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).

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.

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.

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.

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.

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 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.

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.

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.