# 15. OPAMP Gain Bandwidth Product $$GBW$$¶

## 15.1. Objective¶

The objective of this activity is to explore a key parameter that affects the performance of op-amps at high frequencies. The parameter is the gain-bandwidth product ($$GBW$$) or unity gain bandwidth.

## 15.2. Notes¶

In this tutorials we use the terminology taken from the user manual when referring to the connections to the Red Pitaya STEMlab board hardware. Extension connector pins used for -3.3V and +3.3V voltage supply are show in the documentation here. Bode analyzer application is used to measure frequency response of op-amp circuit.

## 15.3. Materials¶

• Red Pitaya STEMlab
• OPAMP: 1x OP97
• Resistor: 2x 100 $$\Omega$$
• Resistor: 1x 47 $$k \Omega$$
• Resistor: 1x 100 $$k \Omega$$
• Resistor: 2x 10 $$k \Omega$$

## 15.4. Background¶

The forward gain, $$G$$ is defined as the gain of the op-amp when a signal is fed differentially into the amplifier with no negative feedback applied. This gain is ideally infinite at all frequencies, but in a real op-amp is finite, and depends on the frequency. At low frequency the gain is maximum, decreases linearly with increasing frequency, and has a value of one at the frequency commonly referred to as the unity gain or cut-off frequency $$f_{c}$$ (in equation form, $$G(f_c)=1$$). For the OP97 op-amp, the unity gain frequency is 900 kHz, the open-loop gain at this frequency is simply one. This is also the Closed-Loop Bandwidth or the maximum frequency when the feedback is configured with a closed loop gain of 1. $$G_f$$ is defined as the gain-bandwidth product, $$GBW$$, and for all input frequencies this product is constant and equal to $$f_c$$. The gain can be specified as a simple number (magnitude) or in dB(decibel).

\begin{align}\begin{aligned}GBW = gain * f_c\\or\\GBW = gain * BW\end{aligned}\end{align}

where $$f_c$$ is cut-off frequency (at $$f_c$$ gain drops by -3dB ( -3dB = $$1 / \sqrt{2}$$ drob in signal amplitude)) and $$BW$$ frequency bandwidth in this case given as $$BW = f_c$$

Figure 1, from the OP97 datasheet, graphically illustrates this relationship. When feedback is provided, as in an inverting amplifier, the gain is given by G = –R2/R1; however, it must be recognized that the magnitude of this gain can never exceed the gain as given by the gain-bandwidth product.

Figure 1: OP97 Open-Loop Gain, Phase vs. Frequency

Note

You should remember following: The maximum frequency of “normal operation” (i.e frequency bandwidth, i.e cut-off frequency, i.e frequency where gain drops by 3dB) of your op-amp circuit (amplifier) in non-inverting or inverting configuration will be ALWAYS dependent on the GAIN. If you chose higher gain the frequency bandwidth will be lower and vice versa.

For example: If we want to have inverting amplifier with gain = 100 based on OP97 then our frequency bandwidth will be given as:

$BW = GBW_{OP927}/100 = 900 kHz / 100 = 9 kHz$

for gain = 1000

$BW = GBW_{OP927}/1000 = 900 kHz / 1000 = 900 Hz$

## 15.5. Procedure¶

Build the circuit shown in figure 2 on your solderless breadboard to measure the frequency response of a inverting amplifier configured with a closed loop gain of 1000. Because the circuit’s gain is so high, the circuit needs to be driven with a very small input signal. In order to generate small signal with the STEMlab generator, a 1/1000 voltage divider, [R3 - (R4||R1)], is used to reduce the 2V p2p sine signal to 2 mV p2p at the inverting amplifier input. R4 and R1 are effectively in parallel due to the “virtual ground” at pin 2. The parallel combination of R4 and R1 will be 50 Ω which with the 47 KΩ R3 results in a divider ratio close to 1/1000.

Figure 2: Inverting amplifier with gain of 1000

Warning

Before connecting the circuit to the STEMlab -3.3V and +3.3V pins double check your circuit. The -3.3V and +3.3V voltage supply pins do not have short circuit handling and they can be damaged in case of short circuit.

1. Set up the filter circuit as shown in figure 2 on your solderless breadboard, with the component values R1 = R4 = 100 Ω, R2 = 100 kΩ and R3 = 47 kΩ.
2. Start the Bode analyzer application. The Bode analyzer application will make a frequency sweep in such way it will generate sine signal on $$OUT1$$ within frequency range selected by us(in settings menu). $$IN1$$ input signal is directly connected to $$OUT_1$$ following that $$IN1=V_{in}$$. $$IN2$$ is connected on the other side of the filter and from that $$IN2=V_{out}$$. Bode analyzer application will then for each frequency step take the ratio of $$IN1/IN2$$ ($$V_{in}/V_{out}$$) and calculate frequency response.
3. In the Bode analyzer settings menu set for:
• start frequency: 100 Hz
• end frequency: 20 kHz
• number of steps: 50
• scale: Log
• select RUN button

Figure 3: Typical OP97 Bode Plot Gain = 1000

From the figure 3 we can see that theory and OP97 datasheet data are consistent with the measurement. At gain = 1000 the BW is 900Hz. Set R2 to 10 kΩ, R3 = 4.7 kΩ, repeat the measurements and observe the results.

Figure 3: Typical OP97 Bode Plot Gain = 100

As we can see from the figure 4 for x10 less gain the BW is increased x10 confirming the equation $$GBW = BW * Gain = const$$.