Nullor

Key takeaways

Nullor belongs to engineering; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Nullor to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Nullor from memory before moving on to harder problems.

Reference excerpt

A nullor is a theoretical two-port network consisting of a nullator at its input and a norator at its output. Nullors represent an ideal amplifier, having infinite current, voltage, transconductance and transimpedance gain. Its transmission parameters are all zero, that is, its input–output behavior is summarized with the matrix equation

( v 1 i 1 ) = ( 0 0 0 0 ) ( v 2 i 2 ) {\displaystyle {\begin{pmatrix}v_{1}\\i_{1}\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}{\begin{pmatrix}v_{2}\\i_{2}\end{pmatrix}}}

In negative-feedback circuits, the circuit surrounding the nullor determines the nullor output in such a way as to force the nullor input to zero. Inserting a nullor in a circuit schematic imposes mathematical constraints on how that circuit must behave, forcing the circuit itself to adopt whatever arrangements are needed to meet the conditions. For example, an ideal operational amplifier can be modeled using a nullor, and the textbook analysis of a feedback circuit using an ideal op-amp uses the mathematical conditions imposed by the nullor to analyze the circuit surrounding the op-amp.

Example: voltage-controlled current sink

Figure 1 shows a voltage-controlled current sink. The sink is intended to draw the same current iOUT regardless of the applied voltage VCC at the output. The value of current drawn is to be set by the input voltage vIN. Here the sink is to be analyzed by idealizing the op amp as a nullor. Using properties of the input nullator portion of the nullor, the input voltage across the op amp input terminals is zero. Consequently, the voltage across reference resistor RR is the applied voltage vIN, making the current in RR simply vIN/RR. Again using the nullator properties, the input current to the nullor is zero. Consequently, Kirchhoff's current law at the emitter provides an emitter current of vIN/RR. Using properties of the norator output portion of the nullor, the nullor provides whatever current is demanded of it, regardless of the voltage at its output. In this case, it provides the transistor base current iB. Thus, Kirchhoff's current law applied to the transistor as a whole provides the output current drawn through resistor RC as

i OUT = v IN R R − i B {\displaystyle i_{\text{OUT}}={\frac {v_{\text{IN}}}{R_{\text{R}}}}-i_{\text{B}}}

where the base current of the bipolar transistor iB is normally negligible provided the transistor remains in active mode. That is, based upon the idealization of a nullor, the output current is controlled by the user-applied input voltage vIN and the designer's choice for the reference resistor RR. The purpose of the transistor in the circuit is to reduce the portion of the current in RR supplied by the op-amp. Without the transistor, the current through RC would be iOUT = (VCC − vIN)/RC, which interferes with the design goal of independence of iOUT from VCC. Another practical advantage of the transistor is that the op amp must deliver only the small transistor base current, which is unlikely to tax the op amp's current delivery capability. Of course, only real op amps are current-limited, not nullors. The remaining variation of the current with the voltage VCC is due to the Early effect, which causes the β of the transistor to change with its collector-to-base voltage VCB according to the relation β = β0(1 + VCB/VA), where VA is the so-called Early voltage. Analysis based upon a nullor leads to the output resistance of this current sink as Rout = rO(β + 1) + RC, where rO is the small-signal transistor output resistance given by rO = (VA + VCB)/iout. See current mirror for the analysis. Use of the nullor idealization allows design of the circuitry around the op-amp. The practical problem remains of designing an op-amp that behaves like a nullor.

References

Illustrations

Worked examples

Example 1 — a first encounter with Nullor

Start with the simplest possible case. Write down what Nullor claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In engineering, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Nullor before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Nullor ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Nullor

In research: Nullor appears in engineering research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Nullor in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Nullor is common in secondary-school and first-year university syllabi. It links to neighbouring topics Analog circuits, Control theory, Electrical components, so understanding it makes those chapters shorter.
In everyday life: Look for Nullor outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Nullor in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nullor means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Nullor out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Nullor in simple terms?

A nullor is a theoretical two-port network consisting of a nullator at its input and a norator at its output. Nullors represent an ideal amplifier, having infinite current, voltage, transconductance and transimpedance gain.

Why does Nullor matter?

Because it connects several engineering ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Nullor?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Nullor.