In a 4-bit R–2R ladder digital-to-analog converter (DAC), the op-amp is used in an inverting summing configuration. Because of negative feedback, what potential does the operational amplifier maintain at its inverting (−) input, often called the virtual ground?

Introduction / Context:
R–2R ladder DACs are popular because they use only two resistor values and interface neatly with an operational amplifier. A key idea is the “virtual ground” at the inverting input when the op-amp is configured with negative feedback. Understanding this node's behavior is crucial to predicting DAC linearity and output voltage.

Given Data / Assumptions:

• 4-bit R–2R ladder network drives an op-amp in inverting summing mode.
• Ideal op-amp assumptions: very high open-loop gain, input bias currents negligible for concept, and negative feedback present.
• Noninverting (+) input is tied to ground.

Concept / Approach:

With high open-loop gain and negative feedback, the op-amp drives its output so that the voltage difference between its inputs is approximately zero. If the noninverting input is grounded, the inverting input node is held near 0 V. This is called a virtual ground because it is at ground potential without being physically connected to ground, enabling each DAC switch current to sum linearly through the feedback network.

Step-by-Step Solution:

Verification / Alternative check:

Analyze nodal currents: the small-signal condition V(−) ≈ 0 V ensures linear superposition of each bit's current. SPICE simulations or bench measurements confirm the inverting input hovers very close to 0 V.

Why Other Options Are Wrong:

5 V (or any fixed supply value) is incorrect; the inverting node is not tied to VCC. A variable voltage set directly by the digital code at V(−) is also incorrect; the code sets currents, not the summing node voltage in an ideal case. “None of the above” is therefore wrong.

Common Pitfalls:

Confusing virtual ground with a real ground; forgetting that finite op-amp gain and bias currents cause a tiny error but do not change the conceptual result.

Final Answer:

zero volts