11.2. Network analysis of linear circuits - harmonic regime

The harmonic dynamics of a linear circuit can be evaluated in Fourier domain, or using complex numbers to represent harmonic functions,

\[\begin{split}\begin{aligned} v(t) & = V_{max} \cos (\Omega t + \varphi_v) = \text{re} \{ V_{max} e^{i (\Omega t + \varphi_v)} \} = \\ & = \sqrt{2} V \cos (\Omega t + \varphi_v) = \sqrt{2} \, \text{re} \{ V e^{i (\Omega t + \varphi_v)} \} = \sqrt{2} \, \text{re} \{ v e^{i \Omega t} \} \\ i(t) & = I_{max} \cos (\Omega t + \varphi_i) = \text{re} \{ I_{max} e^{j (\Omega t + \varphi_i)} \} = \\ & = \sqrt{2} I \cos (\Omega t + \varphi_i) = \sqrt{2} \, \text{re} \{ I e^{j (\Omega t + \varphi_i)} \} = \sqrt{2} \, \text{re} \{ i e^{j \Omega t} \} \\ \end{aligned}\end{split}\]

having anticipated the definition Definition 11.1 of effective tension \(V\) and current \(I\).

11.2.1. Power

Instantaneous power.

(11.1)\[\begin{split}\begin{aligned} P(t) & = v(t) i(t) = \\ & = V_{max} I_{max} \cos (\Omega t ) \cos (\Omega t - \varphi_i) = \\ & = \frac{1}{2} V_{max} I_{max} \left[ \cos \varphi_i + \cos ( 2 \Omega t ) \right] \\ \end{aligned}\end{split}\]

having used Werner’s formula,

\[\cos x \cos y = \frac{1}{2} \left[ \cos(x-y) + \cos(x+y) \right] \ .\]

and the property \(\cos(-x) = \cos x\).

Average power on a period. Over a period \(T = \frac{1}{f} = \frac{2 \pi}{\Omega}\)

\[\overline{P} = \frac{1}{T} \int_{t=t_0}^{t_0+T} P(t) \, dt = \frac{V_{max} I_{max}}{2} = V I\ ,\]

as the integral of the harmonic term with period \(\frac{T}{2}\) of the instantaneous power (11.1) is identically zero, and with the definition of the effective voltage and current

Definition 11.1 (Effective voltage and current in AC)

Effective voltage and currents

\[V := \frac{V_{max}}{\sqrt{2}} \quad , \quad I := \frac{I_{max}}{\sqrt{2}} \ , \]

are defined as those voltage and current in DC providing the same value of average power.

Complex power. Complex power of a dipole with impedence \(Z\), \(v = Z i\)

\[\begin{split}\begin{aligned} S & := v i^* = |v|e^{j \varphi_v} |i| e^{-j \varphi_i} = |v| |i| e^{j(\varphi_v - \varphi_i)} = \\ & = Z i i^* = Z |i|^2 = (R + j X ) |i|^2 = |Z||i|^2 e^{j \varphi_Z} = P + j Q \ , \end{aligned}\end{split}\]

with the active power \(P\) and the reactive power \(Q\)

\[\begin{split}\begin{aligned} P & = \text{re}\{ S \} && = && |S| \cos \varphi_Z && = && \dots \\ Q & = \text{im}\{ S \} && = && |S| \sin \varphi_Z && = && \dots \\ \end{aligned}\end{split}\]