AC Circuits (RLC Series)

TL;DR

When resistors, inductors, and capacitors are connected in series to

Electromotive Force and Internal Resistance

TL;DR

Electromotive Force (EMF) is the maximum potential difference a source can provide, even when no current flows. Internal resistance in a power source causes some energy to be lost, reducing the actual voltage available at its terminals. This internal resistance makes the terminal voltage drop as more current is drawn from the source.

1. The Mental Model

Imagine a perfect battery as a pump that always pushes water with the same force (EMF). But inside the battery there's a narrow pipe (internal resistance) that resists the water flow, so less "pressure" (voltage) comes out of the battery when you're using a lot of water.

2. The Core Material

When we talk about a power source like a battery or a generator, we often think of it as having a fixed voltage. However, in reality, every power source has some inherent resistance inside it. This is called internal resistance, and it affects the voltage you actually get out of the source.

Let's break down the key terms:

2.1 Electromotive Force (EMF, $\mathcal{E}$)

EMF is the total energy per unit charge that a power source can provide. Think of it as the "strength" of the source when it's just sitting there, not powering anything. It's the maximum potential difference across its terminals when no current is flowing (i.e., in an open circuit). EMF is measured in Volts (V).

2.2 Internal Resistance (r)

This is the opposition to the flow of charge within the power source itself. No power source is perfectly efficient; some energy is always lost as heat due to this internal resistance when current flows. Internal resistance is measured in Ohms ($\Omega$).

2.3 Terminal Voltage (V)

This is the actual voltage you measure across the terminals of the power source when it's connected to a circuit and providing current. Because of internal resistance, the terminal voltage is always less than the EMF when current is flowing.

Here's the crucial relationship:

Terminal Voltage (V) = EMF ($\mathcal{E}$) - (Current (I) $\times$ Internal Resistance (r))

So, $V = \mathcal{E} - Ir$.

This equation shows that as the current (I) drawn from the source increases, the voltage drop across the internal resistance ($Ir$) increases, and consequently, the terminal voltage (V) decreases. If no current flows ($I=0$), then $V = \mathcal{E}$, meaning the terminal voltage equals the EMF.

You can also think of the total resistance in the circuit as the

Operational Amplifiers (Op-Amps)

TL;DR

Op-amps are versatile integrated circuits that amplify voltage differences, forming the building blocks for many analog circuits. They work by having very high input impedance, low output impedance, and a huge open-loop gain. You'll typically use them with negative feedback to precisely control their behavior.

1. The Mental Model

Imagine a super-smart, tiny amplifier that tries incredibly hard to keep its two input terminals at the exact same voltage. If there's even a tiny difference, it'll dump out a huge voltage at its output to correct it. We tame this zealous behavior with feedback.

2. The Core Material

An operational amplifier, or op-amp, is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. They're called "operational" because they used to be used to perform mathematical operations in analog computers.

What's Inside an Op-Amp (Simplified)

You don't need to know the transistors and resistors making up an op-amp. What matters are its ideal characteristics:

• Infinite input impedance: No current flows into its input terminals.
• Zero output impedance: It can supply any current without its output voltage dropping.
• Infinite open-loop gain: Even a tiny input difference makes the output saturate (hit its max/min voltage).
• Infinite bandwidth: It responds instantly to changes.
• Zero input offset voltage: If inputs are identical, output is zero.

In reality, op-amps are very good approximations of these ideals. We often use the "golden rules" for analysis:

1. No current flows into the input terminals. (Due to high input impedance)
2. The voltage difference between the input terminals is zero. (Due to massive gain, negative feedback forces them to be equal).

The Op-Amp Pins

A typical op-amp (like the ubiquitous 741) has 8 pins, but you primarily care about these:

• Non-inverting input (+): The output moves in the same direction as this input.
• Inverting input (-): The output moves in the opposite direction as this input.
• Output (OUT): Where the amplified signal comes out.
• Positive power supply (V+): Connects to your positive voltage rail.
• Negative power supply (V-): Connects to your negative voltage rail (often ground or a negative rail).

Understanding Feedback

Because of the op-amp's ridiculously high open-loop gain, connecting it without fe