Exercises: The operational amplifier

Draw the symbol for an operational amplifier and label its five basic connections: inverting input, non-inverting input, output, and the two power supply terminals. Explain what the $+$ and $-$ signs next to the inputs mean and why they should not be confused with the supply polarity.

You can do it on paper and send a photo with your answer.

Topic challenge. An ideal operational amplifier (op amp) is characterized by four properties that greatly simplify circuit analysis.

1. a) List the four characteristics of the ideal op amp. For each one, state which op amp quantity it refers to (open-loop gain, input resistance, output resistance and bandwidth) and what value it is assumed to have.

1. b) Explain in your own words what each of them implies when analyzing a circuit with an op amp. For example, an infinite input resistance implies that no current flows into the inverting and non-inverting inputs.

What will be assessed: that you mention the four characteristics with their correct ideal value and that you explain the practical consequence of each one in circuit analysis.

In an op-amp with negative feedback, the "virtual short circuit" applies: the voltage between the two inputs is practically zero. Explain why the huge open-loop gain forces both inputs to remain at the same voltage when the circuit is stable, and deduce from this that no current enters them either.

Without feedback, the operational amplifier (op-amp) works as a comparator. Explain what voltage the output delivers when the non-inverting input ($V^+$) is greater than the inverting input ($V^-$) and what voltage it delivers when the opposite happens. State the type of saturation (positive or negative) in each case, taking into account that the output stays close to $\pm V_{CC}$, and give a practical application of the comparator (for example, a light or temperature level detector).

Look at the circuit with an ideal operational amplifier, in which $R_1 = 10\ \text{k}\Omega$ and $R_2 = 47\ \text{k}\Omega$:

Identify the inverting configuration and answer:

1. a) Which input of the operational amplifier does the signal $V_e$ reach?

1. b) Which resistors make up the circuit and what is the function of each one?

1. c) Calculate the voltage gain $A_v = \dfrac{V_{sal}}{V_e}$ of the circuit.

1. d) Explain what the sign of the gain you have obtained means physically.

The figure shows two circuits with an ideal operational amplifier, A and B. Both use the resistors $R_1$ and $R_2$. $V_e$ is the input voltage and $V_s$ the output voltage.

1. a) Identify which of the two circuits is the non-inverting one and write its gain $A_v = \frac{V_s}{V_e}$ as a function of $R_1$ and $R_2$.

1. b) Explain why the minimum gain of this circuit is $1$.

1. c) Compare it with the inverting circuit in terms of the phase of the output signal with respect to the input signal.

Challenge of the unit. The voltage follower is an operational amplifier in a non-inverting configuration taken to the limit: the output is connected directly to the inverting input and the signal is applied to the non-inverting input.

1. a) State what its voltage gain is.

1. b) Explain what a circuit that "does not amplify" is used for. Justify your answer in terms of input impedance and output impedance, and of whether or not it loads the previous stage.

1. c) Give a practical example of use in which it is helpful, stating what is connected to its input and what is connected to its output.

You will be assessed on: the correct gain, the explanation of the impedance-matching function, and a realistic, well-justified example of use.

Challenge of the unit. An (ideal) operational amplifier is set up as an inverting summer with two inputs, $V_1$ and $V_2$. Each input reaches the inverting input through its own input resistor, and both resistors are equal, with value $R_1$. The non-inverting input is connected to ground and the feedback resistor, between the output and the inverting input, is $R_2$.

1. a) Applying the properties of the ideal operational amplifier (zero voltage between its inputs and zero current through them) and Kirchhoff's current law at the inverting input node, derive the expression for the output voltage $V_s$ as a function of $V_1$, $V_2$, $R_1$ and $R_2$.

1. b) Calculate $V_s$ if $R_1 = 10\ \text{k}\Omega$, $R_2 = 10\ \text{k}\Omega$, $V_1 = 0{,}5\ \text{V}$ and $V_2 = 0{,}3\ \text{V}$.

1. c) Explain a practical application of this circuit, for example as an audio signal mixer: what each input represents and what happens to the output.

You will be assessed on: the correct derivation of the expression, the numerical calculation with its units, and the clarity of the explanation of the application.

An inverting amplifier has $R_1 = 10\ \text{k}\Omega$ and $R_2 = 47\ \text{k}\Omega$. a) Calculate its gain. b) Calculate the output voltage for an input of $0{,}5\ \text{V}$. c) State the phase shift between the input and output signals.

A non-inverting amplifier has $R_1 = 10\ \text{k}\Omega$ and $R_2 = 22\ \text{k}\Omega$. a) Calculate its gain with $A_v = 1 + \frac{R_2}{R_1}$. b) Calculate the output for an input of $1\ \text{V}$. c) Calculate the gain if $R_2$ is short-circuited and state the name of this circuit configuration.

The figure shows an inverting amplifier powered at $\pm 15\ \text{V}$. a) Calculate its gain. b) Calculate the output voltage, marked with the question mark, for the specified input. c) Explain why the inverting input acts as a "virtual earth" even though it is not connected to earth.

The figure shows a non-inverting amplifier powered at $\pm 15\ \text{V}$.

1. a) Calculate its gain, marked with the question mark.

1. b) Calculate the output voltage for the indicated input.

1. c) Explain what advantage this circuit offers over the inverting one when the signal source is weak.

An op-amp powered at $\pm 12\ \text{V}$ operates in open loop as a comparator: the inverting input is set to $3\ \text{V}$ using a voltage divider and the non-inverting input receives the signal from a sensor. a) Describe the output when the signal is $2\ \text{V}$, $4\ \text{V}$, and exactly $3\ \text{V}$. b) Explain why in this setup it makes no sense to talk about "gain".

The figure shows the waveforms of a comparator powered at $\pm 12\ \text{V}$ whose reference is obtained from a divider with two $10\ \text{k}\Omega$ resistors between $12\ \text{V}$ and earth. a) Calculate the reference voltage, marked with the question mark. b) Explain in which sections of the input the output is at $+12\ \text{V}$ and in which ones at $-12\ \text{V}$, knowing that the signal enters via the non-inverting terminal. c) Reason what would happen if the inputs were swapped.

An inverting amplifier has a gain of $-20$ and is powered at $\pm 12\ \text{V}$. a) Calculate the theoretical output for inputs of $0{,}2\ \text{V}$, $0{,}5\ \text{V}$, and $1\ \text{V}$. b) State which of these outputs are real and which saturate, explaining why the output can never exceed the supply voltage.

An inverting summer with an ideal operational amplifier has two inputs, $V_1$ and $V_2$, each with a resistor of $10\ \text{k}\Omega$, and a feedback resistor of $10\ \text{k}\Omega$.

1. a) Calculate the output voltage for $V_1 = 1\ \text{V}$ and $V_2 = 2\ \text{V}$.

1. b) Repeat the calculation with a feedback resistor of $22\ \text{k}\Omega$.

1. c) Explain how to make one input "weigh" twice as much as the other in the sum.

You need to condition the signal from a sensor whose internal output impedance is $R_s = 2{,}2\ \text{k}\Omega$ and whose signal delivers $v_s = 0{,}4\ \text{V}$. A voltage gain with a magnitude of 15 is required to drive an acquisition stage, supplying the operational amplifier with $\pm 15\ \text{V}$ (saturation voltage $V_{sat} = \pm 13{,}5\ \text{V}$).

1. a) A classmate proposes using an inverting amplifier with standard E12 series resistors, setting $R_1 = 10\ \text{k}\Omega$ and $R_2 = 150\ \text{k}\Omega$, connecting the sensor directly to the input. Calculate the actual gain that would be obtained taking $R_s$ into account and justify the relative error incurred with respect to the gain of $-15$.
2. b) Redesign the inverting circuit by choosing new standard values from the E12 series for $R_1$ and $R_2$ so that the resulting actual gain (considering $R_s$) is as close as possible to $-15$.
3. c) With the components of your design, predict the output voltage $v_o$ for $v_s = 0{,}4\ \text{V}$ and for $v_s = -1{,}1\ \text{V}$, justifying whether the op-amp enters saturation in either case.

An industrial sensor with high output impedance delivers a DC signal between $0\ \text{V}$ and $0{,}5\ \text{V}$, and must be connected to an analog-to-digital converter (ADC) whose input range is $0\ \text{V}$ to $5{,}5\ \text{V}$. You have an operational amplifier powered at $\pm 12\ \text{V}$ (with saturation at $\pm 10{,}5\ \text{V}$) and standard E12 series resistors (from $1\ \text{k}\Omega$ to $100\ \text{k}\Omega$).

1. a) Justify why an inverting topology would degrade the measurement due to loading and sign effects, and demonstrate why the non-inverting configuration solves both drawbacks.
2. b) Design the non-inverting stage by selecting standard E12 values for $R_1$ and $R_2$ that maximize the dynamic range of the ADC without exceeding it in any case (guaranteeing $V_{out} \le 5{,}5\ \text{V}$ for $V_{in} = 0{,}5\ \text{V}$).
3. c) With the chosen standard values, predict the actual output range for the sensor input range and calculate the percentage full-scale error with respect to the $5{,}5\ \text{V}$ target. Reason whether there is any risk of op-amp saturation.

To process a $100\ \text{kHz}$ sinusoidal signal with an amplitude of $0{,}3\ \text{V}$, a non-inverting amplifier system with an overall voltage gain of $A_v = +10$ powered at $\pm 15\ \text{V}$ is required. You have identical operational amplifiers with a gain-bandwidth product $GBP = 1\ \text{MHz}$ and a saturation voltage of $\pm 13{,}5\ \text{V}$. Two design options are considered:

• Option A: A single non-inverting stage with $A_v = +10$.

• Option B: Two cascaded inverting stages with $A_{v1} = -4$ and $A_{v2} = -2{,}5$.

1. a) Calculate the $-3\ \text{dB}$ cutoff frequency of each stage in both options and justify which of the two alternatives allows the $100\ \text{kHz}$ signal to be amplified without undesirable attenuation.
2. b) If each op-amp exhibits an input offset voltage of $V_{os} = 3\ \text{mV}$, calculate the worst-case DC component that will appear at the final output at idle ($v_{in} = 0\ \text{V}$) for both options.
3. c) In view of the bandwidth and offset, justify in a reasoned manner which design option is technically superior for this application.

A temperature sensor with a sensitivity of $10\ \text{mV/}^\circ\text{C}$ ($0\ \text{V}$ at $0^\circ\text{C}$) monitors an enclosure. The analog signal presents high-frequency noise with an amplitude of $\pm 15\ \text{mV}$. It is required to drive a heating relay using an operational amplifier powered between $0\ \text{V}$ and $12\ \text{V}$ (saturation levels $0\ \text{V}$ and $12\ \text{V}$), so that the heating turns on ($V_{out} = 12\ \text{V}$) if the temperature drops below $19^\circ\text{C}$ and turns off ($V_{out} = 0\ \text{V}$) when it exceeds $22^\circ\text{C}$.

1. a) Predict the behavior that the relay will experience if a simple comparator without hysteresis is implemented with a threshold at $20{,}5^\circ\text{C}$ in the presence of said noise.
2. b) Design the appropriate comparator with hysteresis (Schmitt trigger): draw the schematic, justify which input receives the sensor, and calculate the reference voltage and the resistors needed to precisely set the two switching thresholds at $19^\circ\text{C}$ and $22^\circ\text{C}$.
3. c) Plot the transfer curve $V_{out}$ versus $V_{in}$, clearly indicating the threshold values and the direction of the hysteresis loop.

A three-channel active audio mixer is designed using an operational amplifier powered at $\pm 15\ \text{V}$ ($V_{sat} = \pm 13{,}5\ \text{V}$). The following specifications are established: gain for channel 1 ($v_1$) of $-1$, gain for channel 2 ($v_2$) of $-2$, and gain for channel 3 ($v_3$) of $-0{,}5$. Furthermore, to avoid overloading the audio preamplifiers, the input resistance of any channel must be at least $10\ \text{k}\Omega$.

1. a) Design the inverting summing topology by calculating the minimum standard values of resistors $R_1$, $R_2$, $R_3$, and feedback resistor $R_f$ that satisfy the impedance condition.
2. b) If at a given instant the signals simultaneously reach their peak values: $V_1 = 1{,}5\ \text{V}$, $V_2 = 4{,}0\ \text{V}$, and $V_3 = 3{,}0\ \text{V}$, calculate the theoretical output and justify whether the signal will undergo clipping due to saturation.
3. c) If the feedback resistor $R_f$ is accidentally unsoldered, leaving an open circuit, deduce in a reasoned manner what voltage will be recorded at the op-amp output in the presence of the signals from section b.

A measurement bridge with an equivalent Thévenin resistance $R_{th} = 5\ \text{k}\Omega$ generates a signal between $0\ \text{mV}$ and $100\ \text{mV}$. This signal must be conditioned to be fed into an analog-to-digital converter with a full-scale range from $0\ \text{V}$ to $5\ \text{V}$.

1. a) A technician directly connects an inverting amplifier with a nominal gain of $-50$ using $R_1 = 2\ \text{k}\Omega$ and $R_f = 100\ \text{k}\Omega$ (followed by a unity inverting buffer). Calculate the actual voltage obtained for an input of $100\ \text{mV}$ and determine the percentage error introduced by the bridge loading effect.
2. b) Design a non-inverting stage that completely eliminates the loading effect: calculate the standard E12 series values for $R_1$ and $R_2$ that provide the closest gain to 50 without exceeding $5\ \text{V}$ at the output.
3. c) If the chosen op-amp has an input offset voltage $V_{os} = 1{,}5\ \text{mV}$, predict the output voltage when the bridge is balanced ($V_{in} = 0\ \text{mV}$) and calculate what percentage of the ADC range is wasted due to this error.

In a laboratory test bench, four circuits with 741 operational amplifiers ($V_{CC} = \pm 15\ \text{V}$, $V_{sat} = \pm 13{,}5\ \text{V}$) are tested, yielding anomalous results compared to the ideal model:

• Circuit 1: With a theoretical gain of $-10$, when inputting a $2\ \text{V}$ peak sine wave at $1\ \text{kHz}$, the output is flatly clipped at $+13{,}5\ \text{V}$ and $-13{,}5\ \text{V}$.

• Circuit 2: Connected as a voltage follower ($A_v = 1$), when applying a $10\ \text{V}$ amplitude square wave at $500\ \text{kHz}$, the output is a symmetrical triangular wave with rising and falling slopes of $0{,}5\ \text{V}/\mu\text{s}$.

• Circuit 3: An inverting amplifier with $R_1 = 1\ \text{M}\Omega$ and $R_f = 10\ \text{M}\Omega$, with its input connected to ground ($0\ \text{V}$), delivers a DC voltage of $-0{,}8\ \text{V}$ at the output instead of $0\ \text{V}$. When reducing the resistors to $1\ \text{k}\Omega$ and $10\ \text{k}\Omega$, the output becomes practically zero.

• Circuit 4: With a gain of $A_v = 100$, it amplifies correctly at $100\ \text{Hz}$, but at $200\ \text{kHz}$ the actual gain drops below 5 without the signal undergoing clipping.

For each of the four circuits: identify the real op-amp parameter that causes the phenomenon, analytically justify the numerical result, and propose a practical design solution.

A student sets up an inverting amplifier with a nominal gain $A_v = -10$ powered at $\pm 15\ \text{V}$ ($V_{sat} = \pm 13{,}8\ \text{V}$), using $R_1 = 10\ \text{k}\Omega$ and $R_2 = 100\ \text{k}\Omega$. When connecting a DC input $v_i = 0{,}5\ \text{V}$, the voltmeter constantly reads $v_o = -13{,}8\ \text{V}$, regardless of how $v_i$ varies.

To locate the fault, they measure DC voltages with respect to ground at several points with the circuit powered on:

• Non-inverting pin ($V^+$): $0{,}00\ \text{V}$.

• Inverting pin ($V^-$): $+0{,}49\ \text{V}$.

• Input node before $R_1$: $+0{,}50\ \text{V}$.

1. a) Analyze these readings in light of the virtual short-circuit concept and deduce with reasoning which of these faults has occurred: short circuit in $R_2$, swapped inverting and non-inverting pins, or open track/broken resistor in the feedback $R_2$.
2. b) If, instead of those measurements, $V^+ = 0{,}00\ \text{V}$, $V^- = 0{,}00\ \text{V}$, and $v_o = -13{,}8\ \text{V}$ had been recorded for $v_i = 0{,}5\ \text{V}$, justify what internal or assembly cause would explain that situation.
3. c) Design a two-step protocol using a multimeter (in voltmeter and ohmmeter modes) to unambiguously verify the condition of the feedback path before energizing the board.

Topic Challenge. Design and simulate an automatic cooling system for an equipment cabinet in Tinkercad Circuits, contextualized with real meteorological data from your area.

1. Data collection: Go to the AEMET website and check the maximum temperature recorded ($T_{\text{máx}}$) at the station closest to your municipality over the past month. This value will set your critical trigger threshold.

2. Analog circuit design: Build a circuit in Tinkercad that coordinates:

3. An operational amplifier (LM358) in comparator configuration, which compares the thermal sensor signal against the $T_{\text{máx}}$ reference.

4. A BJT transistor (NPN) biased for switching (cutoff/saturation) to drive a DC motor (fan) from the op-amp output.

5. A silicon diode ($V_D \approx 0,7\text{ V}$) connected in antiparallel with the motor to suppress inductive spikes (flyback diode).

6. Calculation and verification: Analytically determine the value of the base resistor $R_B$ that ensures saturation ($V_{CE} \le 0,2\text{ V}$) and verify with the virtual multimeter that the voltages match your calculations.

7. Technical report: Write a brief document using LibreOffice Writer that includes the public link to the circuit, the technical report with the calculations, and the justification for the chosen threshold.

Evaluation criteria:

• Functional and correct integration of the three blocks (protection diode, switching BJT, and op-amp comparator).

• Precision in the theoretical biasing calculations and their comparison with the simulation.

• Coherent justification of the design based on the real data obtained from AEMET.