHz} \]
The capacitive reactance is:
\[ X_C = \frac{1}{2\pi f C} \]
\[ X_C = \frac{1}{2\pi(1000)(1 \times 10^{-6})} \]
\[ X_C \approx 159 \ \Omega \]
So at 1 kHz, a \(1 \ \mu F\) capacitor has an impedance magnitude of about:
\[ 159 \ \Omega \]
Its complex impedance is:
\[ Z_C = -j159 \ \Omega \]
At 10 kHz, the same capacitor would have:
\[ X_C \approx 15.9 \ \Omega \]
So increasing the frequency by a factor of 10 decreases the capacitive reactance by a factor of 10.
Capacitive impedance explains why capacitors are useful in many electronic circuits.
Because a capacitor has very high impedance at DC, it can block DC while allowing AC to pass. This is used in AC coupling between amplifier stages.
Capacitors are used in filters because their impedance changes with frequency.
For example:
In power supply circuits, capacitors are often placed between a supply rail and ground. At DC, they do not short the supply. But at high frequencies, they have low impedance, so they help divert noise to ground.
This is why decoupling capacitors are commonly placed close to IC power pins.
An ideal capacitor has only capacitance, but real capacitors also have parasitic resistance and inductance.
A more realistic capacitor impedance model is:
\[ Z \approx R{\text{ESR}} + j\left(\omega L{\text{ESL}} - \frac{1}{\omega C}\right) \]
where:
At low and moderate frequencies, the capacitor behaves mainly capacitively. But at very high frequencies, the parasitic inductance can dominate.
This leads to the self-resonant frequency of the capacitor. At that frequency, the capacitive reactance and inductive reactance cancel:
\[ \omega L_{\text{ESL}} = \frac{1}{\omega C} \]
At self-resonance, the capacitor’s impedance is at a minimum and is mostly resistive, dominated by ESR.
Above the self-resonant frequency, the component may behave more like an inductor than a capacitor.
Capacitive impedance is the AC opposition of a capacitor.
For an ideal capacitor:
\[ Z_C = \frac{1}{j\omega C} = -\frac{j}{\omega C} \]
and its magnitude is:
\[ X_C = \frac{1}{2\pi f C} \]
Key ideas:
