Chapter 7 — Circuits that Remember

This chapter introduces inductors and capacitors, two fundamental circuit elements that exhibit dynamic behavior, unlike resistors. It explains how inductors store energy in a magnetic field and resist changes in current, while capacitors store energy in an electric field and resist changes in voltage. The chapter analyzes the behavior of inductors and capacitors when a DC voltage is suddenly applied, exploring the transient response and the concept of the time constant. It also details how to calculate total inductance and capacitance for series and parallel combinations and concludes by highlighting the duality between inductors and capacitors and their importance in electronic circuits.

Learning Objectives:

• Understand the fundamental properties of inductors and capacitors, including how inductors resist changes in current while capacitors resist changes in voltage.
• Analyze the transient response of RL and RC circuits to DC step inputs, including the significance of the time constant.
• Calculate total inductance and capacitance for series and parallel combinations of components.
• Explain energy storage in inductors (magnetic fields) and capacitors (electric fields) and determine the amount of energy stored.
• Interpret the duality between inductors and capacitors, including their complementary behaviors in circuit analysis.

The preceding chapters have examined circuits with steady DC voltages and currents, where resistors follow Ohm's law consistently. Real-world circuits, however, often involve signals that change over time. Two new components, inductors and capacitors, are essential for these dynamic circuits but behave very differently from resistors.

Inductors and capacitors are the "memory elements" of electronics. While resistors simply resist current flow, inductors and capacitors can store energy and release it later, giving circuits the ability to respond to changes over time. This property makes them fundamental building blocks for many everyday applications, including filtering unwanted signals in audio equipment, storing energy in power supplies, creating timing circuits in computers, and coupling signals between circuit stages.

This chapter explores what happens when inductors and capacitors are connected to a DC voltage that suddenly turns on, a so-called "step" input.[^fn1] These components initially resist change, but in different ways, creating interesting "transient" behaviors before eventually settling into a steady state.

[^fn1]: In a subsequent chapter, we will investigate how inductors and capacitors behave when driven by periodically varying sinusoidal signals.

Both components store energy, but in different forms: inductors store energy in magnetic fields, while capacitors store energy in electric fields.

For clarity when discussing time-varying quantities, the following notation is used throughout:

1. A lowercase letter (like $i$ for current) represents the general case, which may or may not vary with time.
2. A lowercase letter with $(t)$, such as $i(t)$, explicitly shows time dependence.
3. An uppercase letter (like $I$) indicates a constant value that does not change with time.

Section 1 — Step Functions: Modeling Switches in Circuits

Before exploring how inductors and capacitors behave, we need a way to mathematically describe a switch being flipped on or off. In practice, this is straightforward: flip a switch and a circuit turns on. To analyze circuits mathematically, however, we need a precise way to represent this action.

This is where the Heaviside step function (or simply the "step function") comes in. Think of it as a mathematical light switch that turns on at a specific moment in time. Written as $h(t)$, it is defined as:

$$h(t)=\begin{cases} 0 & \text{when } t < 0 \text{ (switch is OFF)} \\ 1 & \text{when } t \geq 0 \text{ (switch is ON)} \end{cases}$$

The step function equals 0 for all time before $t=0$ (representing "OFF"), and equals 1 for all time at and after $t=0$ (representing "ON").

Step functions are particularly useful because they can be combined to model more complex switching behaviors. For example, consider a signal that turns ON at $t=0$ and then turns OFF again at time $t=6\tau$ (where $\tau$ is the time constant introduced in the following sections). This can be represented by combining two step functions: $h(t) - h(t-6\tau)$. The first term $h(t)$ turns the signal ON at $t=0$. The second term $h(t-6\tau)$ would normally turn a signal ON at $t=6\tau$, but because it is subtracted, it turns the signal OFF at that time instead.

Figure: Creating a pulse signal: The blue curve shows $h(t)$ turning ON at $t=0$. The red curve shows $h(t-6\tau)$ turning ON at $t=6\tau$. Subtracting them to obtain $h(t) - h(t-6\tau)$ produces a pulse that turns ON at $t=0$ and OFF at $t=6\tau$.

Section 2 — Ideal Inductor: The Magnetic Energy Storage Element

How Inductors Work: The Magnetic Connection

The inductor is one of the simplest yet most fascinating components in electronics. It is based on a fundamental principle of electromagnetism: when electric current flows through a wire, it creates a magnetic field around that wire.

Figure: When current flows through a wire, it creates a magnetic field that circles around the wire. The field is strongest near the wire (shown by thicker lines) and weakens with distance. The field direction follows the right-hand rule: point the thumb in the direction of current flow, and the curled fingers indicate the direction of the magnetic field.

An inductor takes advantage of this effect by coiling the wire many times. This concentrates the magnetic field, making it much stronger, and allows the inductor to store energy in this magnetic field.

Figure: An inductor is typically a coil of wire, sometimes wrapped around a core material. The magnetic field produced by an inductor is illustrated on the left. The coil shape concentrates the magnetic field inside the coil. To determine its direction, apply the right-hand rule: curl the fingers in the direction of current flow through the coil, and the thumb points in the direction of the magnetic field inside the inductor. Image adapted from www.iqsdirectory.com

Figure: Circuit symbol for an inductor. The loops in the symbol represent a coil of wire.

Inductance: Measuring an Inductor's Strength

Each inductor has a property called inductance, measured in units called henrys (H). Inductance describes how effectively the inductor creates a magnetic field and stores energy when current flows through it. Most practical inductors have values in millihenrys (mH) or microhenrys ($\mu$H).

For a coil-shaped inductor (solenoid), the inductance can be approximated by:

$$L = \frac{\mu N^2 A}{l}$$

Where:

• $L$ is the inductance in henrys (H)
• $\mu$ is the permeability of the core material (a measure of how well it supports magnetic fields)
• $N$ is the number of turns in the coil
• $A$ is the cross-sectional area of the coil
• $l$ is the length of the coil

This equation reveals several important relationships: inductance increases with the square of the number of turns ($N^2$); larger diameter coils (bigger $A$) have more inductance; shorter coils (smaller $l$) have more inductance; and using a magnetic core material (higher $\mu$) greatly increases inductance.

The Fundamental Property of Inductors

The most important characteristic of an inductor is this: an inductor resists changes in current. This is described mathematically by the voltage-current relationship:

$$v = L \frac{di}{dt}$$

This equation states that the voltage ($v$) across an inductor depends on how quickly the current is changing ($\frac{di}{dt}$); larger inductance ($L$) means larger voltage for the same rate of current change; if the current is constant ($\frac{di}{dt} = 0$), the voltage across the inductor is zero; and if the current is changing rapidly, the voltage can be very large.

This is what makes inductors behave so differently from resistors. While a resistor's voltage depends on the current itself ($v = iR$), an inductor's voltage depends on how quickly the current is changing, giving inductors their characteristic time-dependent behavior.

Inductors in Series and Parallel

Just like resistors, inductors can be combined in series or parallel to obtain different total inductance values.

Inductors in Series

When inductors are connected one after another in series, the total inductance is simply the sum of the individual inductances:

$$L_{total} = L_1 + L_2 + L_3 + \ldots$$

Figure: Inductors in series. The total inductance is $L_{total} = L_1 + L_2$, analogous to resistors in series.

Physically, connecting two inductors in series is like making one longer coil, which increases the total inductance.

Inductors in Parallel

When inductors are connected in parallel (with the same voltage across them), the total inductance follows the reciprocal formula:

$$\frac{1}{L_{total}}=\frac{1}{L_1}+\frac{1}{L_2}+\frac{1}{L_3}+\ldots$$

For two inductors, this simplifies to:

$$L_{total}=\frac{L_1 L_2}{L_1+L_2}$$

Figure: Inductors in parallel. The total inductance follows $\frac{1}{L_{total}}=\frac{1}{L_1}+\frac{1}{L_2}$, analogous to resistors in parallel.

The total inductance of inductors in parallel is always less than the smallest individual inductor, mirroring the behavior of resistors in parallel.

Summary of Inductor Combinations

Inductors in series add directly (like resistors in series), while inductors in parallel follow the reciprocal formula (like resistors in parallel).

Section 3 — What Happens When We Suddenly Apply Voltage to an Inductor?

The Key Property: Inductors Resist Changes in Current

The most important thing to remember about inductors is this: inductors resist changes in current. Not the current itself, but changes in current. This seemingly simple property leads to some fascinating behavior when voltage is suddenly applied or removed.

Note: Unlike a resistor, which immediately allows current proportional to the applied voltage, an inductor initially blocks current flow when voltage is first applied, then gradually allows current to increase.

Physical Explanation: Why Inductors Resist Current Changes

To understand why inductors behave this way, consider what happens when current starts flowing through an inductor:

• When voltage is first applied, current begins to flow and creates a magnetic field.
• This changing magnetic field induces a voltage in the coil itself (self-induction).
• According to Lenz's Law, this induced voltage opposes the change that created it.
• This opposing voltage initially prevents the current from increasing rapidly.

Helpful Analogies to Understand Inductor Behavior

The Flywheel Analogy:
Think of an inductor like a heavy flywheel. It takes significant effort (voltage) to start the flywheel spinning (increase current). Once spinning at constant speed (steady current), it requires no effort to maintain. If the spinning is interrupted suddenly (current reduced), the flywheel resists by pushing back. A heavier flywheel (larger inductance) resists changes more strongly.

The Water Pipe Analogy:
Imagine water flowing through a pipe fitted with a heavy paddle wheel. When the valve is first opened (voltage applied), the wheel's inertia prevents immediate flow, and the flow gradually increases as the wheel speeds up. Once flowing steadily, the wheel spins at constant speed. If the valve is suddenly closed, the wheel's momentum continues to push water through.

Analyzing an RL Circuit Step Response

Consider what happens in a specific circuit when voltage is suddenly applied. The RL circuit (a resistor and inductor connected in series) is shown below.

• The DC voltage source is 1 V.
• The resistor is 100 Ω.
• The inductor is 1 mH (millihenry).
• The switch closes at time $t=0$ (turning the circuit ON).
• The switch opens at time $t=6\tau$ (turning the circuit OFF).

The input voltage can be described mathematically as:

$$v_{in}(t)=V_o \left[h(t)-h(t-6\tau)\right]$$

Figure: An RL circuit with a switch. The behavior of both the current and the voltage across the inductor is observed when the switch closes (at $t=0$) and later opens (at $t=6\tau$).

The Time Constant: How Fast Does Current Build Up?

A key concept for understanding inductor (and capacitor) behavior is the time constant, represented by the Greek letter tau ($\tau$):

$$\tau = \frac{L}{R}$$

Where $L$ is the inductance in henrys and $R$ is the resistance in ohms.

In this example, $\tau = \frac{1 \text{ mH}}{100\ \Omega} = 10 \text{ microseconds}$.

The time constant describes how quickly the circuit responds:

• After $1\tau$, the current reaches about 63% of its final value.
• After $2\tau$, about 86%.
• After $3\tau$, about 95%.
• After $5\tau$, the current is within 1% of its final value.

What Happens When the Switch Closes?

When the switch first closes at $t=0$, the following sequence of events occurs simultaneously:

• The full voltage initially appears across the inductor (the voltage curve jumps to 1 V).
• Initially, almost no current flows (the current curve starts at zero).
• The voltage across the inductor decreases exponentially.
• The current increases exponentially.
• Eventually, the voltage across the inductor drops to zero.
• The current reaches its maximum value of $I = V/R = 1\text{ V}/100\ \Omega = 10\text{ mA}$.

Figure: Behavior of the RL circuit when the switch is closed and later opened. Left graph: The blue line shows the input voltage (1 V when ON, 0 V when OFF). The red line shows the voltage across the inductor. Right graph: The blue line shows the current through the circuit. Notice how the current increases gradually rather than instantly.

Understanding Lenz's Law: The Science Behind an Inductor's Behavior

Lenz's Law explains the fundamental physics at work inside an inductor.

Figure: Lenz's Law in action: When the applied current (blue) starts to increase, it creates a growing magnetic field. This changing magnetic field induces a voltage in the coil that drives an opposing current (red), which tries to prevent the change by creating its own magnetic field in the opposite direction.

Lenz's Law states: When a changing magnetic field induces a current, that current flows in a direction that creates a magnetic field opposing the change that caused it.

In practical terms, inductors "push back" against current changes. When voltage is applied to the circuit:

• The applied current begins to increase (blue in the diagram), creating a growing magnetic field.
• This increasing magnetic field induces a voltage in the coil (self-induction).
• This induced voltage drives a current in the opposite direction (red in the diagram).
• This opposing current creates an opposing magnetic field.
• The opposing field resists the increasing applied field, slowing down the current build-up.

The result is that the current does not increase instantly but instead follows an exponential curve.

The Mathematical Equations

When the switch closes at $t=0$, the current through the circuit grows exponentially:

$$i(t) = \frac{V_o}{R} \left(1-e^{-t/\tau}\right) = \frac{V_o}{R} \left(1-e^{-tR/L}\right)$$

The voltage across the inductor decreases exponentially:

$$v_L(t) = V_o e^{-t/\tau} = V_o e^{-tR/L}$$

Where $V_o$ is the applied voltage (1 V in this example), $R$ is the resistance (100 Ω), $L$ is the inductance (1 mH), $\tau = L/R$ is the time constant (10 microseconds), and $e^{x}$ is the exponential function with base $e \approx 2.718$.

Note: The current builds up gradually following an exponential curve, eventually approaching the final value of $I_{final} = V_o/R$.

Energy Storage in the Inductor

While current is flowing through the inductor, energy is stored in its magnetic field. The energy stored at any moment is:

$$E(t) = \frac{1}{2} L\, i(t)^2$$

Where $E(t)$ is the energy in joules, $L$ is the inductance in henrys, and $i(t)$ is the current at time $t$ in amperes.

In this example, when the current reaches its maximum value of 10 mA, the stored energy is:

$$E = \frac{1}{2} \times 1\ \text{mH} \times (10\ \text{mA})^2 = 50\ \text{nanojoules}$$

This is a small amount of energy (a nanojoule is $10^{-9}$ joules), but larger inductors in power applications can store significant energy.

What Happens When the Switch Opens?

When the switch opens at $t=6\tau$, the energy stored in the inductor does not simply disappear. Inductors resist changes in current, so when the current is interrupted:

• The inductor generates a voltage spike in the opposite direction.
• This voltage attempts to maintain the current flow.
• The current gradually decreases exponentially.
• The energy stored in the magnetic field is released.

In practical circuits, this "back EMF" (electromotive force) can generate very high voltages when current is suddenly interrupted, sometimes causing arcing across switch contacts or damaging components. This is why flyback diodes are often used in inductive circuits to provide a safe path for this energy.

Inductor Behavior in DC Steady State

After the transient period (approximately 5 time constants), the circuit reaches steady state. In DC steady state:

• The current is constant at $I = V/R$.
• Since the current is not changing, $di/dt = 0$.
• The voltage across the inductor is zero ($v = L \times 0 = 0$).
• The inductor effectively behaves like a short circuit (a wire).

Caution: While an inductor behaves like a short circuit in DC steady state, it presents significant opposition to rapidly changing signals. This is why inductors are useful for filtering high-frequency signals while passing DC.

Summary of Inductor Step Response

When a DC voltage is applied to an RL circuit:

• Initially ($t = 0$): current is zero, full voltage appears across the inductor, and no energy is stored.
• During the transient period ($0 < t < 5\tau$): current increases exponentially, voltage across the inductor decreases exponentially, and energy accumulates in the magnetic field.
• In steady state ($t > 5\tau$): current reaches maximum value ($I = V/R$), voltage across the inductor is zero, the inductor behaves like a wire (short circuit), and maximum energy is stored.

Section 4 — Ideal Capacitor: The Electric Energy Storage Element

How Capacitors Work: Storing Electric Charge

A capacitor is another fundamental electronic component that stores energy, but instead of storing it in a magnetic field like an inductor, a capacitor stores energy in an electric field.

Figure: Circuit symbol for a capacitor. The two parallel lines represent the two conducting plates.

The basic structure of a capacitor is simple: two conductive plates (usually metal) separated by an insulating material called a dielectric. When a voltage is applied across these plates, electrons are pushed away from the negative terminal of the voltage source and accumulate on one plate, creating a negative charge on that plate and a positive charge on the other. This separation of charge creates an electric field between the plates, and energy is stored in this electric field.

Figure: A parallel plate capacitor. When connected to a voltage source, electrons accumulate on the negative plate (left) and are pulled away from the positive plate (right), creating an electric field between them.

Capacitance: Measuring a Capacitor's Charge-Storing Ability

Each capacitor has a property called capacitance, measured in units called farads (F). Capacitance describes how much charge a capacitor can store per volt of applied voltage. One farad is an enormous capacitance in practice; most capacitors have values in microfarads ($\mu$F), nanofarads (nF), or picofarads (pF).

For a parallel plate capacitor, the capacitance is:

$$C = \frac{\varepsilon A}{d}$$

Where:

• $C$ is the capacitance in farads (F)
• $\varepsilon$ is the permittivity of the dielectric material
• $A$ is the area of overlap between the plates
• $d$ is the distance between the plates

This equation shows that larger plate area ($A$) increases capacitance, smaller separation distance ($d$) increases capacitance, and a dielectric material with higher permittivity ($\varepsilon$) increases capacitance.

The Fundamental Property of Capacitors

The most important characteristic of a capacitor is this: a capacitor resists changes in voltage. This is described mathematically by the voltage-current relationship:

$$i = C \frac{dv}{dt}$$

This equation states that the current ($i$) through a capacitor depends on how quickly the voltage is changing ($\frac{dv}{dt}$); larger capacitance ($C$) means larger current for the same rate of voltage change; if the voltage is constant ($\frac{dv}{dt} = 0$), the current through the capacitor is zero; and if the voltage is changing rapidly, the current can be very large.

This is what makes capacitors behave so differently from resistors. While a resistor's current depends on the voltage itself ($i = v/R$), a capacitor's current depends on how quickly the voltage is changing.

Helpful Analogies to Understand Capacitor Behavior

The Water Tank Analogy:
Think of a capacitor like a water tank with a flexible rubber membrane in the middle. The water level represents voltage and the flow of water represents current. Pouring water in quickly (rapid voltage change) causes a large flow, but once the water level stabilizes (constant voltage), flow stops. The size of the tank represents capacitance.

The Balloon Analogy:
A capacitor behaves like a balloon: it requires effort (current) to inflate initially. The more it is inflated (higher voltage), the more pressure (charge) it contains. Once fully inflated to a certain pressure, no more air flows in. When released, the stored energy is released.

Capacitors in Series and Parallel

Like inductors and resistors, capacitors can be combined in series or parallel. Their combination rules are the reverse of inductors, which reflects the broader duality between the two components discussed at the end of this chapter.

Capacitors in Parallel

When capacitors are connected in parallel (with the same voltage across each), the total capacitance is simply the sum of the individual capacitances:

$$C_{total} = C_1 + C_2 + C_3 + \ldots$$

Figure: Capacitors in parallel. The total capacitance is $C_{total} = C_1 + C_2$. Connecting capacitors in parallel is physically equivalent to increasing the plate area, which increases capacitance.

Capacitors in Series

When capacitors are connected in series (where the same current must flow through each), the total capacitance follows the reciprocal formula:

$$\frac{1}{C_{total}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\ldots$$

For two capacitors, this simplifies to:

$$C_{total}=\frac{C_1 C_2}{C_1+C_2}$$

Figure: Capacitors in series. The total capacitance follows $\frac{1}{C_{total}}=\frac{1}{C_1}+\frac{1}{C_2}$. Physically, this is equivalent to increasing the separation between plates, which reduces capacitance.

The total capacitance of capacitors in series is always less than the smallest individual capacitor, because series connection is equivalent to increasing the plate separation distance.

Comparison of Component Combination Rules

The following table summarizes how resistors, inductors, and capacitors combine in series and parallel:

Section 5 — What Happens When We Suddenly Apply Voltage to a Capacitor?

The Key Property: Capacitors Resist Changes in Voltage

Just as inductors resist changes in current, capacitors resist changes in voltage. Not the voltage itself, but changes in voltage. When a voltage is suddenly applied to a capacitor, the voltage across it cannot change instantaneously; it must increase gradually.

Note: When voltage is first applied to a capacitor, the capacitor initially looks like a short circuit (allowing current to flow easily), then gradually builds up a voltage across its terminals as it charges, eventually becoming an open circuit (blocking current flow).

Analyzing an RC Circuit Step Response

Consider what happens in a specific circuit when voltage is suddenly applied. The RC circuit (a resistor and capacitor connected in series) is shown below.

• The DC voltage source is 1 V.
• The resistor is 10 kΩ.
• The capacitor is 1 μF.
• The switch closes at time $t=0$ (turning the circuit ON).
• The switch opens at time $t=6\tau$ (turning the circuit OFF).

Figure: An RC circuit with a switch. The behavior of both the current and the voltage across the capacitor is observed when the switch closes (at $t=0$) and later opens (at $t=6\tau$).

The Time Constant: How Fast Does Voltage Build Up?

Just like with inductors, the time constant ($\tau$) describes how quickly the capacitor charges or discharges. For a capacitor:

$$\tau = RC$$

Where $R$ is the resistance in ohms and $C$ is the capacitance in farads.

In this example, $\tau = 10\ \text{k}\Omega \times 1\ \mu\text{F} = 10\ \text{milliseconds}$.

The time constant describes the circuit's response:

• After $1\tau$, the capacitor voltage reaches about 63% of its final value.
• After $2\tau$, about 86%.
• After $3\tau$, about 95%.
• After $5\tau$, the voltage is within 1% of its final value.

What Happens When the Switch Closes?

When the switch first closes at $t=0$:

• The capacitor voltage is zero (it cannot change instantaneously).
• The current is maximum at the start ($I = V/R = 1\ \text{V}/10\ \text{k}\Omega = 0.1\ \text{mA}$).
• The capacitor begins to charge and its voltage increases gradually.
• As the capacitor charges, the current decreases.
• Eventually, the capacitor is fully charged to the source voltage (1 V).
• The current drops to zero and the capacitor effectively becomes an open circuit.

Figure: Behavior of the RC circuit when the switch is closed and later opened. Left graph: The blue line shows the input voltage (1 V when ON, 0 V when OFF). The red line shows the voltage across the capacitor, which increases gradually. Right graph: The blue line shows the current through the circuit, which is highest when the capacitor is charging most rapidly.

Physical Explanation: Why Capacitors Behave This Way

To understand why capacitors behave this way, consider what is physically happening:

• When voltage is first applied, electrons begin to flow toward one plate and away from the other.
• As charge accumulates on the plates, an electric field builds between them.
• This electric field creates a voltage that opposes the applied voltage.
• As more charge accumulates, this opposing voltage increases.
• When the opposing voltage equals the applied voltage, current stops flowing.

The process is similar to filling a water tank through a narrow pipe: the flow is fastest when the tank is empty and decreases as the tank fills.

The Mathematical Equations

When the switch closes at $t=0$, the voltage across the capacitor increases exponentially:

$$v_C(t) = V_o\left(1-e^{-t/\tau}\right) = V_o\left(1-e^{-t/RC}\right)$$

The current through the circuit decreases exponentially:

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

Where $V_o$ is the applied voltage (1 V in this example), $R$ is the resistance (10 kΩ), $C$ is the capacitance (1 μF), $\tau = RC$ is the time constant (10 ms), and $e \approx 2.718$ is the base of the natural logarithm.

Note: For capacitors, current is highest at the beginning and decreases with time, while voltage starts at zero and increases. This is the opposite of an inductor's behavior, and reflects the duality between the two components.

Energy Storage in the Capacitor

While a capacitor is charging, energy is stored in its electric field. The energy stored at any moment is:

$$E(t) = \frac{1}{2} C\, v_C(t)^2$$

Where $E(t)$ is the energy in joules, $C$ is the capacitance in farads, and $v_C(t)$ is the voltage across the capacitor at time $t$.

In this example, when the capacitor is fully charged to 1 V, the stored energy is:

$$E = \frac{1}{2} \times 1\ \mu\text{F} \times (1\ \text{V})^2 = 0.5\ \mu\text{J}$$

What Happens When the Switch Opens?

When the switch opens at $t=6\tau$, the capacitor is fully charged to 1 V. With no path for the charge to flow, the capacitor maintains its voltage indefinitely in an ideal circuit. In a real circuit, charge will eventually leak through the dielectric or other components.

If a resistor is connected across the capacitor (creating a discharge path), the capacitor discharges exponentially with the same time constant $\tau = RC$:

$$v_{cap}(t)= V_o\, e^{-(t-6\tau)/RC}$$

The discharge current is:

$$i(t) = -\frac{V_0}{R}\, e^{-(t-6\tau)/RC}$$

The negative sign indicates that the current flows in the opposite direction during discharge.

Capacitor Behavior in DC Steady State

After the transient period (approximately 5 time constants), the circuit reaches steady state. In DC steady state:

• The capacitor voltage equals the source voltage.
• Since the voltage is not changing, $dv/dt = 0$.
• The current through the capacitor is zero ($i = C \times 0 = 0$).
• The capacitor effectively behaves like an open circuit (a break in the wire).

Caution: While a capacitor behaves like an open circuit in DC steady state, it allows current to flow when signals are changing rapidly. This is why capacitors are useful for blocking DC while passing AC signals.

Displacement Current: Maxwell's Insight

A natural question arises: if no electrons actually cross the dielectric insulator between the plates, how can current be said to flow through a capacitor?

This puzzle led James Maxwell to introduce the concept of "displacement current." He recognized that a changing electric field acts like a current, creating magnetic effects just as a real current would. When a capacitor is charging or discharging, the changing electric field between the plates constitutes what is called displacement current.

This concept proved crucial for Maxwell's equations that unified electricity and magnetism, ultimately predicting the existence of electromagnetic waves such as light and radio waves.

Summary of Capacitor Step Response

When a DC voltage is applied to an RC circuit:

• Initially ($t = 0$): capacitor voltage is zero, current is maximum ($I = V/R$), and no energy is stored.
• During the transient period ($0 < t < 5\tau$): voltage increases exponentially, current decreases exponentially, and energy accumulates in the electric field.
• In steady state ($t > 5\tau$): voltage reaches the source voltage, current drops to zero, the capacitor behaves like an open circuit, and maximum energy is stored.

Chapter Summary

• Inductors and capacitors are energy storage elements. Unlike resistors, which dissipate energy, inductors store energy in a magnetic field and capacitors store energy in an electric field. This stored energy can later be returned to the circuit, giving these components their characteristic memory-like behavior.

• An inductor resists changes in current. The governing relationship $v = L\,di/dt$ means the voltage across an inductor is proportional to the rate of change of current, not the current itself. Constant current produces zero voltage; rapidly changing current produces large voltage.

• A capacitor resists changes in voltage. The governing relationship $i = C\,dv/dt$ means the current through a capacitor is proportional to the rate of change of voltage. Constant voltage produces zero current; rapidly changing voltage produces large current.

• The step function models a closing switch. The Heaviside step function $h(t)$ equals 0 before $t=0$ and 1 at and after $t=0$. Combining two step functions, $h(t) - h(t-t_1)$, models a switch that closes at $t=0$ and opens at $t=t_1$.

• Transient responses are exponential. When a DC voltage is suddenly applied, both RL and RC circuits respond exponentially, governed by the time constant $\tau$. After approximately $5\tau$, the transient has decayed and the circuit is in steady state.

• DC steady state has a simple rule. Once all transients have settled, $di/dt = 0$ for an inductor (voltage across it is zero, so it behaves like a short circuit) and $dv/dt = 0$ for a capacitor (current through it is zero, so it behaves like an open circuit).

• Inductors and capacitors are duals of each other. Every property of one component has a precise counterpart in the other: current and voltage swap roles, series and parallel combination rules are reversed, and the DC steady-state behaviors are opposite. This duality is one of the most elegant structural features of circuit analysis.

• Series and parallel combinations follow opposite rules. Inductors combine like resistors (series: add; parallel: reciprocal sum). Capacitors do the reverse (parallel: add; series: reciprocal sum).

• Looking ahead. The behavior studied here applies to DC step inputs. When inductors and capacitors are driven by sinusoidal signals, a richer set of phenomena emerges, including impedance, resonance, and frequency-selective filtering. These are the subjects of the following chapters.

Key Formulas — Chapter 7