Electronics in a phy ...

The systems studied in this exercise are two-port devices consisting of the linear elements RC, RL and RLC. The relationships between voltages and currents in such two-port devices can be described by a system of differential equations. An analytical solution to this type of system can be obtained by applying the Laplace transform and then solving a system of algebraic equations. If the two-port device is devoid of voltage and current sources, then the current and voltage waveforms in the two-port device will depend only on the initial conditions, i.e. the voltages that occurred on the capacitors and the currents flowing through the inductors at the initial moment. If the system consists of real (lossy) elements, it will reach equilibrium after some time. Before that happens, however, the system is in the so-called transient.

The transient state in an electrical circuit is called the general integral of the homogeneous equation. During the transient state, energy dissipation processes take place in the systems. If the system is passive then after some time the energy stored in the circuit must be dissipated and the transient must disappear.

It can be shown that in a linear, stationary two-port device with lumped parameters, in which at the initial moment the voltages on the capacitors and the currents in the inductances are zero, the following relationship holds:

\( y(s) = K(s) x(s) \)

where x(s) and y(s) are Laplace transforms. The voltages at the input x(t) and at the output of the two-port device y(t) respectively are:

\( x(s) = \int_{0}^{\infty}{e^{-st}x(t)dt} \)

\( y(s) = \int_{0}^{\infty}{e^{-st}y(t)dt} \)

The function K(s) is called the transfer function of the two-port device. The properties of two-port devices can be described in the time domain or in the frequency domain. Both of these descriptions are equivalent to each other. The transfer function of the system K(ω) is identical to the transfer function K(s) assuming that:

\( s = j \omega \)

Two types of characteristics are usually used to describe the properties of a system in the time domain:

• The impulse response of the system k(t),
• Response to a unit step h(t).

The impulse response or time-impulse characteristic of the system is the voltage waveform y(t) caused by the signal x(t) = δ(t), assuming zero initial conditions (i.e. energy stored in inductors and capacitors), δ(t) -  Dirac's delta. Since the Laplace transform of the Dirac's delta is:

\( \mathcal{L} [ \delta (t) ] = 1 \)

so:

\( y(s) = K(s) \)

so we can compute the k(t) functions using the inverse Laplace transform: 

\( k(t) = \mathcal{L}^-1 [ K(s) ] \) .

It follows that the transfer function of the two-port device  K(s) is the Laplace transform of its impulse response k(t). Knowing the impulse response of the two-port device, we can calculate its response to any signal x(t) using the convolution operation:

\( y(t) = k(t) \otimes x(t) \)

The response to a unit step or the time-unit characteristic of the system is the voltage waveform y(t) caused by the Heavyside step function x(t) = 1(t). Function 1(t) is defined as:

\( 1(t) = \left\{{ {0, for \ t<0} \atop {1, for \ t \geq 0} }\right. \)

The Laplace transform of the Hevyside function is:

\( \mathcal{L} [1(t)] = \frac{1}{s} \)

so: \( y(s) = \frac{K(s)}{s} \),

that is: \( \mathcal{L} [h(t)] = \frac{K(s)}{s} \).

The Laplace transform of the system response to a unit step is the quotient K(s)/s, where K(s) - system transmittance.