Perform a full nodal analysis of the 3-Op-Amp Instrumentation Amplifier.

Circuit Description

• Stage 1: Two identical non-inverting amplifiers with a shared gain resistor $R_g$ between their outputs
• Stage 2: Precision difference amplifier with matched resistors

Analysis of First Stage

For the left op-amp (Op-Amp 1):

Let $V_{o1}$ be the output of the left buffer.

Since $V_+ = V_1$ (high impedance input), and virtual short gives $V_- = V_1$:

Current through $R_g$: $$I_g = \frac{V_{o1} - V_{o2}}{R_g}$$

This current also flows through $R_1$ (left side): $$I_g = \frac{V_{o1} - V_1}{R_1}$$

For the right op-amp (Op-Amp 2):

Similarly: $$I_g = \frac{V_2 - V_{o2}}{R_1}$$

Solving for $V_{o1}$ and $V_{o2}$:

From the current equations: $$\frac{V_{o1} - V_1}{R_1} = \frac{V_{o1} - V_{o2}}{R_g}$$ $$\frac{V_2 - V_{o2}}{R_1} = \frac{V_{o1} - V_{o2}}{R_g}$$

Subtracting these equations: $$\frac{V_{o1} - V_1 - V_2 + V_{o2}}{R_1} = 0$$ $$V_{o1} - V_{o2} = V_2 - V_1 + I_g R_g$$

Since $I_g R_g = (V_{o1} - V_{o2})$, and combining: $$V_{o1} - V_{o2} = (V_2 - V_1)\left(1 + \frac{2R_1}{R_g}\right)$$

Analysis of Second Stage

Standard difference amplifier with matched resistors $R_3 = R_4$: $$V_{out} = \frac{R_4}{R_3}(V_{o1} - V_{o2})$$

For matched resistors (gain = 1 for difference stage): $$V_{out} = V_{o1} - V_{o2}$$

Final Result

Combining both stages: $$V_{out} = (V_2 - V_1)\left(1 + \frac{2R_1}{R_g}\right)$$

Advantages Verified: 1. Input impedance at $V_1$ and $V_2$ → ∞ (op-amp inputs) 2. Single resistor $R_g$ sets entire gain 3. Differential measurement with high CMRR

Analog PID Controller Design

PID Control Equation

$$u(t) = K_p e(t) + K_i \int e(t)\,dt + K_d \frac{de(t)}{dt}$$

Where:

• $e(t)$: Error signal (setpoint - measured)
• $K_p$: Proportional gain
• $K_i$: Integral gain
• $K_d$: Derivative gain

Circuit Implementation

Stage 1: Error Signal Generation

• Use difference amplifier to compute $e(t) = V_{setpoint} - V_{sensor}$

Stage 2: P, I, D Computation 1. Proportional: Simple inverting amplifier with gain $K_p$ 2. Integral: Miller integrator with $K_i = 1/(R_i C_i)$ 3. Derivative: Practical differentiator with $K_d = R_d C_d$

Stage 3: Summing Stage

• 3-input summing amplifier combines P, I, D outputs
• Each input can have independent gain weighting

Design Specifications:

• Input range: 0-10V
• Output range: 0-10V
• Adjustable gains via potentiometers
• Include anti-windup for integrator

Testing Procedure: 1. Apply step input change 2. Observe settling time and overshoot 3. Tune $K_p$, $K_i$, $K_d$ using Ziegler-Nichols method

Applications:

• Temperature control
• Motor speed control
• Liquid level control
• Pressure regulation