Determine F12 And F21 For The Following Configurations

Determining F12 and F21 for Various Op-Amp Configurations

The concept of feedback is fundamental to the operation and performance of operational amplifiers (op-amps). Understanding how to determine the feedback factors, specifically F12 and F21, is crucial for analyzing and designing op-amp circuits. These factors play a significant role in calculating gain, stability, input and output impedance, and overall performance. This comprehensive guide will delve into the meaning of F12 and F21, explain their importance, and illustrate how to determine them for various common op-amp configurations.

Understanding Feedback Factors: F12 and F21

In the context of op-amp circuits, feedback refers to the process of taking a portion of the output signal and feeding it back to the input. This feedback can be positive (in-phase) or negative (out-of-phase). Negative feedback is generally used in linear op-amp applications because it provides stability, reduces distortion, and allows for precise control over the circuit's gain.

F12 and F21 are components of a general feedback network represented as a two-port network. They are defined as follows:

F12: The reverse transfer function. It represents the fraction of the output signal that is fed back to the input. In simpler terms, it quantifies how much of the output voltage influences the input voltage. F12 is often expressed as the ratio of the voltage fed back to the input (Vf) to the output voltage (Vo): F12 = Vf / Vo.
F21: The forward transfer function. Although less directly involved in feedback per se, understanding F21 helps in complete network analysis. It relates the input current to the output current within the feedback network. It's a crucial parameter when dealing with current-feedback amplifiers or when the impedance of the feedback network plays a significant role. F21 is expressed as the ratio of the output current (Io) to the input current (Ii): F21 = Io / Ii. This is generally less important for voltage feedback op-amp configurations.

Why are F12 and F21 Important?

Understanding F12 and F21 provides several key benefits:

Gain Calculation: The closed-loop gain of an op-amp circuit is heavily dependent on the feedback factor F12. Knowing F12 allows you to accurately predict the circuit's gain for different input signals.
Stability Analysis: The amount of feedback significantly impacts the stability of an op-amp circuit. Excessive positive feedback can lead to oscillations. F12 helps assess the stability margin by influencing the loop gain.
Input and Output Impedance Control: Negative feedback alters the input and output impedances of the op-amp circuit. F12 is essential in determining these impedance values.
Circuit Design and Optimization: By understanding the relationship between F12, F21, and circuit performance, designers can optimize op-amp circuits to meet specific requirements.

General Method for Determining F12

The most common method to determine F12 involves the following steps:

Identify the Feedback Network: Isolate the part of the circuit that constitutes the feedback network. This typically consists of resistors and other components that connect the output to the input.
Apply a Test Voltage: Apply a test voltage (Vo) at the output of the op-amp.
Calculate the Feedback Voltage (Vf): Determine the voltage (Vf) that appears at the input of the op-amp due to the applied test voltage (Vo). This usually involves using voltage divider rules or circuit analysis techniques.
Calculate F12: Calculate F12 using the formula: F12 = Vf / Vo.

General Method for Determining F21

Determining F21 follows a similar principle, focusing on current relationships:

Identify the Feedback Network: As with F12, identify the feedback network.
Apply a Test Current: Inject a test current (Ii) into the input of the feedback network.
Calculate the Output Current (Io): Determine the current (Io) that flows out of the output of the feedback network due to the applied test current (Ii).
Calculate F21: Calculate F21 using the formula: F21 = Io / Ii.

Determining F12 and F21 for Common Op-Amp Configurations

Let's apply these methods to determine F12 and F21 for several common op-amp configurations:

1. Inverting Amplifier

Circuit Description: The inverting amplifier configuration consists of an op-amp, an input resistor (R1), and a feedback resistor (Rf). The input signal is applied to the inverting input through R1, and the feedback network consists of Rf connected between the output and the inverting input. The non-inverting input is grounded.

Diagram of an Inverting Amplifier with R1 and Rf labelled
Determining F12:
1. Feedback Network: Rf is the primary component of the feedback network.
2. Apply Test Voltage: Apply a test voltage Vo at the output.
3. Calculate Vf: Because the inverting input of the op-amp is a virtual ground, the voltage at that node is approximately 0V. This virtual ground is created due to the high open-loop gain of the op-amp and the negative feedback. Therefore, the feedback voltage Vf is actually the voltage at the inverting input. However, to analyze it through Rf, we consider that the Vf will be equal to the voltage drop across R1 when the output voltage is applied. To determine the feedback voltage, we perform voltage division on the input circuit. Since the non-inverting (+) terminal of the Op-Amp is connected to ground, the inverting (-) terminal will tend towards 0V (virtual ground). This means the voltage Vf at the inverting terminal will be approximately 0V. The formula for voltage division, Vf = Vo * (R1 / (R1 + Rf)), might seem applicable, but the "virtual ground" approximation simplifies this analysis greatly. The current through Rf is (Vo - Vf) / Rf and the current through R1 is (Vf - Vin) / R1. Since Vf is approximately 0V, we can say the voltage drop across R1 is approximately equal to the input voltage Vin multiplied by -1, and is related to Vo as follows:
  
  Vf ≈ 0V This is the key to understanding the inverting amplifier's F12. Therefore, the change in Vf is virtually zero for any change in Vo. However, we must relate changes in Vf with changes in Vo, and that relationship comes through Rf. Think of it this way: an infinitesimally small change in Vo is reflected in an infinitesimally small change in the current flowing through Rf, which then causes an infinitesimally small adjustment to the virtual ground.
4. Calculate F12:
  
  F12 = Vf / Vo ≈ 0 / Vo ≈ 0
  
  A more nuanced analysis is needed here. While the virtual ground approximation is powerful, it hides some subtleties. The inverting amplifier's negative feedback forces the inverting input to be close to 0V. The difference between the inverting input and true ground is what is modulated by the output voltage. This means that the actual amount of voltage feedback is related to the open-loop gain (Aol) of the op-amp. If the output voltage changes by ΔVo, the voltage at the inverting input will change by ΔVf = ΔVo / Aol. Therefore, a more accurate (though less practically useful) F12 is:
  
  F12 = Vf/Vo = 1/Aol
  
  Because Aol is typically very large (10^5 or more), F12 is very small, approaching zero. The practical takeaway is that the feedback is very strong in maintaining the virtual ground, making the ideal F12 approximately zero for hand calculations and simulations except when considering stability margins.
Determining F21:
1. Feedback Network: As before, Rf is the feedback component.
2. Apply Test Current: Inject a test current Ii into the inverting input.
3. Calculate Io: The current Io flowing out of the output is determined by the feedback resistor Rf and the applied input current Ii. Since the inverting input is at virtual ground, almost all of Ii must flow through Rf. The voltage drop across Rf is Ii Rf. The current that must flow from Vo through Rf to maintain virtual ground is Ii. Therefore, Io can be considered the negative of Ii (ideally).
  
  Io ≈ -Ii
4. Calculate F21:
  
  F21 = Io / Ii ≈ -Ii / Ii ≈ -1
  
  The inverting amplifier's F21 is ideally -1. This signifies that, in terms of current within the feedback network, the input current is effectively inverted and appears at the output. However, in reality the high gain nature of op-amp means that the changes to Ii are negligible compared to the output capability

2. Non-Inverting Amplifier

Circuit Description: The non-inverting amplifier configuration also uses an op-amp, an input resistor (R1), and a feedback resistor (Rf). However, the input signal is applied directly to the non-inverting input. The feedback network consists of Rf connected between the output and the inverting input, and R1 connected between the inverting input and ground.

Diagram of a Non-Inverting Amplifier with R1 and Rf labelled
Determining F12:
1. Feedback Network: R1 and Rf form the feedback network.
2. Apply Test Voltage: Apply a test voltage Vo at the output.
3. Calculate Vf: The feedback voltage Vf is the voltage at the inverting input. This is determined by the voltage divider formed by R1 and Rf:
  
  Vf = Vo * (R1 / (R1 + Rf))
4. Calculate F12:
  
  F12 = Vf / Vo = R1 / (R1 + Rf)
Determining F21:
1. Feedback Network: R1 and Rf still form the relevant network.
2. Apply Test Current: Inject a test current Ii into the inverting input node. This current will split between R1 and Rf.
3. Calculate Io: The current through Rf will generate a corresponding change in output voltage. Calculating the exact relationship between Ii and Io is more complex than for the inverting amplifier, as the input impedance looking into the non-inverting input of the op-amp is very high, and Io will primarily be driven by the op-amp's internal circuitry. As a result, a direct, simple relationship between Ii and Io is difficult to establish without considering the op-amp's internal parameters. The change in Vo will depend not only on Ii, but also on the op-amp's ability to supply current. The change in Vo can be expressed as Io * Zout, where Zout is the output impedance of the op-amp.
4. Calculate F21: Without specifying detailed op-amp parameters, a precise F21 is challenging to calculate. However, the magnitude will be very small.
  
  F21 = Io / Ii ≈ Difficult to Express Simply
  
  The takeaway: Calculating the F21 for a non-inverting amplifier is not straightforward and generally requires a more complex analysis considering the op-amp's internal characteristics.

3. Voltage Follower (Buffer)

Circuit Description: The voltage follower, also known as a buffer, is a special case of the non-inverting amplifier where Rf = 0 and R1 = ∞ (open circuit). The output is directly connected to the inverting input.

Diagram of a Voltage Follower/Buffer Op-Amp circuit
Determining F12:
1. Feedback Network: The direct connection between the output and the inverting input is the feedback network.
2. Apply Test Voltage: Apply a test voltage Vo at the output.
3. Calculate Vf: Since the output is directly connected to the inverting input, the feedback voltage Vf is equal to the output voltage Vo.
  
  Vf = Vo
4. Calculate F12:
  
  F12 = Vf / Vo = Vo / Vo = 1
  
  The voltage follower has a feedback factor of 1. This means that 100% of the output voltage is fed back to the input.
Determining F21:
1. Feedback Network: Direct Connection
2. Apply Test Current: Inject a test current Ii into the inverting input.
3. Calculate Io: Similar to the non-inverting amplifier, relating the injected Ii to Io requires considering the op-amp's output characteristics. The injected current will influence the small voltage difference between the non-inverting and inverting inputs, prompting the op-amp to adjust the output current to maintain Vf ≈ Vin. This is very complex to express simply.
4. Calculate F21:
  
  F21 = Io / Ii ≈ Difficult to Express Simply
  
  Like the non-inverting amplifier, calculating the F21 precisely requires in-depth knowledge of the op-amp's internal parameters.

4. Summing Amplifier

Circuit Description: The summing amplifier takes multiple input voltages and produces an output voltage that is a weighted sum of the inputs. It typically uses multiple input resistors (R1, R2, R3, etc.) connected to the inverting input, and a feedback resistor (Rf).

Diagram of a Summing Amplifier with multiple input resistors
Determining F12:
1. Feedback Network: Rf connected between the output and the inverting input acts as the feedback network.
2. Apply Test Voltage: Apply a test voltage Vo at the output.
3. Calculate Vf: Since the inverting input is at virtual ground (approximately 0V), Vf ≈ 0V. As with the inverting amplifier, while the absolute voltage is close to zero, the change in voltage is modulated by Vo.
4. Calculate F12:
  
  F12 = Vf / Vo ≈ 0
  
  Similar to the inverting amplifier, the F12 is practically zero due to the virtual ground, but more accurately 1/Aol.
Determining F21:
1. Feedback Network: Rf
2. Apply Test Current: Inject a test current Ii into the inverting input node. This Ii represents the sum of currents flowing from the input resistors (R1, R2, etc.) plus the injected current.
3. Calculate Io: To maintain the virtual ground, almost all of this combined input current must flow through Rf. The current Io is, therefore, approximately the negative of the total current entering the inverting input, dominated by the Ii we injected.
4. Calculate F21:
  
  F21 = Io / Ii ≈ -1
  
  Like the inverting amplifier, the F21 is ideally -1, reflecting the inversion of current within the feedback loop to maintain the virtual ground.

5. Differential Amplifier

Circuit Description: The differential amplifier amplifies the difference between two input voltages. It typically uses two input resistors (R1 and R2) and a feedback network consisting of two resistors (Rf1 and Rf2).

Diagram of a Differential Amplifier circuit
Determining F12:
1. Feedback Network: The feedback network consists of Rf1 and Rf2. The key is how these components are arranged to feed back a portion of the output voltage to the inverting input.
2. Apply Test Voltage: Apply a test voltage Vo at the output.
3. Calculate Vf: This is more complex than previous examples because the voltage at the inverting input is dependent on both the output voltage and the voltage at the non-inverting input (V2). Assuming the resistor networks are balanced (R1 = R2 and Rf1 = Rf2), the voltage at the inverting input due to the output can be found using voltage division. However, we need to account for the effect of V2.
  
  Let's say that R1 = R2 = R and Rf1 = Rf2 = Rf. Due to the balanced nature of the network, a change in Vo will cause a proportional change in the voltage at the inverting input relative to the voltage at the non-inverting input.
  
  Therefore, if the circuit is designed such that Rf/R = desired gain, F12 is approximately 0 because the op-amp corrects very quickly to minimize the differential voltage. In a perfect op-amp Vf is identical to V2, hence no change in Vf occurs as Vo is modulated, as changes to Vo are quickly corrected by the Op-Amp's high gain and negative feedback.
4. Calculate F12:
  
  F12 = Vf / Vo ≈ 0
  
  Because the op-amp's gain is so high, F12 is approximately zero. However, a more accurate assessment would again include the open loop gain in the denominator, and this analysis assumes a perfectly balanced resistor bridge.
Determining F21:
1. Feedback Network: Rf1 and Rf2.
2. Apply Test Current: This is again complicated. Injecting a test current Ii into the inverting input will have effects on both inputs of the op-amp due to the resistor network and the high-gain differential amplification.
3. Calculate Io: Determining the direct relationship between Ii and Io requires a detailed analysis of the entire circuit, including the influence of V2 and the op-amp's internal characteristics.
4. Calculate F21:
  
  F21 = Io / Ii ≈ Difficult to Express Simply
  
  A simple, general expression for F21 is difficult to obtain without significant circuit simplification and assumptions or detailed knowledge of the op-amp’s characteristics.

Key Considerations and Limitations

Ideal Op-Amp Assumptions: The above analyses often rely on ideal op-amp assumptions, such as infinite open-loop gain, infinite input impedance, and zero output impedance. In real-world scenarios, these parameters are finite and can affect the accuracy of the calculated F12 and F21 values.
Op-Amp Specifications: For precise calculations, especially when dealing with F21, it's crucial to consider the op-amp's datasheet specifications, including open-loop gain, input bias current, input offset voltage, and output impedance.
Frequency Effects: At higher frequencies, the op-amp's open-loop gain decreases, and the impedance of capacitors and inductors in the circuit becomes significant. This can alter the feedback network and affect the values of F12 and F21.
Stability Analysis: Understanding F12 is critical for performing stability analysis using techniques like Bode plots and Nyquist plots. The gain margin and phase margin, which are derived from these plots, indicate the circuit's stability. F12 is indirectly incorporated into these analyses.
Simulation: Circuit simulation software (e.g., SPICE) can be used to accurately determine F12 and F21 for complex op-amp circuits. Simulation allows you to account for non-ideal op-amp characteristics and frequency effects.

Practical Applications

The understanding of F12 and F21 is crucial in a variety of applications:

Precision Instrumentation: In precision instrumentation amplifiers, a precise knowledge of F12 is essential for achieving accurate gain and minimizing errors.
Filter Design: In active filter design, the feedback network determines the filter's frequency response. F12 helps to accurately predict the filter's cutoff frequency, bandwidth, and Q-factor.
Oscillator Design: Positive feedback is used to create oscillators. Understanding the feedback conditions and F12 helps ensure stable oscillation at the desired frequency.
Control Systems: Op-amps are used in control systems to regulate various parameters, such as temperature, pressure, and flow rate. F12 is important for designing stable and responsive control loops.
Audio Amplifiers: In audio amplifiers, negative feedback reduces distortion and improves linearity. F12 helps to optimize the feedback network for the best audio performance.

Conclusion

Determining F12 and F21 is fundamental to understanding and designing op-amp circuits. While simplified analyses often rely on ideal op-amp assumptions, a deeper understanding of these feedback factors, and the limitations of these assumptions, allows for more accurate predictions of circuit behavior and performance. By considering op-amp specifications, frequency effects, and stability analysis techniques, engineers can design robust and reliable op-amp circuits for a wide range of applications. Although precise calculation of F21 can be complex, particularly for non-inverting configurations, the focus on F12 provides a powerful tool for analyzing and optimizing op-amp circuits with negative feedback. Utilizing circuit simulation software and careful consideration of component values will aid in practical implementations and ensure optimal circuit performance. The interplay between theoretical understanding and practical application is key to mastering op-amp circuit design.