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

Contents

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Key learnings:

DC Gain Definition: DC gain is the ratio of the steady-state output to the steady-state input of a control system when given a step input.
Transfer Function: A transfer function represents the relationship between a control system’s input and output using the Laplace transform.
Final Value Theorem: The final value theorem helps find the DC gain by evaluating the transfer function at zero for continuous systems.
Continuous vs. Discrete Systems: DC gain calculations differ between continuous (using G(s)) and discrete systems (using G(z)), but the principles remain similar.
Practical Examples: Examples of first-order systems illustrate how to apply these concepts to find the DC gain in real scenarios.

What is a Transfer Function?

A transfer function shows how the input signal of a control system affects the output signal. A block diagram helps visualize this by using blocks for the transfer function and arrows for the input and output signals.

A transfer function is a useful way to represent a linear time-invariant system. It is mathematically expressed using complex variables.

In a control system, an input signal, called excitation or cause, goes through a transfer function to produce an output 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 transfer function of a control system is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions.

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

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

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

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.

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