Question

Critically evaluate different techniques used to solve problems on series-parallel R, L, C circuits using A.C. theory.

Answer 1

Series circuit

Figure 1: RLC series circuit

• V, the voltage source powering the circuit
• I, the current admitted through the circuit
• R, the effective resistance of the combined load, source, and components
• L, the inductance of the inductor component
• C, the capacitance of the capacitor component

In this circuit, the three components are all in series with the voltage source. The governing differential equation can be found by substituting into Kirchhoff's voltage law (KVL) the constitutive equation for each of the three elements. From the KVL,

where V_R, V_L and V_C are the voltages across R, L and C respectively and V(t) is the time-varying voltage from the source.

Substituting

,

into the equation above yields:

For the case where the source is an unchanging voltage, taking the time derivative and dividing by L leads to the following second order differential equation:

This can usefully be expressed in a more generally applicable form:

α and ω₀ are both in units of angular frequency. α is called the neper frequency, or attenuation, and is a measure of how fast the transient response of the circuit will die away after the stimulus has been removed. Neper occurs in the name because the units can also be considered to be nepers per second, neper being a unit of attenuation. ω₀ is the angular resonance frequency.

For the case of the series RLC circuit these two parameters are given by

A useful parameter is the damping factor, ζ, which is defined as the ratio of these two; although, sometimes α is referred to as the damping factor and ζ is not used.

In the case of the series RLC circuit, the damping factor is given by

The value of the damping factor determines the type of transient that the circuit will exhibit.

The differential equation for the circuit solves in three different ways depending on the value of ζ. These are underdamped (ζ < 1), overdamped (ζ > 1) and critically damped (ζ = 1). The differential equation has the characteristic equation

The roots of the equation in s are

The general solution of the differential equation is an exponential in either root or a linear superposition of both,

The coefficients A₁ and A₂ are determined by the boundary conditions of the specific problem being analysed. That is, they are set by the values of the currents and voltages in the circuit at the onset of the transient and the presumed value they will settle to after infinite time.

Overdamped response

The overdamped response (ζ > 1) is

The overdamped response is a decay of the transient current without oscillation.

Underdamped response

The underdamped response (ζ < 1) is

By applying standard trigonometric identities the two trigonometric functions may be expressed as a single sinusoid with phase shift,

The underdamped response is a decaying oscillation at frequency ω_d. The oscillation decays at a rate determined by the attenuation α. The exponential in α describes the envelope of the oscillation. B₁ and B₂ (or B₃ and the phase shift φ in the second form) are arbitrary constants determined by boundary conditions. The frequency ω_d is given by

This is called the damped resonance frequency or the damped natural frequency. It is the frequency the circuit will naturally oscillate at if not driven by an external source. The resonance frequency, ω₀, which is the frequency at which the circuit will resonate when driven by an external oscillation, may often be referred to as the undamped resonance frequency to distinguish.

Critically damped response

The critically damped response (ζ = 1) is

The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting. D₁ and D₂ are arbitrary constants determined by boundary conditions.

The series RLC can be analyzed for both transient and steady AC state behavior using the Laplace transform.^[16] If the voltage source above produces a waveform with Laplace-transformed V(s) (where s is the complex frequency s = σ + jω), the KVL can be applied in the Laplace domain:

where I(s) is the Laplace-transformed current through all components. Solving for I(s):

And rearranging, we have

Laplace admittance[edit]

Solving for the Laplace admittance Y(s):

Simplifying using parameters α and ω₀ defined in the previous section, we have

Poles and zeros[edit]

The zeros of Y(s) are those values of s such that Y(s) = 0:

The poles of Y(s) are those values of s such that Y(s) → ∞. By the quadratic formula, we find

The poles of Y(s) are identical to the roots s₁ and s₂ of the characteristic polynomial of the differential equation in the section above.

General solution[edit]

For an arbitrary V(t), the solution obtained by inverse transform of I(s) is:

• In the underdamped case, ω₀ > α:

  

• In the critically damped case, ω₀ = α:

  

• In the overdamped case, ω₀ < α:

  

where ω_r = √α² − ω₀², and cosh and sinh are the usual hyperbolic functions.

Parallel circuit

Figure 2. RLC parallel circuit
V – the voltage source powering the circuit
I – the current admitted through the circuit
R – the equivalent resistance of the combined source, load, and components
L – the inductance of the inductor component
C – the capacitance of the capacitor component

The properties of the parallel RLC circuit can be obtained from the duality relationship of electrical circuits and considering that the parallel RLC is the dual impedance of a series RLC. Considering this, it becomes clear that the differential equations describing this circuit are identical to the general form of those describing a series RLC.

For the parallel circuit, the attenuation α is given by

and the damping factor is consequently

Likewise, the other scaled parameters, fractional bandwidth and Q are also reciprocals of each other. This means that a wide-band, low-Q circuit in one topology will become a narrow-band, high-Q circuit in the other topology when constructed from components with identical values. The fractional bandwidth and Q of the parallel circuit are given by

Notice that the formulas here are the reciprocals of the formulas for the series circuit, given above.

Frequency domain

The complex admittance of this circuit is given by adding up the admittances of the components:

The change from a series arrangement to a parallel arrangement results in the circuit having a peak in impedance at resonance rather than a minimum, so the circuit is an anti-resonator.

The graph opposite shows that there is a minimum in the frequency response of the current at the resonance frequency when the circuit is driven by a constant voltage. On the other hand, if driven by a constant current, there would be a maximum in the voltage which would follow the same curve as the current in the series circuit.