Gain–bandwidth product

This quantity is commonly specified for operational amplifiers, and allows circuit designers to determine the maximum gain that can be extracted from the device for a given frequency (or bandwidth) and vice versa.

When adding LC circuits to the input and output of an amplifier the gain rises and the bandwidth decreases, but the product is generally bounded by the gain–bandwidth product.

If the GBWP of an operational amplifier is 1 MHz, it means that the gain of the device falls to unity at 1 MHz. Hence, when the device is wired for unity gain, it will work up to 1 MHz (GBWP = gain × bandwidth, therefore if BW = 1 MHz, then gain = 1) without excessively distorting the signal. The same device when wired for a gain of 10 will work only up to 100 kHz, in accordance with the GBW product formula. Further, if the maximum frequency of operation is 1 Hz, then the maximum gain that can be extracted from the device is 1{{e|6}}.

We can also analytically show that for frequencies $\backslash omega\; \backslash gg\; \backslash omega\_c${{Clarification needed|reason=\omega_c needs to be specified. It is presumably a cutoff frequency, though that should be specified, as well as what is it a cutoff frequency of? The low cutoff frequency of the op-amp's band pass range?|date=June 2023}} GBWP is constant.

Let $A\_1(\backslash omega)$ be a first-order transfer function given by:

$A\_1(\backslash omega)=\; \backslash frac\{\{\{H\_0\}\}\}\{\{\backslash sqrt\; \{1\; +\; \{\{\backslash left(\; \{\backslash frac\{\backslash omega\; \}\{\{\{\backslash omega\_c\}\}\}\}\; \backslash right)\}^2\}\}\; \}\}$

We will show that:

$\backslash mathit\{GBWP\}\_\{\backslash omega\; \backslash gg\; \{\backslash omega\_c\}\}\; =\; \{A\_1\}(\backslash omega\; )\backslash cdot\backslash omega\; \backslash approx\; \backslash mathrm\{const.\}$

Proof: We will expand $A\_1$ using Taylor series and retain the constant and first term, to obtain:

$\backslash mathit\{GBWP\}\; =\; \{A\_1\}(\backslash omega\; )\backslash cdot\backslash omega\; =\; \backslash frac\{\{\{H\_0\}\}\}\{\{\backslash sqrt\; \{1\; +\; \{\{\backslash left(\; \{\backslash frac\{\backslash omega\; \}\{\{\{\backslash omega\_c\}\}\}\}\; \backslash right)\}^2\}\}\; \}\}\backslash cdot\backslash omega\; \backslash simeq\; \backslash frac\{\{\{H\_0\}\}\}\{\{\backslash sqrt\; \{\{\{\backslash left(\; \{\backslash frac\{\backslash omega\; \}\{\{\{\backslash omega\_c\}\}\}\}\; \backslash right)\}^2\}\}\; \}\}\; \{\{\; \{\{\{\backslash left(\backslash omega-\; \{\backslash frac\{\backslash omega\_c^2\; \}\{\{\{2\backslash omega\}\}\}\}\; \backslash right)\}\}\}\; \}\}$

= {H_0}\cdot{\omega_c} \left(1- \frac{\omega_c^2 }{2\omega^2} \right) = \mathrm{const.}

Example for $\backslash omega\; =\; 5\backslash cdot\; \backslash omega\_c$

$\backslash mathit\{GBWP\}\backslash \; =\; \backslash frac\{\{\{H\_0\}\}\}\{\{\backslash sqrt\; \{\backslash frac\{\{\backslash omega\_c^2\; +\; 25\{\backslash omega\; \_c\}^2\}\}\{\{\backslash omega\_c^2\}\}\}\; \}\}\backslash cdot5\{\backslash omega\_c\}\; =\; \backslash frac\{5\}\{\{\backslash sqrt\; \{26\}\; \}\}\{H\_0\}\backslash cdot\{\backslash omega\_c\}\; =\; 0.98\backslash cdot\{H\_0\}\backslash cdot\{\backslash omega\_c\}$

Note that the error in this case is only about 2%, for the constant term, and using the second term, $\backslash left(1-\; \backslash frac\{\backslash omega\_c^2\; \}\{2\backslash omega^2\}\; \backslash right)$

, the error drops to .06%.

For transistors, the current-gain–bandwidth product is known as the {{math|f_T}} or transition frequency.

{{cite book

| title = Principles of transistor circuits: introduction to the design of amplifiers, receivers, and digital

| edition = 9th

| author = Stanley William Amos and Mike James

| publisher = Newnes

| year = 2000

| isbn = 978-0-7506-4427-3

| page = 169

| url = https://books.google.com/books?id=mfec2Zw_b7wC&pg=PA169

}}

{{cite book

| title = Fundamentals of semiconductor devices

| author = M K Achuthan and K N Bhat

| publisher = Tata McGraw-Hill Education

| year = 2007

| isbn = 978-0-07-061220-4

| page = 408

| url = https://books.google.com/books?id=REQkwBF4cVoC&pg=PA408

}}

It is calculated from the low-frequency (a few kilohertz) current gain under specified test conditions, and the cutoff frequency at which the current gain drops by 3 decibels (70% amplitude); the product of these two values can be thought of as the frequency at which the current gain would drop to 1, and the transistor current gain between the cutoff and transition frequency can be estimated by dividing {{math|f_T}} by the frequency. Usually, transistors must be used at frequencies well below {{math|f_T}} to be useful as amplifiers and oscillators.Martin Hartley Jones A practical introduction to electronic circuits, Cambridge University Press, 1995 {{ISBN|0-521-47879-0}} page 148 In a bipolar junction transistor, frequency response declines owing to the internal capacitance of the junctions. The transition frequency varies with collector current, reaching a maximum for some value and declining for greater or lesser collector current.

• [http://masteringelectronicsdesign.com/an-op-amp-gain-bandwidth-product/ "Op-amp gain-bandwidth-product"] masteringelectronicsdesign.com

{{DEFAULTSORT:Gain-bandwidth product}}

Category:Electronic amplifiers