Electrical Systems

In electrical engineering, the analysis of networks composed of passive linear components — resistors (R), capacitors (C), and inductors (L) — is fundamental. These components are organized into circuits where the relationship between input and output can be analyzed using transfer functions. The transfer function of an RLC circuit, which relates the voltage across a capacitor to the input voltage, can be derived using Kirchhoff's laws.

To derive the transfer function, consider an RLC circuit with the components connected in series. Kirchhoff's Voltage Law (KVL) states that the sum of all voltages around a closed loop is zero. For an RLC circuit, this translates to the following integro-differential equation, assuming zero initial conditions:

Here, V(t) is the input voltage, V_R(t)=i(t)R is the voltage across the resistor, V_C(t) = is the voltage across the capacitor, where  q(t) is the charge on the capacitor, and V_L(t) = is the voltage across the inductor.

Current i(t) is related to charge q(t) by . Substituting these relationships into the KVL equation gives:

Writing the equation in voltage form, and taking the Laplace transform of both sides, and assuming all initial conditions are zero, the following equation is obtained:

Rearranging terms to solve for the transfer function:

This transfer function represents the voltage across the capacitor in response to the input voltage in the frequency domain. The impedance, analogous to resistance but applicable to capacitors and inductors, plays a key role in defining the transfer function. Additionally, Kirchhoff's Current Law (KCL) can also be used to derive transfer functions via nodal analysis. This law states that the sum of currents entering a node must equal the sum of currents leaving the node. In the RLC circuit, the total current is the sum of the current through the capacitor and the current through the series combination of the resistor and inductor. By applying KCL and simplifying, one can achieve the same transfer function.

KVL and KCL are powerful tools in circuit analysis, enabling the derivation of transfer functions that succinctly describe the dynamic behavior of electrical networks in the frequency domain. These methods are essential for the design and analysis of complex circuits in various applications.