Understanding Capacitor, Capacitive Reactance and Impedance

Understanding Capacitor

1. Characteristics of a Capacitor

An ideal capacitor is a theoretical electronic component used in circuit analysis. It's a "pure" capacitor, meaning it only exhibits the property of capacitance and has none of the parasitic effects found in real-world capacitors.

Note: An ideal capacitor has zero Equivalent series resistance (ESR) and zero Equivalent series inductance (ESL)

2 Capacitance (Charge-Voltage Relationship)

The defining equation for capacitance (C) is the ratio of the electric charge (Q) stored on one plate to the voltage (V) across the capacitor.

${\displaystyle Q=C\cdot <!--\; \cdot \; -->V}$

Or, can be rearranged as:

${\displaystyle C=\frac{Q}{V}}$

Where,

• C: Capacitance (measured in Farads)
• Q: Charge (measured in Coulombs)
• V: Voltage (measured in Volts)

2.1 SI Unit of capacitance: Farad

A Farad (F) is the standard unit of capacitance.

What is 1 Farad? A capacitor has a capacitance of 1 Farad when 1 Coulomb of electrical charge is stored on its plates by a potential difference of 1 Volt.

3. Current-Voltage Relationship

The most important equation for circuit analysis defines the current (I) flowing through the capacitor. The current is proportional to the rate of change of the voltage across it, not the voltage itself.

${\displaystyle I(t)=\frac{dV(t)}{dt}}$

Where,

• C: Capacitance (in Farads)
• I(t): Current as a function of time (in Amperes)
• ${\displaystyle \frac{dV(t)}{dt}}$: The derivative of voltage with respect to time (how fast the voltage is changing, in Volts per second)

4. Energy Storage

An ideal capacitor stores energy (E) within the electric field between its plates. This stored energy is given by:

${\displaystyle E=\frac{1}{2}C\cdot <!--\; \cdot \; -->V^2}$

5. AC Circuit Behavior (Impedance)

In an AC circuit, a capacitor's "resistance" to the flow of current is called capacitive reactance (X_C), which is a component of its total impedance (Z).

Capacitive reactance of a capacitor, (X_C), is given by:

${\displaystyle X_C=\frac{1}{\mathrm{\omega <!--\; \omega \; -->}C}}$

The impedance (Z) of an ideal capacitor is:

${\displaystyle Z=-<!--\; -\; -->j\times <!--\; \times \; -->X_C}$

Or,

${\displaystyle Z=\frac{1}{j\mathrm{\omega <!--\; \omega \; -->}C}}$

Where,

• j: The imaginary unit ( ${\displaystyle \sqrt{-<!--\; -\; -->1}}$)
• ω: Angular frequency (in radians per second, ${\displaystyle \mathrm{\omega <!--\; \omega \; -->}=2\mathrm{\pi <!--\; \pi \; -->}f}$)
• f: Frequency (in Hertz)

5.4 Impedance as function of "frequency" (rearranging the equation above):

The Impedance equation above can be written in another form. Let’s see how to do that step by step.

Step 1: Substitute "j" and multiply and divide RHS by ${\displaystyle \sqrt{-<!--\; -\; -->1}}$

${\displaystyle \Rightarrow <!--\; \Rightarrow \; -->\frac{1}{\sqrt{-<!--\; -\; -->1}\times <!--\; \times \; -->\mathrm{\omega <!--\; \omega \; -->}\times <!--\; \times \; -->C}\times <!--\; \times \; -->\left(\frac{\sqrt{-<!--\; -\; -->1}}{\sqrt{-<!--\; -\; -->1}}\right)}$

Step 2:

${\displaystyle \Rightarrow <!--\; \Rightarrow \; -->\frac{1}{\mathrm{\omega <!--\; \omega \; -->}\times <!--\; \times \; -->C}\times <!--\; \times \; -->\left(\frac{\sqrt{-<!--\; -\; -->1}}{\sqrt{-<!--\; -\; -->1}\times <!--\; \times \; -->\sqrt{-<!--\; -\; -->1}}\right)}$

Step 3:

${\displaystyle \Rightarrow <!--\; \Rightarrow \; -->\frac{1}{\mathrm{\omega <!--\; \omega \; -->}\times <!--\; \times \; -->C}\times <!--\; \times \; -->\left(\frac{\sqrt{-<!--\; -\; -->1}}{-<!--\; -\; -->1}\right)}$

Step 4:

${\displaystyle \Rightarrow <!--\; \Rightarrow \; -->-<!--\; -\; -->1\times <!--\; \times \; -->\frac{\sqrt{-<!--\; -\; -->1}}{\mathrm{\omega <!--\; \omega \; -->}\times <!--\; \times \; -->C}}$

Step 6: Substituting "j" back

${\displaystyle \Rightarrow <!--\; \Rightarrow \; -->-<!--\; -\; -->\frac{j}{\mathrm{\omega <!--\; \omega \; -->}C}}$

Step 7: Substiting "ω"

${\displaystyle \Rightarrow <!--\; \Rightarrow \; -->-<!--\; -\; -->\frac{j}{2\mathrm{\pi <!--\; \pi \; -->}fC}}$