6.1 Capacitor Characteristics and Behavior

Capacitors store electric charge and energy, making them fundamental building blocks in circuit analysis. This topic covers how capacitance works, what affects it physically, and how capacitors behave when placed in DC circuits with resistors. You'll need a solid grasp of these ideas before moving into more complex AC analysis later in the course.

Capacitance and its properties

Fundamental concepts of capacitance

Capacitance measures a component's ability to store electric charge. It's measured in farads (F), though you'll almost always see microfarads ($\mu F$), nanofarads ($nF$), or picofarads ($pF$) in practice since one farad is an enormous amount of capacitance.

The defining relationship is:

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

where $C$ is capacitance, $Q$ is the stored charge in coulombs, and $V$ is the voltage across the capacitor. More capacitance means more charge stored for the same voltage.

The energy stored in a capacitor is:

$E = \frac{1}{2}CV^2$

Notice that energy scales with the square of voltage. Doubling the voltage quadruples the stored energy.

Combining capacitors follows rules that are the opposite of resistors:

• Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + ...$
• Parallel: $C_{eq} = C_1 + C_2 + C_3 + ...$

A quick way to remember: capacitors in parallel share the same voltage and their plates effectively add together, so capacitances add directly. In series, the total capacitance is always less than the smallest individual capacitor.

Factors affecting capacitance

For a parallel-plate capacitor, capacitance is determined by:

$C = \frac{\varepsilon_0 \varepsilon_r A}{d}$

where $A$ is the plate area, $d$ is the distance between plates, $\varepsilon_0$ is the permittivity of free space ($8.854 \times 10^{-12}$ F/m), and $\varepsilon_r$ is the relative permittivity (dielectric constant) of the material between the plates.

Three things increase capacitance:

• Larger plate area (more surface to accumulate charge)
• Smaller plate separation (stronger electric field for the same voltage)
• Higher dielectric constant (the dielectric material reduces the electric field between plates, allowing more charge to accumulate for the same applied voltage)

Some typical dielectric constants: vacuum is 1, air is ~1.0006, paper is ~3.5, ceramic ranges from ~6 to several thousand, and mica is ~5–7.

Capacitor structure and properties

Physical structure of capacitors

Every capacitor has the same basic structure: two conductive plates separated by an insulating dielectric material. The differences between capacitor types come down to what that dielectric is and how the plates are arranged.

• Ceramic capacitors use ceramic dielectrics. They're small, cheap, and work well at high frequencies, but typically have lower capacitance values (picofarads to low microfarads).
• Electrolytic capacitors use a thin metal oxide layer as the dielectric, which allows very high capacitance in a compact package (up to thousands of $\mu F$). They're polarized, meaning you must connect them with the correct polarity or risk damage.
• Film capacitors use thin plastic films (polyester, polypropylene) as dielectrics. They offer good stability and low losses, making them reliable for precision applications.

Electrical characteristics of capacitors

Real capacitors aren't ideal. Here are the key specs you'll encounter:

• Voltage rating: The maximum voltage the capacitor can handle before the dielectric breaks down. Always choose a capacitor rated above your circuit's operating voltage.
• Capacitance tolerance: How much the actual value can deviate from the labeled value (e.g., ±5%, ±10%).
• Temperature coefficient: Describes how capacitance drifts as temperature changes. Some ceramic types can shift significantly with temperature.
• Equivalent Series Resistance (ESR): Every real capacitor has some internal resistance. Lower ESR is better, especially in power supply filtering.
• Parasitic inductance: At high frequencies, the leads and internal structure of a capacitor act like a small inductor. This creates a self-resonant frequency above which the capacitor actually behaves more like an inductor than a capacitor. The impedance of a real capacitor reaches a minimum at this resonant frequency and then rises.

Capacitor behavior in DC circuits

Charging process of capacitors

When you connect an uncharged capacitor to a DC voltage source through a resistor, the following happens:

1. At $t = 0$, the capacitor has no voltage across it, so the full source voltage $V$ appears across the resistor. Current is at its maximum: $i(0) = V/R$.

2. Current flows onto the capacitor plates, and voltage across the capacitor begins to rise.

3. As $v_C$ increases, the voltage across the resistor ($V - v_C$) decreases, which reduces the current.

4. The current and voltage follow exponential curves until the capacitor is fully charged to $V$ and current drops to zero.

The governing equations are:

$v(t) = V\left(1 - e^{-t/RC}\right)$

$i(t) = \frac{V}{R}\, e^{-t/RC}$

The fundamental relationship tying current to voltage for any capacitor is $i = C\frac{dv}{dt}$. This tells you that current through a capacitor is proportional to the rate of change of voltage, not the voltage itself. That's why current is highest at the start (voltage is changing fastest) and drops to zero once the capacitor is fully charged (voltage is constant).

Discharging process of capacitors

When a capacitor charged to $V_0$ is disconnected from the source and connected across a resistor:

1. At $t = 0$, the full capacitor voltage appears across the resistor, driving an initial current of $i(0) = -V_0/R$. (The negative sign indicates current flows in the opposite direction compared to charging.)
2. As charge leaves the plates, the voltage drops and the current magnitude decreases.
3. Both voltage and current decay exponentially toward zero.

The equations are:

$v(t) = V_0\, e^{-t/RC}$

$i(t) = -\frac{V_0}{R}\, e^{-t/RC}$

The energy released during discharge can be tracked using $E = \frac{1}{2}Cv^2(t)$ at any instant. All of the stored energy is eventually dissipated as heat in the resistor.

RC circuit analysis

Time constant and its significance

The time constant of an RC circuit is:

$\tau = RC$

where $R$ is in ohms and $C$ is in farads, giving $\tau$ in seconds.

What $\tau$ tells you physically:

• After $1\tau$, a charging capacitor reaches ~63.2% of its final voltage (or a discharging capacitor drops to ~36.8% of its initial voltage).
• After $2\tau$: ~86.5%. After $3\tau$: ~95.0%. After $4\tau$: ~98.2%. After $5\tau$: ~99.3%.
• The convention is that the transient is essentially complete after $5\tau$.

A larger $\tau$ means a slower response. For example, $R = 10\,k\Omega$ and $C = 100\,\mu F$ gives $\tau = 1$ second, so the circuit takes about 5 seconds to fully charge. Change $R$ to $1\,k\Omega$ and the circuit settles in about 0.5 seconds.

Transient response analysis

The transient response is the circuit's behavior as it transitions from one steady state to another. Two standard characterizations:

• Step response: How the circuit reacts to a sudden change in input voltage (e.g., a switch closing to connect a DC source). The charging and discharging equations above are the step response of an RC circuit.
• Impulse response: How the circuit reacts to a very brief pulse. For a series RC circuit, the impulse response across the capacitor is an exponential decay.

To analyze any RC transient, follow these steps:

1. Find the initial condition $v(0)$: What's the capacitor voltage just before the change occurs? (Remember, capacitor voltage can't change instantaneously.)
2. Find the final condition $v(\infty)$: What will the capacitor voltage settle to after a long time? (Replace the capacitor with an open circuit for DC steady state.)
3. Calculate $\tau = RC$ using the resistance "seen" by the capacitor.
4. Write the complete response: $v(t) = v(\infty) + [v(0) - v(\infty)]\,e^{-t/\tau}$

This general formula works for both charging and discharging, regardless of the initial and final voltages. Sketching the voltage and current waveforms is a good way to verify your answer makes physical sense.