Lecture 12: Capacitors and First-order RC Systems

What this lecture covers

Every element studied so far (resistors, voltage sources, digital gates) is memoryless: the output depends only on the current input. This lecture introduces the capacitor, an element whose behavior depends on the past, and uses a surprising demo — a chain of two inverters whose output lags the input — to motivate why. It then shows where a capacitor physically arises inside a MOSFET, derives the capacitor's element law from first principles, and solves the resulting first-order differential equation for an RC circuit using the "particular plus homogeneous" method that will be reused throughout the rest of the course.

After watching, you can explain why inverter delay exists, state the capacitor's element law (i = C dv/dt), confirm that a capacitor still obeys the lumped matter discipline, and solve RC dvc/dt + vc = vI for a step input and known initial state using the three-step method: particular solution, homogeneous solution, then apply the initial condition.

Key ideas

• Memoryless vs. memory elements: resistors, voltage sources, and prior digital gates have outputs determined entirely by the current input; capacitors introduce dependence on the circuit's past (state).
• Inverter delay: cascading two inverters and observing the intermediate node shows the transition is not instantaneous — it rises gradually and crosses VT a bit later, delaying the final output; this delay is what limits how fast digital circuits can be clocked.
• Gate capacitance (CGS): the physical structure of a MOSFET (metal gate, thin oxide, silicon substrate) forms a parallel-plate capacitor between gate and source, which is the physical origin of the observed delay.
• Capacitor element law: C = epsilon*A/D (plate area, separation, and dielectric permittivity); charge q = Cv; current i = dq/dt = C dv/dt (for constant C); stored energy E = (1/2)Cv^2.
• Capacitors still obey the lumped matter discipline: dq/dt = 0 for all time only if both plates (net zero charge) are included inside the element boundary.
• First-order RC differential equation: for a Thevenin source vI through resistance R charging a capacitor C, the node equation gives RC dvc/dt + vc = vI.
• Particular + homogeneous solution method: find any solution to the full equation (particular), find the general solution to the equation with the drive set to zero (homogeneous, via the characteristic equation), sum them, then use the initial condition to solve for the remaining constant.
• Time constant tau = RC: governs how quickly the exponential term decays or rises in the solution.

Walkthrough

A surprising two-inverter demo (2:00)

The lecture chains two inverter circuits and applies a square wave at input A, expecting B to be a clean inverted square wave and C to match A. This sets up the expectation based on everything covered so far about memoryless digital elements.

The waveforms don't match expectations (4:50)

Zooming in on an oscilloscope trace reveals that the intermediate node B does not transition sharply; it rises gradually rather than instantaneously. Overlaying the output C on the input A shows C transitions slightly later than A — a delay that nothing covered so far in the course can explain.

Why the "itty-bitty" delay matters (9:23)

The lecture connects this small delay to why faster processors require new hardware generations: the delay is the fundamental limit on how fast a digital circuit can be clocked. It attributes the delay to the gradual rise of B reaching the MOSFET's threshold voltage VT slightly later than an instantaneous transition would.

Finding the capacitor inside the MOSFET (12:25)

The lecture dissects a MOSFET's physical structure — p-type silicon substrate, n-type drain and source regions, a thin oxide layer, and a metal gate — and shows that applying a positive gate voltage draws electrons into a channel beneath the oxide, forming two charge-separated regions on either side of a thin insulator. This is structurally a capacitor, CGS, sitting between the gate and source, unintentionally present in the device and responsible for the delay seen in the demo.

The linear capacitor element (17:20)

The lecture formalizes the capacitor as a primitive circuit element with plates of area A, separation D, and dielectric permittivity epsilon, giving C = epsilon*A/D in farads. It raises and resolves an apparent contradiction with the lumped matter discipline (which requires zero net charge buildup inside an element): because a capacitor's two plates carry equal and opposite charge, as long as both plates are included within the element's boundary, the net dq/dt is still zero, so the discipline holds.

Element law, energy, and a live charge demo (22:35)

The lecture states the core relationships: q = Cv, i = dq/dt = C dv/dt (for constant capacitance), and stored energy E = (1/2)Cv^2. A large, charged 250-volt capacitor is then discharged live in front of the class (with appropriate safety precautions) to demonstrate that a capacitor really does hold charge and energy over time, unlike a resistor.

Setting up and solving the RC differential equation (43:15)

Modeling the rest of the circuit as a Thevenin source (vI in series with R) driving the capacitor, the node method gives RC dvc/dt + vc = vI. The lecture solves this for t >= 0 given a constant input VI and initial capacitor voltage V0, using the three-step method: the particular solution vCP = VI (found by trial), the homogeneous solution vCH = A*e^(-t/tau) (found by assuming Ae^st, forming the characteristic equation RCs + 1 = 0, and solving for s = -1/RC, with tau = RC), and then applying the initial condition vC(0) = V0 to find A = V0 - VI. The total solution is vC(t) = VI + (V0 - VI)*e^(-t/tau), which the lecture illustrates with two example waveforms: a capacitor with initial charge decaying to zero with no input, and a capacitor with zero initial charge rising exponentially toward VI when a step input is applied.

Before you watch

• Be comfortable with the inverter circuit and MOSFET threshold-voltage switching behavior from earlier digital-circuits lectures.
• Recall the node method for writing circuit equations.
• A basic familiarity with solving first-order linear differential equations is helpful, though the lecture walks through the method from scratch.

Check your understanding

1. Why does the intermediate node voltage B in the two-inverter demo rise gradually instead of switching instantaneously, and how does this explain the delay seen at output C?
2. How does the lecture reconcile the capacitor's stored charge with the lumped matter discipline's requirement that dq/dt = 0 for all time within an element?
3. What are the three steps of the particular-plus-homogeneous method used to solve RC dvc/dt + vc = vI?
4. What role does the characteristic equation RCs + 1 = 0 play in finding the homogeneous solution?
5. Given vC(t) = VI + (V0 - VI)*e^(-t/tau), what does the solution look like if V0 = 0 and a step input VI is applied? What if VI = 0 and the capacitor starts with V0 charge?