**Nikola Šerman,** prof. emeritus *Linear Theory*

nserman@fsb.hr *Transfer Function Concept Digest*

**4. TRANSFER FUNCTION**

According to (3.5), function *f(t)* *n*-th derivative L-transform generally depends on the function itself as well as on its initial conditions.

When initial conditions are equal to zero, i.e. when it holds:

Function *f(t)* *n*-th derivative L-transform becomes an algebraic product of *F(s)* and the *n*-th power of * s*:

In the time domain relationship between output and input is defined by a differential equation (2.1) . However, when moved to the frequency domain by (4.2) the same relationship takes a simple algebraic fraction form. This form is called *transfer function*.

It can be expressed as:

*either*

*:*

**s****or**

*poles*

*s*(roots of the denominator) and zeros

_{1}….s_{m}*s*(roots of the nominator) of the transfer function

_{1}….s_{n}**:**

*H(s)*Due to the transfer function definition (4.3) or (4.4), the output ** Y(s)** L-transform can be obtained by a simple multiplication of the transfer function

*with the input*

**H(s)***L-transform:*

**X(s)**This simple frequency domain algebraic relationship between input, output and the transfer function makes the transfer function concept such a powerful tool in dealing with complex linear dynamic systems.