How to Find the DC Gain of a Transfer Function (Examples Included)

Electrical4u

Field: Basic Electrical

China

What Is A Transfer Function

What is a Transfer Function?

A transfer function describes the relationship between the output signal of a control system and the input signal. A block diagram is a visualization of the control system that uses blocks to represent the transfer function and arrows representing the different input and output signals.

The transfer function is a convenient representation of a linear time-invariant dynamical system. Mathematically the transfer function is a function of complex variables

For any control system, there is a reference input known as excitation or cause that operates through a transfer function to produce an effect resulting in a controlled output or response.

Thus, the cause and effect relationship between output and input is linked to each other through a transfer function. In a Laplace Transform, if the input is represented by $R(s)$ and the output is represented by $C(s)$ .

The control system transfer function is defined as the Laplace transform ratio of the output variable to the Laplace transform of the input variable, assuming that all initial conditions are zero.

$\begin{align*} G(s)=\frac{C(s)}{R(s)}\end{align*}$

What is DC Gain?

The transfer function has many useful physical interpretations. The steady-state gain of a system is simply the ratio of the output and the input in steady-state represented by a real number between negative infinity and positive infinity.

When a stable control system is stimulated with a step input, the response at a steady-state reaches a constant level.

The term DC gain is described as the ratio of amplitude between the response of the steady-state and the step input.

DC gain is the ratio of the magnitude of the response to the steady-state step to the magnitude of the step input. The final value theorem demonstrates that DC gain is the value of the transfer function assessed at 0 for stable transfer functions.

Time Response of First Order Systems

The order of a dynamic system is the order of the highest derivative of its governing differential equation. First-order systems are the simplest dynamic systems to analyze.

To understand the concept of steady-state gain or DC gain, consider a general first-order transfer function.

$\begin{align*}G(s)=\frac{G(s)}{R(s)} = \frac{b_{0}}{s+ a_{0}}\end{align*}$

$G(s)$ can also be written as

$\begin{align*}\frac{K}{\tau s+1} = \frac{b_{0}}{s+a_{0}}\end{align*}$

Here,

$\begin{align*} a {0}=\frac{1}{\tau} \; \; \; \; b {0}=\frac{K}{\tau} \end{align*}$

$\tau$ is called the time constant. K is called the DC gain or steady-state gain

How to Find the DC Gain of a Transfer Function

DC gain is the ratio of the steady-state output of a system to its constant input, i.e., steady-state of the unit step response.

To find the DC gain of a transfer function, let us consider both continuous and discrete Linear Transform Inverse (LTI) systems.

Continuous LTI system is given as

(1) $\begin{equation*} G(s)=\frac{Y(s)}{U(s)}\end{equation*}$

Discrete LTI system is given as

$\begin{equation*} G(z)=\frac{Y(z)}{U(z)}\end{equation*}$

Use final value theorem to compute the steady-state of the unit step response.

(3) $\begin{equation*} L\left ( y_{step(t)} \right )=G(s)\frac{1}{s}\end{equation*}$

(4) $\begin{equation*}DC\; \; Gain = \lim_{t\rightarrow \infty }y_{step(t)}\end{equation*}$

(5) $\begin{equation*} DC\; \; Gain = \lim_{s\rightarrow 0 }s\left [ G(s)\frac{1}{s} \right ]\end{equation*}$

$G(s)$ is stable and all poles lie on the left hand side

Hence,

(6) $\begin{equation*}DC\; \; Gain = \lim_{s\rightarrow 0 }s\left [ G(s)\right ]\end{equation*}$

The formula of the final value theorem used for a continuous LTI system is

(7) $\begin{equation*}\frac{y(\infty)}{u(\infty)} = G(s)_{s=0}=G(0)\end{equation*}$

The formula of the final value theorem used for a discrete LTI system is

(8) $\begin{equation*}\frac{y(\infty)}{u(\infty)} = G(z)_{z=1}=G(1)\end{equation*}$

In both cases, if the system has an integration the result will be $\infty$ .

The DC gain is the ratio between the steady-state input and the steady-state derivative of the output can be obtained via differentiation of the obtained output. It is nearly same for both continuous and discrete system.

Differentiation in the Continuous Domain

In the continuous system or ‘s’ domain, the equation (1) is differentiated by multiplying the equation by ‘s’.

(9) $\begin{equation*}\frac{\dot{Y(s)}}{U(s)}= sG(s)\end{equation*}$

where $\dot{Y(s)}$ is the Laplace transform of $\dot{y(t)}$

Differentiation in the Discrete Domain

Th derivative in the discrete domain can be obtained by a first difference.

(10) $\begin{equation*}\dot{y(k)}=\frac{y_{k}-y_{k-1}}{T}\end{equation*}$

(11) $\begin{equation*}\dot{Y(z)}=\frac{Y(z)-z^{-1}Y(z)}{T}\end{equation*}$

(12) $\begin{equation*}\dot{Y(z)}=Y(z)\left [\frac{ ^{1-z^{-1}}}{T} \right ]\end{equation*}$

(13) $\begin{equation*}\dot{Y(z)}=Y(z)\left [\frac{z-1}{T_{z}} \right ]\end{equation*}$

Thus to differentiate in the discrete domain, we need to multiply $\frac{z-1}{T_{z}}$

Numerical Examples To Find DC Gain

Example 1

Consider the continuous transfer function,

$\begin{align*} H(s) =\frac{Y(s)}{U(s)} = \frac{12}{(s+2)(s+10)}\end{align*}$

To find the DC gain (steady-state gain) of the above transfer function, apply the final value theorem

$\begin{align*}\lim_{t\rightarrow \infty}y(t)= \lim_{s\rightarrow 0}s\times \frac{12}{(s+2)(s+10)}\end{align*}$

$\begin{align*}\lim_{t\rightarrow \infty}y(t)= \lim_{s\rightarrow 0}s\times \frac{12}{2\times 3}=2\end{align*}$

Now the DC gain is defined as the ratio of steady state value to the applied unit step input.

DC Gain = $\frac{2}{1}=2$

Hence it is important to note that the concept of DC Gain is applicable only to those systems which are stable in nature.

Example 2

Determine the DC gain for the equation

$\begin{align*}G(s)=\frac{K}{\tau s+1}\end{align*}$

The step response of the above transfer equation is

$\begin{align*}y_{step}(t)=L^{-1}\left [\frac{K}{(\tau s+1)s} \right ]\end{align*}$

$\begin{align*}y_{step}(t)=L^{-1}\left [ K\left ( \frac{1}{s}-\frac{\tau }{\tau s+1} \right ) \right ]\end{align*}$

Now, apply the final value theorem to find the DC gain.

$\begin{align*}y_{ss}=\lim_{t\rightarrow \infty }y_{step}(t)= \lim_{s\rightarrow 0}\frac{K}{(\tau s+1)s}s = K\end{align*}$

Statement: Respect the original, good articles worth sharing, if there is infringement please contact delete.

Give a tip and encourage the author!

Topics:

Power Systems

Control Systems

About experts

Electrical4u

China

Recommended

Causes and Preventive Measures of Fire and Explosion in Oil Circuit Breakers

Causes of Fire and Explosion in Oil Circuit Breakers When the oil level in an oil circuit breaker is too low, the oil layer covering the contacts becomes too thin. Under the effect of the electric arc, the oil decomposes and releases flammable gases. These gases accumulate in the space beneath the top cover, mixing with air to form an explosive mixture, which can ignite or explode under high temperature. If the oil level inside the tank is too high, the released gases have limited space to expan

Felix Spark

11/06/2025

THD Measurement Error Standards for Power Systems

Error Tolerance of Total Harmonic Distortion (THD): A Comprehensive Analysis Based on Application Scenarios, Equipment Accuracy, and Industry StandardsThe acceptable error range for Total Harmonic Distortion (THD) must be evaluated based on specific application contexts, measurement equipment accuracy, and applicable industry standards. Below is a detailed analysis of key performance indicators in power systems, industrial equipment, and general measurement applications.1. Harmonic Error Standar

Edwiin

11/03/2025

Busbar-Side Grounding for 24kV Eco-Friendly RMUs: Why & How

Solid insulation assistance combined with dry air insulation is a development direction for 24 kV ring main units. By balancing insulation performance and compactness, the use of solid auxiliary insulation allows passing insulation tests without significantly increasing phase-to-phase or phase-to-ground dimensions. Encapsulation of the pole can address the insulation of the vacuum interrupter and its connected conductors.For the 24 kV outgoing busbar, with the phase spacing maintained at 110 mm,

Dyson

11/03/2025

How Vacuum Tech Replaces SF6 in Modern Ring Main Units

Ring main units (RMUs) are used in secondary power distribution, directly connecting to end-users such as residential communities, construction sites, commercial buildings, highways, etc.In a residential substation, the RMU introduces 12 kV medium voltage, which is then stepped down to 380 V low voltage through transformers. The low-voltage switchgear distributes electrical energy to various user units. For a 1250 kVA distribution transformer in a residential community, the medium-voltage ring m

James

11/03/2025