Calculate Loop Gain Using the Voltage Injection Method

Key Takeaways

• Loop gains are vital in determining the stability and transient response of control systems. A loop gain is the product of all gains present in a loop. Calculating loop gain in a negative feedback system is helpful in assessing the overall stability of the closed-loop control process.

• The closed-loop transfer function of the system can be given by , where G(s) is the open-loop gain and H(s) is the gain of the feedback loop.

• For measured loop gain to be approximately equal to real loop gain, two conditions must be satisfied: Condition 1:Z₁(s)>>Z₂(s) and Condition      

Steering wheels are closed-loop control systems

Most days, you most likely come across some sort of control system without even noticing it. Your house thermostat, the steering wheel of your car, or an industrial robot are just a few examples of control systems that impact our daily lives. 

The ultimate aim of a control system is to control and regulate the output to its desired value or state. A control system is said to be excellent when it takes immediate action to settle down the system to the desired output without any lag or transients and is able to keep the system stable throughout operation.

Calculating loop gains is a way to determine the stability and transient response of a control system. Loop gain is the product of all gains present in a loop, and calculating loop gain in a negative feedback system can help you assess the overall stability of the closed-loop control process. 

Closed-Loop Control Systems

Among open-loop and closed-loop control systems, closed-loop systems with negative feedback are most commonly used. In negative feedback closed-loop systems, the output is fed back to check its deviation from the reference or desired value. The error—the difference between the desired output and actual output—controls the action of the control system, depending on whether the error is positive or negative. 

The Voltage Injection Method to Calculate Loop Gain

Consider a negative feedback control system. The closed-loop transfer function of the system can be given by the following equation, where G(s) is the open-loop gain and H(s) is the gain of the feedback loop. The loop gain is given by the product term G(s)H(s). 

We can determine the stability of a closed-loop system from its transfer function by calculating the gain margin and phase margin, or from Nyquist or Bode plots. 

In practical closed-loop systems that use negative feedback loops to maintain the output voltage at the desired value—such as DC choppers and voltage regulators—it is ideal to calculate the loop gain of the feedback system experimentally, rather than rely on analytical models and plots. 

The loop gain calculated from models neglects the effect of parasitics or disturbances that occur in the real-time system. It is a good engineering practice to experimentally measure the loop gain of the closed-loop to validate the design and its stability. The voltage injection method is one such approach, where the loop gain is calculated by applying a test voltage. 

The Voltage Injection Method 

The voltage injection method utilizes a test voltage to measure the loop gain of a system. By using a voltage regulator, this method establishes the determination of the phase and magnitude of the loop gain with only voltage measurements. 

The test voltage is injected into the loop, keeping the loop closed. This approach holds the operating points undisturbed, and the loop gain determined is more accurate, as the loop remains closed throughout the measurement. By using an injection transformer, the test voltage is given to an appropriate injection point in the feedback loop. The non-linearities and saturation in the measurement are avoided by keeping the test voltage value low. The injection transformer secondary is connected across a resistor, R, to apply the test voltage.  A value of 10 Ω is inserted as R in DC-DC converters and regulated power supplies. This set up keeps the DC bias operating point unchanged, even when test voltage is applied to the loop. The loop gain is calculated with the following equation, where V1 and V2 are the voltages.

In the voltage injection method, using the appropriate injection point is very important and is dependent on two impedances, Z1(s) and Z2(s), where Z1(s) is the impedance looking forward around the feedback loop and Z2(s) is the impedance looking backward from the injection point. For the measured loop gain to be approximately equal to real loop gain, two conditions must be satisfied:

Similar to the voltage injection method, there is a current injection method where a test current is injected at an appropriate injection point, instead of test voltage. The appropriate injection point in the current injection method should satisfy the opposite of the condition given in equation 3 (above) and can be expressed with the following: 

The stability of voltage regulators in electronic circuits and DC-DC converters in PhotoVoltaic, fuel cell, or battery-powered systems reflect on the output performance of the complete system. The traditional methods of analytical models and plots become obsolete, as they neglect parasitic effects and disturbances. 

The voltage injection method and current injection method are excellent, experimental approaches to calculate the loop gain of voltage regulators and DC-DC converters. With these methods, you can determine the loop gain with very little error—ensuring the stability of your control system.