Lecture 8: Dependent Sources and Amplifiers

What this lecture covers

This lecture answers two connected questions: why do circuits need amplification at all, and how do you build a device that amplifies? It opens by motivating amplification from three angles: recovering a small signal buried in noise, the low-noise amplifier stage found in virtually every cell phone and wireless radio, and a proof that even digital systems require built-in amplification to satisfy the static discipline's stricter output thresholds.

The lecture then introduces the dependent source, a new circuit element whose output depends on some other voltage or current elsewhere in the circuit, and uses a voltage-controlled current source to build the course's first amplifier circuit. Applying the node method to this circuit produces a boxed equation the lecture says will reappear constantly for the rest of 6.002, and the resulting device is shown to amplify, even though it is not yet linear.

Key ideas

• Why amplify (analog): a small signal corrupted by noise during transmission can become unreadable; amplifying the signal before or during transmission makes the same absolute noise proportionally less disruptive.
• Low-noise amplifier (LNA): the first stage after a receiving antenna in cell phones and wireless cards, which boosts a tiny received signal before further analog or digital processing.
• Why amplify (digital): because the static discipline requires stricter output thresholds (VOH, VOL) than input thresholds (VIH, VIL), any digital gate must amplify its input to produce a valid output; the minimum required amplification is (VOH - VOL) / (VIH - VIL), which must exceed one.
• Dependent source: a circuit element, drawn as a diamond, whose output voltage or current is a function of some other voltage or current in the circuit (its control variable), rather than being fixed like an independent source.
• Voltage-controlled current source (VCCS): a dependent current source whose output current is a function of a control voltage elsewhere in the circuit; other combinations include current-controlled current sources, current-controlled voltage sources, and voltage-controlled voltage sources.
• Dependent sources make circuits nonlinear: unlike independent sources, a dependent source's value depends on another circuit variable, so circuits containing them are generally solved with the same node-method techniques used for nonlinear circuits.
• A first amplifier: a supply VS, resistor RL, and a voltage-controlled current source with ID = K/2 * (VI - 1)^2 for VI >= 1 (and ID = 0 otherwise) produces an output V0 = VS - (K*RL/2) * (VI - 1)^2, a boxed result the lecture says recurs throughout the course.
• Amplification without linearity: a small change in input voltage can produce a much larger change in output voltage (for example, 0.1V in producing roughly 1V change out), which is amplification, even though the transfer curve is a nonlinear parabola rather than a straight line.
• Model breakdown at saturation: the ideal dependent-source model would let the output swing negative while the source still supplies current, implying the device generates power, which is unphysical for a passive device; in practice the real device stops behaving like an ideal current source and the output curve flattens out instead.

Walkthrough

Motivating amplification (5:23)

The lecture opens by defining an amplifier abstractly as a box that turns a small input signal into a larger output, often with a separate power port. It shows that a 1 mV signal buried in 10 mV of noise is unreadable, but the same signal amplified to 100 mV before the same noise is added remains recognizable, motivating amplification for noisy analog transmission.

Amplification in real devices and in digital systems (9:40)

A physical cell phone is opened to show its antenna feeding into a low-noise amplifier (LNA) stage before further processing. The lecture then proves that digital systems also require amplification: because output thresholds VOH/VOL are stricter (further apart) than input thresholds VIH/VIL under the static discipline, any valid digital circuit must stretch a valid input swing into a larger valid output swing, giving a minimum required amplification of (VOH - VOL) / (VIH - VIL) > 1.

Introducing the dependent source (19:11)

The dependent source is introduced as a new circuit element with a control port and an output port, where the output value is a function of the control variable rather than fixed. The lecture names the four types by which variable controls which: voltage-controlled current source (VCCS), current-controlled current source (CCCS), current-controlled voltage source (CCVS), and voltage-controlled voltage source (VCVS), and shows how such sources are drawn compactly once the control connection is implied.

Analyzing simple circuits with dependent sources (23:34)

Two worked examples apply the node method to circuits containing a VCCS. In the first, a current source whose output is K/VR feeds back into the same node that sets VR, producing the nonlinear result VR = sqrt(K*R). In the second, the control voltage is fed from elsewhere in the circuit, and both examples confirm that the node method works unchanged for circuits with dependent sources, just as it does for other nonlinear circuits.

Building and analyzing the amplifier circuit (31:09)

A circuit with supply VS, resistor RL, and a VCCS with ID = K/2 * (VI - 1)^2 (for VI >= 1; zero otherwise) is analyzed with the node method, yielding V0 = VS - (K*RL/2) * (VI - 1)^2. Plotting V0 against VI shows the output stays flat at VS while VI is below 1, then drops in a curve as VI increases past 1. The lecture identifies this as amplification, since a small change in VI produces a larger change in V0, confirmed with a numeric example (VS = 10V, K = 2 mA/V^2, RL = 5 kOhm) where a 0.1V change in input produces roughly a 1V change in output, and demonstrated live on an oscilloscope.

Why the ideal model breaks down (49:06)

The lecture points out that the mathematical formula alone would let V0 go negative while the dependent current source still supplies current in the same direction, implying the device is generating power, which is impossible for a passive device. In a real device, the ideal current-source model breaks down as V0 approaches zero, and the device instead behaves more like a resistor, so the actual output curve flattens near zero rather than continuing to fall.

Before you watch

• Watch Lectures 6 and 7 first; this lecture builds directly on nonlinear analysis and the node method applied to nonlinear circuits.
• Recall the static discipline's four voltage thresholds (VOH, VOL, VIH, VIL) from Lecture 4, since they are used to prove digital circuits require amplification.

Check your understanding

1. Why does amplifying a signal before adding noise make the signal easier to recover, even though the same absolute noise is present?
2. How does the static discipline's requirement that VOH > VIH and VOL < VIL imply that every valid digital gate must provide amplification?
3. What distinguishes a dependent source from an independent source, and how does the voltage-controlled current source in this lecture fit that definition?
4. In the amplifier example, why does a 0.1V change in VI produce roughly a 1V change in V0, and why is this still called amplification even though the circuit is nonlinear?
5. Why can't a real, passive dependent-source device actually follow the ideal mathematical curve all the way to large negative output voltages?