Equations
Equations page banner

Equations Library

Electronics related equations and more.

Loading equations

Freshly launched content! We'll be adding more on a regular basis.

 Question, comment or suggestion?

Electronics Reference

Operational Amplifiers (Basic Circuits)

Operational Amplifier (Op-Amp) Parameters

Common-Mode Rejection Ratio
latex!encoded:base64,XHRleHRybXtDTVJSfT1cZnJhY3tcdGV4dHJte0F9X3tvbH19e1x0ZXh0cm17QX1fe2NtfX09MjBcO1x0ZXh0cm17bG9nfVxsZWZ0ICggXGZyYWN7XHRleHRybXtBfV97b2x9fXtcdGV4dHJte0F9X3tjbX19IFxyaWdodCApIFw7XDtcOyAoXHRleHRybXtkQn0p
Aol - Open-Loop Differential Voltage Gain
Acm - Common-Mode Gain
Input Bias Currentlatex!encoded:base64,XHRleHRybXtJfV97XHRleHRybXtCSUFTfX09XGZyYWN7MX17Mn1cbGVmdCAoIFx0ZXh0cm17SX1fMStcdGV4dHJte0l9XzJ9IFxyaWdodCAp
Input Offset Currentlatex!encoded:base64,XHRleHRybXtJfV97XHRleHRybXtPU319PVxsZWZ0IHwgXHRleHRybXtJfV8xLSBcdGV4dHJte0l9XzJ9IFxyaWdodCB8
Offset Voltagelatex!encoded:base64,XHRleHRybXtWfV97XHRleHRybXtPU319PVx0ZXh0cm17SX1fe1x0ZXh0cm17T1N9fVx0ZXh0cm17Un1fe1x0ZXh0cm17SU59fQ==
Error Output Voltagelatex!encoded:base64,XHRleHRybXtWfV97XHRleHRybXtPVVR9X3tcdGV4dHJteyhFUlJPUil9fX09XHRleHRybXtBfV97dn1cdGV4dHJte0l9X3tcdGV4dHJte09TfX1cdGV4dHJte1J9X3tcdGV4dHJte0lOfX0=
Slew Ratelatex!encoded:base64,XHRleHRybXtTbGV3XDtSYXRlfSA9IFxmcmFje1xEZWx0YSBcdGV4dHJte1Z9X3tcdGV4dHJte09VVH19fXtcRGVsdGEgXHRleHRybSB0fT1cZnJhY3srXHRleHRybXtWfV97XHRleHRybXtNQVh9fS0oLVx0ZXh0cm17Vn1fe1x0ZXh0cm17TUFYfX0pfXtcRGVsdGEgXHRleHRybSB0fSBcO1w7XDsgKFx0ZXh0cm17Vn0vXG11IFx0ZXh0cm17c30p

Common-Mode Rejection Ratio (CMRR)

In an op-amp, the desired input signal can only occur on one input. Signals may occur on the two inputs, however they must have different polarities. The purpose of this is that if unwanted signals such as noise occur on both input lines with the same polarity, they are cancelled by the op-amp so that they won’t be amplified and don’t appear on the output. The measure of an op-amp’s ability to reject common-mode signals is called common-mode rejection ratio (CMRR). Aol is the open-loop differential voltage gain and Acm is the common-mode gain of the op-amp.

Input Bias Current (IBIAS)


Ideally, there should be no current flowing through the op-amp’s inputs. However, in reality there is this input bias current that is required by the inputs of the op-amp to bias the first stage of the op-amp. Input bias current is the average of the two input currents, I1 and I2.

Input Offset Current (IOS)

Ideally, input bias currents should be equal but in reality, they are not. The difference between the two input bias currents, which is an absolute value, is called input offset current.

Offset Voltage (VOS)


The product of the input offset current and the input impedance of the op-amp.

Error Output Voltage, VOUT(ERROR)


Amplified offset voltage by the op-amp’s gain.

Slew Rate


The maximum rate of change of the op-amp’s output in response to a step input voltage.

 > Operational Amplifier (Op-Amp) Parameters

Noninverting Amplifier Configuration

Feedback Circuit Attenuationlatex!encoded:base64,XHRleHRybXtCfT1cZnJhY3tcdGV4dHJte1J9X3tpfX17XHRleHRybXtSfV97aX0rXHRleHRybXtSfV97Zn19
Voltage Gainlatex!encoded:base64,XHRleHRybXtBfV97Y2x9X3tcdGV4dHJteyhOSSl9fX09MStcZnJhY3tcdGV4dHJte1J9X3tmfX17XHRleHRybXtSfV97aX19
Input Impedancelatex!encoded:base64,XHRleHRybXtafV97aW59X3tcdGV4dHJteyhOSSl9fX09XGxlZnQgKCAxKyBcdGV4dHJte0F9X3tvbH1cdGV4dHJte0J9IFxyaWdodCApXHRleHRybXtafV97aW59

Zin - Open-loop input impedance of the op-amp
Output Impedancelatex!encoded:base64,XHRleHRybXtafV97b3V0fV97XHRleHRybXsoTkkpfX19PVxmcmFje1x0ZXh0cm17Wn1fe291dH19ezErIFx0ZXh0cm17QX1fe29sfVx0ZXh0cm17Qn19

Zout - Open-loop internal output impedance of the op-amp

Feedback Circuit Attenuation (B)

The attenuation (B) caused by the feedback circuit composed of Rf and Ri.

Voltage Gain, Acl(NI)


The closed-loop gain of noninverting amplifier configuration which is reciprocal of the attenuation of the feedback circuit.

Input Impedance, Zin(NI)

Closed-loop input impedance of noninverting amplifier configuration where Aol is the open-loop gain of the op-amp, B is the attenuation, and Zin is the open-loop input impedance of the op-amp.

Output Impedance, Zout(NI)

Output impedance of noninverting amplifier configuration with negative feedback where Zout is the open-loop internal output impedance of the op-amp.

 > Noninverting Amplifier Configuration

Voltage-Follower Configuration

Voltage Gainlatex!encoded:base64,XHRleHRybXtBfV97Y2x9X3tcdGV4dHJteyhWRil9fX09MQ==
Input Impedancelatex!encoded:base64,XHRleHRybXtafV97aW59X3tcdGV4dHJteyhWRil9fX09XGxlZnQgKCAxKyBcdGV4dHJte0F9X3tvbH0gXHJpZ2h0IClcdGV4dHJte1p9X3tpbn0=

Zin - Open-loop input impedance of the op-amp
Output Impedancelatex!encoded:base64,XHRleHRybXtafV97b3V0fV97XHRleHRybXsoVkYpfX19PVxmcmFje1x0ZXh0cm17Wn1fe291dH19ezErIFx0ZXh0cm17QX1fe29sfX0=

Zout - Open-loop internal output impedance of the op-amp

Voltage Gain, Acl(VF)

Voltage follower configuration is just like a noninverting amplifier configuration with its output fed back to its inverting input by a straight connection making its attenuation equal to 1. Since the gain of a noninverting amplifier configuration is just the reverse of its attenuation, the voltage gain of a voltage follower is also 1, which means there is no gain.

Input Impedance, Zin(VF)

The same input impedance formula of noninverting amplifier configuration but with B equal to 1 and greater input impedance.

Output Impedance, Zout(VF)

The same output impedance formula of noninverting amplifier configuration but with B equal to 1 and much smaller output impedance.

 > Voltage-Follower Configuration

Inverting Amplifier Configuration

Feedback Circuit Attenuationlatex!encoded:base64,XHRleHRybXtCfT1cZnJhY3tcdGV4dHJte1J9X3tpfX17XHRleHRybXtSfV97aX0rXHRleHRybXtSfV97Zn19
Voltage Gainlatex!encoded:base64,XHRleHRybXtBfV97Y2x9X3tcdGV4dHJteyhJKX19fT0tXGZyYWN7XHRleHRybXtSfV97Zn19e1x0ZXh0cm17Un1fe2l9fQ==
Input Impedancelatex!encoded:base64,XHRleHRybXtafV97aW59X3tcdGV4dHJteyhJKX19fSBcY29uZyBcdGV4dHJte1J9X3tpfQ==
Output Impedancelatex!encoded:base64,XHRleHRybXtafV97b3V0fV97XHRleHRybXsoSSl9fX09XGZyYWN7XHRleHRybXtafV97b3V0fX17MSsgXHRleHRybXtBfV97b2x9XHRleHRybXtCfX0=

Zout - Open-loop internal output impedance of the op-amp

Feedback Circuit Attenuation (B)

The attenuation (B) caused by the feedback circuit composed of Rf and Ri.

Voltage Gain, Acl(I)


Closed-loop voltage gain of inverting amplifier configuration which is the ratio of Rf to Ri.

Input Impedance, Zin(I)

Inverting amplifier input impedance is equal to Ri because the inverting input is at virtual ground and the input source sees Ri to ground.

Output Impedance, Zout(I)

The same output impedance formula of noninverting amplifier configuration.

 > Inverting Amplifier Configuration

Op-Amp Frequency and Phase Responses

Open-Loop:

Bandwidth (BW)

In general, bandwidth is the difference between the upper critical frequency (fcu) and lower critical frequency (fcl) of an amplifier. Since the fcl of an op-amp is zero, its bandwidth is equal to its fcu.
latex!encoded:base64,XHRleHRybXtCV309XHRleHRybXtmfV97Y3V9XHRleHRybXtcO1w7KHVwcGVyXDtjcml0aWNhbFw7ZnJlcXVlbmN5KX0=
Internal RC lag circuit attenuation

Op-amps internal RC lag circuit attenuation. The RC lag circuits inside an op-amp causes roll-off in gain as frequency increases.
latex!encoded:base64,XGZyYWN7XHRleHRybXtWfV97b3V0fX17XHRleHRybXtWfV97aW59fT1cZnJhY3sxfXtcc3FydHsxK1xmcmFje1x0ZXh0cm17Zn1eezJ9fXtcdGV4dHJte2Z9X3tjfV57Mn19fX0=
Open-Loop Gain (Aol)

The open-loop gain of an op-amp is the product of the midrange open-loop gain (Aol(mid)) and the internal RC lag circuit attenuation.
latex!encoded:base64,XHRleHRybXtBfV97b2x9PVxmcmFje1x0ZXh0cm17QX1fe29sX3tcdGV4dHJteyhtaWQpfX19fXtcc3FydHsxK1xmcmFje1x0ZXh0cm17Zn1eezJ9fXtcdGV4dHJte2Z9X3tjfV57Mn19fX0=
Phase Shift (θ)

A phase shift created between the input signal and the output signal because of the delay caused by the internal RC lag circuit.
latex!encoded:base64,XFRoZXRhID0gLVx0ZXh0cm17dGFufV57LTF9XGxlZnQgKCBcZnJhY3tcdGV4dHJte2Z9fXtcdGV4dHJte2Z9X3tjfX0gXHJpZ2h0ICk=

Closed-Loop:

Critical Frequency, fc(cl)

The closed-loop critical frequency of an op-amp. The negative feedback extends the critical frequency of an op-amp.
latex!encoded:base64,XHRleHRybXtmfV97Y197KGNsKX19PVx0ZXh0cm17Zn1fe2NfeyhvbCl9fVxsZWZ0ICggMStcdGV4dHJte0J9IFx0ZXh0cm17QX1fe29sX3tcdGV4dHJteyhtaWQpfX19IFxyaWdodCAp
Bandwidth (BWcl)

The closed-loop bandwidth of an op-amp. As the negative feedback increases the limit of an op-amp’s critical frequency, it also extends the bandwidth of the op-amp.
latex!encoded:base64,XHRleHRybXtCV31feyhjbCl9PVx0ZXh0cm17Qld9X3sob2wpfVxsZWZ0ICggMStcdGV4dHJte0J9IFx0ZXh0cm17QX1fe29sX3tcdGV4dHJteyhtaWQpfX19IFxyaWdodCAp
Unity-gain Bandwidth

Bandwidth which is equal to the frequency at which the open-loop gain of an op-amp is unity or 0dB.
latex!encoded:base64,XHRleHRybXtmfV97XHRleHRybXtUfX09XHRleHRybXtBfV97KGNsKX1cdGV4dHJte2Z9X3tjX3soY2wpfX0=
 > Op-Amp Frequency and Phase Responses

Comparator with Hysteresis (Schmitt Trigger)

Upper Trigger Point Voltagelatex!encoded:base64,XHRleHRybXtWfV97XHRleHRybXtVVFB9fT1cZnJhY3tcdGV4dHJte1J9X3syfX17XHRleHRybXtSfV97MX0rXHRleHRybXtSfV97Mn19XGxlZnQgKCArXHRleHRybXtWfV97XHRleHRybXtPVVR9X3tcdGV4dHJteyhNQVgpfX19IFxyaWdodCAp
Lower Trigger Point Voltagelatex!encoded:base64,XHRleHRybXtWfV97XHRleHRybXtMVFB9fT1cZnJhY3tcdGV4dHJte1J9X3syfX17XHRleHRybXtSfV97MX0rXHRleHRybXtSfV97Mn19XGxlZnQgKCAtXHRleHRybXtWfV97XHRleHRybXtPVVR9X3tcdGV4dHJteyhNQVgpfX19IFxyaWdodCAp
Hysteresis Amountlatex!encoded:base64,XHRleHRybXtWfV97XHRleHRybXtIWVN9fT1cdGV4dHJte1Z9X3tcdGV4dHJte1VUUH19LVx0ZXh0cm17Vn1fe1x0ZXh0cm17TFRQfX0=

Upper Trigger Point Voltage (VUTP)

To make a comparator insusceptible to noise, a technique that uses positive feedback called hysteresis can be used. The upper trigger point voltage is a reference level wherein the output switches to maximum negative voltage from the maximum positive voltage when the input exceeds the upper trigger point.

Lower Trigger Point Voltage (VLTP)


The lower trigger point voltage is a reference level wherein the output switches to maximum positive voltage from the maximum negative voltage when the input goes below the lower trigger point.

Hysteresis Amount (VHYS)


The hysteresis amount is determined by the difference between the VUTP and the VLTP.

 > Comparator with Hysteresis (Schmitt Trigger)

Summing Amplifier Configuration

latex!encoded:base64,XHRleHRybXtWfV97XHRleHRybXtPVVR9fT0tXGxlZnQgKCBcZnJhY3tcdGV4dHJte1J9X3tmfX17XHRleHRybXtSfV97MX19IFx0ZXh0cm17Vn1fe1x0ZXh0cm17SU59X3sxfX0gKyBcZnJhY3tcdGV4dHJte1J9X3tmfX17XHRleHRybXtSfV97Mn19IFx0ZXh0cm17Vn1fe1x0ZXh0cm17SU59X3syfX0gKyBcZnJhY3tcdGV4dHJte1J9X3tmfX17XHRleHRybXtSfV97M319IFx0ZXh0cm17Vn1fe1x0ZXh0cm17SU59X3szfX0gKyBcY2RvdHMgKyAgXGZyYWN7XHRleHRybXtSfV97Zn19e1x0ZXh0cm17Un1fe259fSBcdGV4dHJte1Z9X3tcdGV4dHJte0lOfV97bn19IFxyaWdodCAp

A summing amplifier is an op-amp configuration that can add or mix two or more input signals. Basically, it’s like an inverting amplifier with more input signals and resistors. Its output voltage is proportional to the negative of the algebraic sum of its input voltages.

 > Summing Amplifier Configuration

Integrator Configuration

latex!encoded:base64,XHRleHRybXtPdXRwdXRcO1ZvbHRhZ2VcO1Nsb3BlOlw7XDtcO30gXGZyYWN7XERlbHRhIFx0ZXh0cm17Vn1fe1x0ZXh0cm17T1VUfX19e1xEZWx0YSBcdGV4dHJte3R9fSA9IC1cZnJhY3tcdGV4dHJte1Z9X3tcdGV4dHJte0lOfX19e1x0ZXh0cm17Un1fe2l9XHRleHRybXtDfX0=

An integrator is an op-amp configuration that simulates mathematical integration. Its output voltage is proportional to the input voltage integrated over time.

 > Integrator Configuration

Differentiator Configuration

latex!encoded:base64,XHRleHRybXtWfV97XHRleHRybXtPVVR9fT0tXGxlZnQgKCBcZnJhY3tcdGV4dHJte1Z9X3tcdGV4dHJte0N9fX17XHRleHRybXt0fX0gXHJpZ2h0ICkgXHRleHRybXtSfV97Zn0gXHRleHRybXtDfSBcO1w7XDtcO1w7XDtcOyAoXHRleHRybXtWfV97XHRleHRybXtDfX19PVx0ZXh0cm17Vn1fe1x0ZXh0cm17SU59fX0p

A differentiator is an op-amp configuration that simulates mathematical differentiation. It produces an output that is proportional to the rate of change of the input voltage.

 > Differentiator Configuration
Make Bread with our CircuitBread Toaster!

Get the latest tools and tutorials, fresh from the toaster.

What are you looking for?