FPGA Based Digital World - Second Order Circuits

Second Order Circuits

A second order circuit is characterized by a second-order differential equation.It usually consists of two energy storage elements.

Natural Response of Source-Free Series RLC Circuit

A series RLC circuit is shown below.

Suppose at t=0,

v(0) =V0,

i(0)= I0

Applying KVL in the loop,

It can be rewritten as

due to

Let

where A and s are contants to be determined.

And solving the probleme with below steps,

we get below characteristic equation.

And the two roots of the above characteristic equation(also called natural frequencies, in Np/s) are

where

is the neper freqency or damping factor,

is the resonant frequency or stictly undamped natural frequency.

Finally, the natural response of the series RLC circuit is a linear combination of the two roots'solution, as below.

where A1 and A2 are determined from the initial values.

There're 3 types of solutions depending on the parameters of R,L and C, as depicted below.

1 Overdamped Case()

In this case, both roots s1 and s2 are negative, and the response decays and approaches zero as t increases, as shown below.

The initial current of the inductor and the initial voltage of the capacitor(or A1 and A2) determine the initial shape of the curve.

2 Critically Damped Case()

In this case, the natural response is

3 Underdamped Case()

In this case, the roots are written as

where

is the damped natural frequency.

And the natural response is

And an example curve is below.

Another method to solve the circuit is to rewrite the original equation as

since

And it has similar solution, but helps to understand the analog to the mechanical vibration system.

Step Response of Series RLC Circuit

For below circuit,

its step response is

where Vt(t) is the transient response, the same as in the natural repsonse of source-free one,

and Vss(t) is the steady-state response.

Steady-State Response of Series RLC Circuit to Sinusoidal Excitation

For below circuit,

suppose V1(s) is a sinusoidal signal input, calculate the output voltage v(t).

In s domain,it could be calculated as

where Xs is the total impdedance of R1, C1 and L1,

Let's find its characteristics in frequency domain.

In frequency domain, the impedance becomes

Clearly, the repsonse depends on the frequency.

When

the imaginary part of the impedance becomes zero.And the frequency satisfying this condition is called the resonance frequency, where

, rad/s

At resonance,

1) the series LC combination acts like a shortcut, and the impedance reaches minimum;

2) the voltage Vs and current Is are in phase;

3) the voltage across the inductance and capacitor can be much more than the source voltage(voltage magnification effect).

Because at resonance, the voltage across the inductance and capacitor are in anti-phase and equal in magnitude.

The frequency reponse is shown below.

This can also be explained graphically in phasor domain.

Steady-State Response of Parallel RLC Circuit to Sinusoidal Excitation

For below circuit,

the admittance is

In frequency domain, the admittance becomes

So it has the same resonance frequency as in the series RLC circuit, i.e.,

At resonance frequency,

1) the LC combination acts like open circuit, and the admittance reaches minimum;

2) the voltage Vs and current Is are in phase;

3) the current through the inductance and capacitor can be much more than the source current

The explanation in phasor domain is shown below.

Reference

1. Fundamentals of Electric Circuits(Fifth Edition), Charles K. Alxander and Mathew N.O. Sadiku, China Machine Press, 2013.2

2. LCR Parallel Circuits, www.learnabout-electronics.org

3. LCR Series Circuits, www.learnabout-electronics.org


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