The fundamental passive linear circuit elements are the resistor (R), capacitor (C) and inductor (L) or coil. These circuit elements can be combined to form an electrical circuit in four distinct ways: the RC circuit, the RL circuit, the LC circuit and the RLC circuit with the abbreviations indicating which components are used. RC and RL are one of the most basics examples of electric circuits and yet they are very rich in content. The manner in which voltage or current varies with time is referred as time response. We are going to determine currents and voltages that arise when energy is either acquired or released by an inductor or capacitor in response to a change in a voltage or current source. 
restart;
add(k^3, k = 1 .. n);

We consider first the response of the three basic idealized passive circuit elements (resistance, inductance, and capacitance) to a steady state sinusoidal excitation.

    Resistance     Inductance     Capacitance

 

Sinusoidal voltage applied

Suppose that each isolated pure circuit elements R, C, and L, is subject to applied sinusoidal voltage given by
\[ v(t) = V_m \cos \left( \omega t - \phi \right) H(t) , \]
where H(t) is the Heaviside step function that represents the switching operation. For the resistance, the current is simply
\[ i_R (t) = \frac{v}{R} = \frac{V_m}{R}\, \cos \left( \omega t - \phi \right) H(t) . \]
The current flowing through the capacitance is
\[ i_C (t) = C\,\frac{{\text d}v}{{\text d}t} = -\omega C V_m \, \sin \left( \omega t - \phi \right) H(t) = \omega C V_m \, \cos \left( \omega t - \phi + \frac{\pi}{2} \right) H(t) . \]
The current in the inductance will be given by the solution of the differential equation
\[ v(t) = L\, \frac{{\text d}i_L}{{\text d}t} , \]
which is
\[ i_L (t) = \frac{1}{L} \int_0^t v(\tau )\,{\text d} \tau = \frac{V_m}{\omega L} \left[ \sin \phi + \cos \left( \omega t - \phi - \frac{\pi}{2} \right) \right] H(t) . \]
This formula shows that the wave form of the current consists of two components: a direct current of magnitude Vm/(ωL) sinφ and a sinusoidal current \( \left( V_m/\omega /L \right) \cos \left( \omega t - \phi - \pi /2 \right) . \) The resultant is an oscillating sinusoidal current but not an alternating one (its average is zero). Thus, the existence of an alternating sinusoidal voltage across a pure inductance does not necessarily imply that the current will also be an alternating sinusoidal lagging π/2 behind the voltage; this will be true, however, only is φ = 0, i.e., if the switch is closed when the voltage is a maximum.

 

Sinusoidal current applied

Suppose that each isolated pure circuit elements R, C, and L, is subject to applied sinusoidal current given by
\[ i(t) = I_m \cos \left( \omega t - \phi \right) H(t) . \]
For the resistor,
\[ v_R (t) = I_m R \,\cos \left( \omega t - \phi \right) H(t) , \]
and for the inductor,
\[ v_L (t) = I_m L \omega \,\cos \left( \omega t - \phi + \pi /2 \right) H(t) . \]
For capacitor, we have
\[ v_C (t) = \frac{I_m}{\omega C} \left[ \sin \phi + \cos \left( \omega t - \phi - \pi /2 \right) \right] H(t) . \]
The latter shows that the voltage across the capacitor is not a pure sinusoidal unless φ = 0.

 

Real circuits

In practice, lumped circuit elements designed to behave as pure resistances, inductances, and capacitances must inherently have some combination of all three properties. Since inductance and capacitance depend essentially on the geometry of the element, it is possible to design elements in which these parameters are negligibly small. Resistance, however, is the one parameter which is difficult to eliminate in inductors and capacitors since it is an inherent property of their constituent elements. Therefore, inductors and capacitors are usually come with a combination with resistors.

The major difference between RC and RL circuits is that the RC circuit stores energy in the form of the electric field while the RL circuit stores energy in the form of magnetic field. Another significant difference between RC and RL circuits is that RC circuit initially offers zero resistance to the current flowing through it and when the capacitor is fully charged, it offers infinite resistance to the current. While the RL circuit initially opposes the current flowing through it but when the steady state is reached it offers zero resistance to the current across the coil. Let’s examine each one carefully. Since the voltages and currents of the basic RL and RC circuits are described by first order differential equations, these basic RL and RC circuits are called the first order circuits. 

Error, unable to execute add
Then we try sum: 
restart;
sum(k^3, k = 1 .. n);
\[ 1/4\, \left( n+1 \right) ^{4}-1/2\, \left( n+1 \right) ^{3}+1/4\, \left( n+1 \right) ^{2} \]
Then we find its compact formula with simplify command 
simplify(%)
\[ \frac{\left( n+1 \right)^2 n^2}{4} \]
(a*b)/c+13*d \[ {\frac {ab}{c}}+13\,d \]
Two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix S such that
Theorem: If λ is an eigenvalue of a square matrix A, then its algebraic multiplicity is at least as large as its geometric multiplicity.    ▣
Let x1, x2, … , xr be all of the linearly independent eigenvectors associated to λ, so that λ has geometric multiplicity r. Let xr+1, xr+2, … , xn complete this list to a basis for ℜn, and let S be the n×n matrix whose columns are all these vectors xs, s = 1, 2, … , n. As usual, consider the product of two matrices AS. Because the first r columns of S are eigenvectors, we have
\[ {\bf A\,S} = \begin{bmatrix} \vdots & \vdots&& \vdots & \vdots&& \vdots \\ \lambda{\bf x}_1 & \lambda{\bf x}_2 & \cdots & \lambda{\bf x}_r & ?& \cdots & ? \\ \vdots & \vdots&& \vdots & \vdots&& \vdots \end{bmatrix} . \]
Now multiply out S-1AS. Matrix S is invertible because its columns are a basis for ℜn. We get that the first r columns of S-1AS are diagonal with &lambda's on the diagonal, but that the rest of the columns are indeterminable. Now S-1AS has the same characteristic polynomial as A. Indeed,
\[ \det \left( {\bf S}^{-1} {\bf AS} - \lambda\,{\bf I} \right) = \det \left( {\bf S}^{-1} {\bf AS} - {\bf S}^{-1} \lambda\,{\bf I}{\bf S} \right) = \det \left( {\bf S}^{-1} \left( {\bf A} - \lambda\,{\bf I} \right) {\bf S} \right) = \det \left( {\bf S}^{-1} \right) \det \left( {\bf A} - \lambda\,{\bf I} \right) \det \det \left( {\bf S} \right) = \det \left( {\bf A} - \lambda\,{\bf I} \right) \]
because the determinants of S and S-1 cancel. So the characteristic polynomials of A and S-1AS are the same. But since the first few columns of S-1AS has a factor of at least (x - λ)r, so the algebraic multiplicity is at least as large as the geometric.    ◂