Question

Design an op amp-based low pass filter with a cut off frequency of \(500 \mathrm{Hz}\) and a passband gain of 1 using a \(50 \mathrm{nF}\) capacitor. a) Draw your circuit, labeling the component values and output voltage. b) If the value of the feedback resistor in the filler is changed but the value of the resistor in the forward path is unchanged, what characteristic of the filter is changed?

Short answer

Passband gain changes with feedback resistor value change, not the cut-off frequency.

Step-by-step solution

Identify the type of filter

Since the problem mentions a low pass filter with a specified cut-off frequency, we need to use an RC low pass filter design. This type of filter will allow signals with a frequency lower than the cut-off frequency to pass with little attenuation and attenuate signals with frequencies higher than the cut-off frequency.

Calculate the required resistor value

For a first-order low pass filter, the cut-off frequency is given by the formula:\[ f_c = \frac{1}{2\pi RC} \]Rearranging for \(R\), we have:\[ R = \frac{1}{2\pi f_c C} \]Substitute \( f_c = 500 \text{ Hz} \) and \( C = 50 \times 10^{-9} \text{ F} \):\[ R = \frac{1}{2 \times \pi \times 500 \times 50 \times 10^{-9}} = 636.62 \text{ ohms} \]So, choose a standard resistor value close to this, such as \( 680 \text{ ohms} \).

Design the op-amp circuit

1. Label your op-amp with input at the inverting terminal which connects through the resistor R = 680 ohms.2. Connect a feedback loop from the output to the inverting terminal through a resistor. 3. Connect a resistor with the same value as the feedback resistor (to set gain = 1) from the non-inverting terminal to ground.4. Place the capacitor \(C = 50 \text{ nF}\) in parallel with the resistor connected to the input (i.e., between inverting input and ground).5. The output label should be \( V_{out} \).

Explore the effect of changing feedback resistor

In this op-amp configuration, if the feedback resistor changes while keeping the input resistor constant, the gain of the filter will change. Specifically, the passband gain (which is given by the ratio of the feedback resistor to the input resistor) will be modified, but the cut-off frequency remains the same.

Key concepts

RC Filter Design

A low pass filter, like the one in this exercise, utilizes a combination of resistors and capacitors to filter out high-frequency signals while allowing low-frequency signals to pass. The simplicity of an **RC Filter Design** stems from just two primary components: a resistor (R) and a capacitor (C). For a basic RC circuit:

• The resistor is connected in series with the input signal.
• The capacitor is connected in parallel with the output, creating a path for high-frequency components to bypass the output.

The key principle is that the capacitor's impedance decreases with increasing frequency, thus shorting high frequencies to the ground and allowing lower frequencies to pass through to the output. A simple RC filter design is often straightforward, but careful values selection is crucial. This ensures the filter performs as intended at the desired cut-off frequency.

Cut-off Frequency Calculation

For any **RC Low Pass Filter Design**, determining the cut-off frequency is vital. The cut-off frequency (\( f_c \)) is where the signal is reduced by 3 dB, or approximately 70.7% of its original amplitude. The formula for calculating this is:\[f_c = \frac{1}{2 \pi RC}\]This relationship highlights how both resistance and capacitance directly influence which frequencies will pass. For the given problem, substituting the components' values:

• Resistance (\( R \)) = 636.62 ohms
• Capacitance (\( C \)) = 50 nF

Thus:\[R = \frac{1}{2 \pi \times 500 \times 50 \times 10^{-9}} \approx 636.62 \text{ ohms}\]However, since resistors typically come in standard values, rounding to 680 ohms is practical. Adjustments like these occasionally affect performance slightly but ensure that components are easily available and cost-effective.

Op-Amp Feedback Resistor

In the design of an op-amp low pass filter, the feedback resistor plays a critical role. It determines the amplification factor, or gain, of the system. Specifically, the feedback resistor (\( R_f \)) in conjunction with another resistor in the input path, formulates the amplification equation:\[A_v = 1 + \frac{R_f}{R_{in}}\]Here, the value of the feedback resistor directly influences the gain. For the exercise at hand, with a desired passband gain of 1, the feedback resistor should be equated with the input resistor to achieve a unity gain configuration. This means that no amplification occurs beyond passing the signal through, keeping it at its original amplitude.If one alters the feedback resistor, while keeping the forward path resistor the same, the filter's gain changes. It becomes:

• Higher with a larger feedback resistor, indicating more signal amplification.
• Lower with a smaller feedback resistor, leading to signal attenuation.

This allows for flexible adjustment of the filter's characteristics without impacting the cut-off frequency.

Passband Gain

The **passband gain** refers to how much a signal is amplified once it passes through a filter, remaining relatively constant for frequencies lower than the cut-off frequency. In this exercise, we aim for a passband gain of 1, ensuring the output signal's amplitude is the same as the input, effectively signifying no gain.Generally, the passband gain is calculated using:\[A_{v} = 1 + \frac{R_f}{R_{in}}\]In our case, this translates to selecting feedback and input resistors of equal values, aligning with a gain value of 1. This design keeps the signal unaltered in amplitude throughout the passband.

• Alterations to the feedback resistor directly impact this gain, either amplifying or attenuating the signal.
• A consistent gain across the passband is crucial for signal accuracy and integrity, maintaining the signal as intended while filtering unnecessary noise.

Understanding the passband gain's role aids in achieving a balance between attenuation and amplification, ensuring a clean and clear final output.