Question

Q) Explain the following,

What are the following :

2nd order or frst order systsms and their difference?

Capacitor and inductor chargung and discharging?

RL and RC and RLC filter?

Answer 1

2^nd order systems contain two energy storage elements i.e. they will contain one capacitor and one inductor. The differential equation characterizing the behaviour of these circuits is of second order i.e. contains a second order derivative term. Further the response of second order systems can be categorized into 3 types i.e. Underdamped, Critically damped and Overdamped.

1^st order circuits contain only one energy storage element i.e either a capacitor or an inductor. The differential equation describing the behaviour of these circuits is of first order i.e. contains a first order derivative term.

2) Charging and discharging of capacitors - Whenever an uncharged capacitor or a capacitor with intially 0 volts on it is connected across a voltage source, it starts charging. The rate of charging depends on R_eqC (time constant) where R_eq is the equivalent resistance across the capacitor. Whenever a capacitor is directly connected across a voltage source without any resistance the capacitor gets instantaneously charged.

The charging equation of a capacitor is given by -

V = V₀*(1 - e)     V₀ = Maximum voltage upto which capacitor can be charged usually taken equal to supply voltage.

The discharging equation of capacitor is given by -

V = V₀* e where V₀ is the initial voltage on capacitor.

In an Inductor we experience another phenomena which is growth and decay of current, An inductor opposes the sudden change of current through it therefore, current either increases gradually or decays gradually. The time constant of RL circuit is given by L/R.

When an initially uncharged inductor is connected across a voltage source through a resistance, the current growth equation is given by -

I = I₀ * (1 - e)                        Where I₀ is the maximum current which can flow in circuit calculated by                                                                     short circuiting the inductor.

The decay of current is given by -

I = I₀ * e                                        Where I₀ is the initial maximum current.

3) RL filter - A RL circuit can be used as a low pass or high pass circuit.

RL high pass circuit consists of a Resistor in series with supply signal and an inductor connected across output. At low frequency inductor will act as short circuit ( Since X_L = 2 * pi * f * L, when f ~0 X_L ~0 ) therefore, will act as short circuit and hence reducing output to zero. At high frequency inductor will behave as an open circuit therefore our signal will be available at the output.

RL low pass circuit will have inductor in series with supply signal and resistance connected across output. At low frequency inductor will behave as short circuit therefore, will allow signal to pass through at high frequency it will act as open circuit therefore, will block the circuit.

RC circuit can also be used as high pass or low pass circuit.

In RC high pass circuit the capacitor is connected in series with supply signal and at high frequency capacitor ideally acts as a short circuit ( X_C = 1/(2* pi* f * C) at f = very high , X_C ~0) therefore, allowing signal to pass through it.

RC low pass circuit consists of resistor connected in series with supply signal and capacitor is connected across the output. At low frequency capacitor acts as open circuit therefore, allowing signal to pass.

RLC filter

RLC circuit can be used as a low pass filter, high pass filter, band stop and band pass filter. For each mode R, L and C components are arranged in a predefined manner.

In RLC low pass filter the components are arranged as follows. At low frequency inductor will act as short circuit and capacitor as open circuit therefore, input signal will be available at the output.

RLC high pass filter, we can say that at high frequency capacitor will behave as short circuit and inductor as open circuit and applied signal will be available at the output.