An operational amplifier, or op-amp, is a high-gain differential amplifier with two inputs and one output. On its own, its output is usually driven hard toward one supply rail or the other. Add negative feedback, however, and the same device can behave like a buffer, a precise amplifier, a signal mixer, an integrator, or a differentiator. The unifying idea is simple: feedback makes the circuit adjust its output until a useful relation holds between the two input voltages.
This chapter develops that idea once, then uses it to derive five practical circuit families. Each derivation starts from the circuit rather than memorized formulas. The final sections show where the ideal model stops being accurate and how to choose components without asking an op-amp to do the impossible.
The Op-Amp and Its Two Inputs
The non-inverting input is labelled \(V_+\), the inverting input \(V_-\), and the output \(V_{\text{out}}\). In the simplest open-loop model, the output is proportional to the difference between the inputs:
\[ V_{\text{out}}=A_{\text{OL}}(V_+-V_-), \]
where \(A_{\text{OL}}\), the open-loop gain, is very large. Real op-amps also have positive and negative supply pins, even when a simplified symbol leaves them out. In the ideal model, the inputs draw no current and the output behaves like a low-impedance voltage source, as long as it stays within its supply-limited range:
Since \(A_{\text{OL}}\) is often tens or hundreds of thousands at low frequency, even a tiny input difference can demand an output far beyond the supply rails. The output then saturates near a rail. This is why an op-amp without feedback is often used as a comparator, not as a predictable linear amplifier.
Negative Feedback and the Two Golden Rules
In a negatively fed-back circuit, some of the output is returned to the inverting input in a way that counteracts a change in the output. If \(V_{\text{out}}\) rises, the feedback raises \(V_-\); the difference \(V_+-V_-\) shrinks, pushing the output back down. This self-correcting action is the source of the circuit's precision.
When an op-amp has effective negative feedback and its output is not saturated, the very large open-loop gain makes the input difference small:
\[ V_-\approx V_+. \]
This is the virtual short rule. It does not say that the input pins are physically connected, and it does not apply when the feedback is positive, broken, or unable to keep the output in its linear range.
The other ideal assumption is that the input terminals draw no current:
\[ I_+=I_-=0. \]
A real op-amp has small input bias currents, but for many first calculations they are negligible. Together, the two rules turn most circuit analysis into ordinary voltage-divider algebra and Kirchhoff's current law.
A Feedback Loop Is a Cause-and-Effect Loop
A useful analysis habit is therefore to check the circuit in this order: identify the feedback path, decide whether it is negative, assume linear operation provisionally, apply \(V_-=V_+\) and zero input current, then verify that the resulting output stays within the op-amp's real limits.
1. Voltage Follower: A Buffer with Gain One
A voltage follower connects the output directly to the inverting input and applies the signal to the non-inverting input. The output voltage follows the input voltage:
The feedback wire gives \(V_-=V_{\text{out}}\), while \(V_+=V_{\text{in}}\). The virtual-short rule then gives
\[ V_{\text{out}}\approx V_-\approx V_+=V_{\text{in}}. \]
A follower does not increase voltage. It is valuable because it separates two parts of a circuit: the source sees the op-amp's high input impedance, while the load is driven by the output stage. For example, a \(10\,\text{k}\Omega\) source resistor connected directly to a \(10\,\text{k}\Omega\) load loses half its voltage. Insert a follower between them and, ideally, almost the full source voltage appears across the load.
This is useful for sensor outputs, voltage dividers, and filters that should not be loaded by the next stage. Check that the chosen op-amp is unity-gain stable; not every device is.
2. Non-Inverting Amplifier
To amplify without reversing polarity, connect \(V_{\text{in}}\) to \(V_+\). A feedback divider sets \(V_-\): \(R_f\) runs from the output to the inverting node, and \(R_g\) runs from that node to ground (or another reference voltage).
Since the input draws no current, the divider is unloaded. Its midpoint voltage is
\[ V_-=V_{\text{out}}\frac{R_g}{R_f+R_g}. \]
Feedback makes \(V_-\approx V_+=V_{\text{in}}\), so
\[ \boxed{\frac{V_{\text{out}}}{V_{\text{in}}} =1+\frac{R_f}{R_g}.} \]
The gain is positive and at least one. For \(R_f=40\,\text{k}\Omega\) and \(R_g=10\,\text{k}\Omega\), the gain is \(5\). An input of \(0.20\,\text{V}\) therefore requests \(1.0\,\text{V}\) at the output, comfortably within the rails of many common op-amps.
The formula describes a ratio relative to ground. If \(R_g\) returns to a reference \(V_{\text{ref}}\) instead, the output is shifted:
\[ V_{\text{out}} =\left(1+\frac{R_f}{R_g}\right)V_{\text{in}} -\frac{R_f}{R_g}V_{\text{ref}}. \]
3. Inverting Amplifier
The inverting amplifier sends the input through \(R_{\text{in}}\) to \(V_-\), returns \(R_f\) from the output to that same node, and connects \(V_+\) to ground or a reference.
The node at \(V_-\) is called a virtual ground: its voltage is approximately zero, but it is not a ground connection. The op-amp input itself takes essentially no current. Kirchhoff's current law at the node therefore says that the current arriving through \(R_{\text{in}}\) must leave through \(R_f\):
\[ \frac{V_{\text{in}}-0}{R_{\text{in}}} =\frac{0-V_{\text{out}}}{R_f}. \]
Hence
\[ \boxed{V_{\text{out}}=-\frac{R_f}{R_{\text{in}}}V_{\text{in}}.} \]
The minus sign means a positive input produces a negative output relative to the reference. If \(R_{\text{in}}=10\,\text{k}\Omega\) and \(R_f=30\,\text{k}\Omega\), an input of \(0.20\,\text{V}\) gives \(-0.60\,\text{V}\). The source sees an input resistance of approximately \(R_{\text{in}}\), unlike the non-inverting configuration's very high input resistance.
4. Summing Amplifier: A Weighted Analog Adder
Add more input branches to the inverting amplifier and each signal contributes current to the same virtual-ground node. With input voltages \(V_1,\ldots,V_n\) and their resistors \(R_1,\ldots,R_n\), Kirchhoff's current law gives
\[ \sum_{i=1}^{n}\frac{V_i}{R_i}+\frac{V_{\text{out}}}{R_f}=0, \qquad \boxed{V_{\text{out}}=-R_f\sum_{i=1}^{n}\frac{V_i}{R_i}.} \]
Equal input resistors give equal weights. If all \(R_i=R\), then
\[ V_{\text{out}}=-\frac{R_f}{R}(V_1+V_2+\cdots+V_n). \]
Setting \(R_f=R\) produces the negative of the sum. Choosing unequal resistor values creates a weighted sum, useful in audio mixers, sensor combinations, and digital-to-analog conversion. The output still has to fit within the available voltage swing; a mathematically valid sum can saturate the physical amplifier.
5. Capacitors Turn Feedback into Calculus
A capacitor's current is proportional to the rate of change of its voltage, \(i_C=C\,dV_C/dt\). Putting a capacitor in the feedback network or the input path therefore makes the output depend on an integral or derivative. The same virtual-ground and current-balance ideas still do the work.
Inverting Integrator
Replace \(R_f\) in the inverting amplifier with a capacitor \(C\). The non-inverting input is grounded, so the inverting node remains at virtual ground. The input resistor current is \(V_{\text{in}}/R\); the capacitor current from the summing node toward the output is \(-C\,dV_{\text{out}}/dt\). Since no current enters the op-amp,
\[ \frac{V_{\text{in}}}{R}+C\frac{dV_{\text{out}}}{dt}=0 \quad\Longrightarrow\quad \boxed{\frac{dV_{\text{out}}}{dt}=-\frac{V_{\text{in}}}{RC}.} \]
Integrating from an initial time \(t_0\) gives the complete relation, including the capacitor's initial condition:
\[ V_{\text{out}}(t) =V_{\text{out}}(t_0)-\frac{1}{RC}\int_{t_0}^{t}V_{\text{in}}(\tau)\,d\tau. \]
For a constant input \(V_0\), the output is a ramp with slope \(-V_0/(RC)\). For example, \(R=100\,\text{k}\Omega\), \(C=10\,\mu\text{F}\), and \(V_0=0.10\,\text{V}\) give \(RC=1\,\text{s}\) and a slope of \(-0.10\,\text{V/s}\).
An ideal integrator has infinite gain at zero frequency. Any input offset, bias current, or tiny DC signal therefore keeps charging the capacitor until the output saturates. A practical integrator places a resistor \(R_f\) in parallel with \(C\). If \(s\) denotes the Laplace variable, its feedback impedance is
\[ Z_f=\frac{R_f}{1+sR_fC}, \qquad \frac{V_{\text{out}}}{V_{\text{in}}} =-\frac{R_f/R}{1+sR_fC}. \]
At low frequency it behaves as an ordinary inverting amplifier with finite gain \(-R_f/R\); above \(f_{\text{low}}=1/(2\pi R_fC)\), it approaches the integrating response. The extra resistor prevents a DC offset from integrating without bound.
Inverting Differentiator
Now put a capacitor \(C\) in the input path and a resistor \(R\) in the feedback path. The virtual-ground node means the input capacitor current is approximately \(C\,dV_{\text{in}}/dt\). That current must flow through \(R\):
\[ C\frac{dV_{\text{in}}}{dt}+\frac{V_{\text{out}}}{R}=0 \quad\Longrightarrow\quad \boxed{V_{\text{out}}=-RC\frac{dV_{\text{in}}}{dt}.} \]
A constant input produces zero output after transients. A rising ramp produces a constant negative output, and a falling ramp produces a constant positive output. For \(R=100\,\text{k}\Omega\), \(C=1\,\mu\text{F}\), and an input slope of \(0.1\,\text{V/s}\), the output is \(-10\,\text{mV}\).
The ideal differentiator is rarely used alone. Since capacitor current increases with frequency, the circuit also amplifies high-frequency noise and can become unstable. A practical differentiator adds a resistor in series with the input capacitor and a small capacitor in parallel with the feedback resistor. Its transfer function is
\[ H(s)=-\frac{R_f}{1+sR_fC_f}\, \frac{sC_{\text{in}}}{1+sR_{\text{in}}C_{\text{in}}}. \]
The series resistor limits high-frequency input current; the feedback capacitor reduces high-frequency gain and improves stability. The circuit approximates differentiation only over a chosen frequency range, rather than differentiating arbitrarily fast changes.
Compare the Five Circuits
| Configuration | Ideal relation | Main use |
|---|---|---|
| Voltage follower | \(V_{\text{out}}=V_{\text{in}}\) | Buffer a source from a load |
| Non-inverting amplifier | \(V_{\text{out}}=(1+R_f/R_g)V_{\text{in}}\) | Amplify without reversing polarity |
| Inverting amplifier | \(V_{\text{out}}=-(R_f/R_{\text{in}})V_{\text{in}}\) | Set gain and invert phase |
| Summing amplifier | \(V_{\text{out}}=-R_f\sum_i V_i/R_i\) | Mix or weight several signals |
| Integrator | \(dV_{\text{out}}/dt=-V_{\text{in}}/(RC)\) | Accumulate a signal over time |
| Differentiator | \(V_{\text{out}}=-RC\,dV_{\text{in}}/dt\) | Respond to a signal's rate of change |
The first four are resistor-feedback circuits; the last two replace one of those resistors with a capacitor. This small change in the feedback or input impedance changes the frequency-dependent gain while the same feedback principles remain in force.
What the Ideal Model Leaves Out
The ideal equations are a starting point, not a promise that any op-amp can produce any output. Before building a circuit, check these limits in the datasheet:
- Output swing and saturation. The output usually cannot reach either supply rail exactly. If a formula demands a voltage outside the allowed swing, the output clips and \(V_-\approx V_+\) no longer holds.
- Input common-mode range. Both inputs must stay within the permitted voltage range relative to the supplies. A valid differential voltage does not guarantee a valid common-mode voltage.
- Gain-bandwidth product. Open-loop gain falls as frequency rises. For a single-pole op-amp, a rough estimate is \(f_{\text{BW}}\approx \text{GBW}/G_N\), where \(G_N\) is the circuit's noise gain. It equals the signal gain in a non-inverting stage; for an inverting stage, \(G_N=1+R_f/R_{\text{in}}\).
- Slew rate. The output cannot change faster than a specified rate. A large, fast signal can distort even when its frequency is below the small-signal bandwidth.
- Input offset and bias current. These small errors can become significant with large resistors, high gain, or an integrator that accumulates DC error.
- Noise and resistor values. Very large resistors increase the effects of current noise, thermal noise, and input bias current; very small resistors waste power and load the output more heavily.
- Stability and capacitive loads. Long cables or large capacitors at the output can reduce phase margin and cause ringing or oscillation. Use an op-amp designed for the load or add an appropriate isolation network.
- Power and layout. Use the correct supply pins and place bypass capacitors close to them. Keep feedback paths short and connect grounds deliberately.
Gain Is Not the Same as Output Range
Suppose a non-inverting stage has gain \(10\), a \(0.5\,\text{V}\) input, and supply rails of \(0\,\text{V}\) and \(5\,\text{V}\). The ideal equation requests \(5\,\text{V}\), but many op-amps cannot reach the positive rail, and the input common-mode range may also be violated. The real design must allow headroom at the input and output, or choose a device whose specified ranges include the intended voltages.
A Reliable Circuit-Design Routine
- Write down the required input range, output range, load, and signal bandwidth.
- Choose the circuit configuration and derive its gain or transfer function.
- Select practical resistor and capacitor values; check loading and the resulting corner frequencies.
- Check output swing, common-mode range, bandwidth, slew rate, noise, and stability in the datasheet.
- Verify the predicted output at the largest positive and negative inputs; make sure neither case saturates.
- Build and measure the circuit, including its response to a step and to the highest-frequency signal of interest.
Summary
Negative feedback makes an op-amp predictable by driving its input difference close to zero. With negligible input current, that gives two powerful analysis rules: \(V_-\approx V_+\) and \(I_+=I_-=0\), as long as the amplifier remains in its linear operating range. A direct feedback wire makes a buffer; resistor networks set amplification and summing; capacitors make the gain depend on time and frequency. The most useful circuit is not the one with the prettiest ideal equation, but the one whose real device, component values, supply rails, and signal range all agree with that equation.