The closed loop transfer function formula defines the exact mathematical relationship between a system's reference input and its controlled output when feedback is applied. For a standard negative feedback system, the direct answer is:
T(s) = G(s) / [1 + G(s)H(s)]
This single equation governs everything from the op-amp buffering a temperature sensor on your workbench to the flight controller stabilizing a drone. Below, we break down the variables, solve two real-world hardware problems with strict unit tracking, and provide a decision framework for selecting the physical components that make this math a reality.
The Core Closed Loop Transfer Function Formula & Symbol Map
To use the formula, you must first map the physical blocks of your circuit to the Laplace-domain variables. The standard negative feedback topology assumes the error signal is the difference between the reference and the feedback signal: E(s) = R(s) - H(s)C(s).
| Symbol | Name | Physical Meaning & Typical Units |
|---|---|---|
| T(s) | Closed-Loop Transfer Function | Ratio of Output to Input [C(s)/R(s)]. Units depend on the system (e.g., V/V, RPM/V). |
| G(s) | Forward Path Gain | The active plant and controller combined (e.g., Motor + Driver). Units: Output/Error. |
| H(s) | Feedback Path Gain | The sensor or voltage divider measuring the output. Units: Feedback/Error. |
| R(s) | Reference Input | The target setpoint (e.g., a 3.3V DAC output commanding a specific speed). |
| C(s) | Controlled Output | The actual physical output being regulated (e.g., Motor shaft speed in rad/s). |
| E(s) | Error Signal | The difference driving the forward path. E(s) = R(s) - H(s)C(s). |
| G(s)H(s) | Loop Gain (L(s)) | The gain around the entire loop. Must be dimensionless. |
Application Boundaries, Assumptions, and Fatal Unit Mistakes
The closed loop transfer function formula is not a universal law of physics; it is a mathematical consequence of specific system properties. According to foundational control theory documented by Control Tutorials for MATLAB and Simulink, the formula strictly applies only when the following assumptions hold:
- Linear Time-Invariant (LTI) System: The components must not change their behavior over time, and the output must scale linearly with the input. (A motor saturating at 12V violates this, which is why we linearize around an operating point).
- Zero Initial Conditions: The Laplace transform assumes the system starts from rest. If your motor is already spinning when the controller engages, the formula alone won't predict the transient response without adding initial condition terms.
- Unilateral Blocks: Signal flow must be strictly forward through G(s) and strictly backward through H(s). Loading effects (where the sensor draws enough current to drag down the plant's output voltage) must be absorbed into the block definitions.
The most common mistake that breaks this formula on the bench is mixing Hertz (Hz) and radians per second (rad/s) when evaluating frequency response by substituting s = jω. The Laplace variable 's' inherently operates in rad/s. If your sensor datasheet specifies a bandwidth of 1 kHz, you must convert it to 6283 rad/s before plugging it into your Bode plot calculations.
Additionally, the loop gain product G(s)H(s) must be strictly dimensionless. If G(s) is a motor constant in (rad/s)/V, your tachometer feedback H(s) must be in V/(rad/s). If your units do not cancel out to a pure number, your block diagram mapping is wrong.
Realistic Answer Magnitudes: In a well-designed analog control loop, the DC loop gain G(0)H(0) is typically > 1000 (60 dB). When G(s)H(s) >> 1, the '1' in the denominator becomes negligible, and the formula collapses to T(s) ≈ 1 / H(s). This is the ultimate goal of feedback: the system's output becomes entirely dependent on the passive, stable feedback network (H), rendering the messy, temperature-drifting active forward path (G) irrelevant.
Worked Problem 1: DC Motor Speed Control Loop
Let's apply the formula to a 12V brushed DC motor driven by an amplifier, with a tachometer providing negative feedback.
Given Parameters:
- Forward path (Motor + Driver): G(s) = Km / (τs + 1)
- Motor steady-state gain: Km = 40 (rad/s) / V
- Motor mechanical time constant: τ = 0.25 seconds
- Feedback path (Tachometer): H(s) = Kt = 0.05 V / (rad/s)
Step 1: Calculate the Loop Gain L(s) = G(s)H(s)
L(s) = [40 / (0.25s + 1)] * 0.05
Unit check: [(rad/s)/V] * [V/(rad/s)] = dimensionless. Correct.
L(s) = 2 / (0.25s + 1)
Step 2: Apply the Closed Loop Formula
T(s) = G(s) / [1 + G(s)H(s)]
T(s) = [40 / (0.25s + 1)] / [1 + (2 / (0.25s + 1))]
Step 3: Algebraic Simplification
Multiply the numerator and denominator by (0.25s + 1) to clear the fractions:
T(s) = 40 / [(0.25s + 1) + 2]
T(s) = 40 / [0.25s + 3]
Step 4: Normalize to Standard First-Order Form
To find the new closed-loop time constant, we must force the constant term in the denominator to equal 1. Divide top and bottom by 3:
T(s) = (40/3) / [(0.25/3)s + 1]
T(s) = 13.33 / [0.083s + 1]
Result Analysis: The open-loop motor spun at 40 (rad/s)/V with a sluggish 0.25s time constant. By closing the loop, we sacrificed raw gain (dropping to 13.33 (rad/s)/V) but slashed the time constant to 0.083s. The motor now reaches its target speed three times faster, which is the classic engineering trade-off of negative feedback.
Worked Problem 2: Op-Amp Non-Inverting Amplifier Hardware
Control theory isn't just for motors; it's the fundamental operating principle of every op-amp circuit. Let's look at a non-inverting amplifier using a precision Texas Instruments OPA211 to scale a sensor signal.
Given Parameters:
- Forward path (Op-Amp Open-Loop Gain): G(s) = AOL = 100,000 V/V (DC)
- Feedback path (Resistor Divider): H(s) = β = R1 / (R1 + R2)
- Resistors: R1 = 1 kΩ, R2 = 9 kΩ
Step 1: Calculate Feedback Factor β
β = 1k / (1k + 9k) = 0.1 V/V (dimensionless)
Step 2: Calculate Loop Gain
AOLβ = 100,000 * 0.1 = 10,000
Step 3: Apply the Formula
T(s) = AOL / [1 + AOLβ]
T(s) = 100,000 / [1 + 10,000]
T(s) = 100,000 / 10,001 = 9.999 V/V
Result Analysis: The ideal textbook gain is 1/β = 1/0.1 = 10 V/V. Our exact closed loop transfer function yields 9.999 V/V. The error is a mere 0.01%. Even if the OPA211's internal transistors age and AOL drops by 20% to 80,000, the new closed-loop gain becomes 80,000 / 8,001 = 9.998 V/V. The output remains locked to the 0.1% tolerance of your physical resistors, proving why we tolerate the gain reduction of feedback.
Rearranged Forms for Component Selection
On the bench, you rarely calculate T(s) from scratch. Usually, you know the target closed-loop behavior T(s) you need, and you must solve for the missing hardware block. Here are the algebraically rearranged forms for rapid component selection:
- Solving for Required Forward Gain G(s):
G(s) = T(s) / [1 - T(s)H(s)]
Use case: You know your desired system bandwidth and your sensor's attenuation, and need to spec the minimum gain-bandwidth product (GBW) of your error amplifier. - Solving for Required Feedback Factor H(s):
H(s) = [1 / T(s)] - [1 / G(s)]
Use case: You have a fixed-gain motor driver and need to calculate the exact voltage divider resistor ratio to achieve a specific RPM-per-Volt scaling. - Solving for Steady-State Error E(s):
E(s) = R(s) * [1 / (1 + G(s)H(s))]
Use case: Calculating how many millivolts of error will remain at the input of your PID controller when driving a constant 10A load.
Decision Tree: Picking the Right Feedback Architecture
Math dictates the topology, but physics dictates the silicon. Use this decision matrix to terminate your design process with a concrete part number based on your loop requirements.
| System Requirement | Loop Gain Constraint | Concrete Component Pick |
|---|---|---|
| Precision DC Sensor Scaling (< 10 Hz bandwidth, microvolt accuracy needed) |
Requires massive DC loop gain (> 120 dB) to drive steady-state error to zero without thermal drift. | TI OPA211 Bipolar input, 1.1 nV/√Hz noise, 130 dB open-loop gain. Costs ~$4.50/unit. |
| High-Speed PWM Motor Drive (> 20 kHz switching, high di/dt environments) |
Requires fast transient recovery and high common-mode rejection to survive PWM noise injection into H(s). | TI DRV8871 Integrated H-bridge with internal current sense (built-in H(s)). Handles up to 3.6A continuous. |
| Ultra-Low Cost Buffering (LED drivers, non-critical threshold switching) |
Loop gain can be low; phase margin is secondary to bill-of-materials cost. | LM358 Dual op-amp, ~$0.10/unit. Warning: Suffers from crossover distortion and slow slew rate (0.6 V/μs). |
If you are designing a custom analog control loop today and are paralyzed by component selection, default to the TI OPAx192 family (e.g., OPA192, OPA2192). In 2026, it remains the ultimate 'get-out-of-jail-free' card for control engineers. It provides rail-to-rail I/O, low offset voltage (25 μV max), and crucially, it is stable across the high capacitive loads typical in DIY sensor wiring. It eliminates the phase-margin headaches and unexpected oscillations that plague older, cheaper parts like the LM358 when closing high-gain loops.
