Electrical resistance, capacitance and inductance 

Principle

When a current is flowing through an electrical resistance, capacitance or inductance, the relation between current and voltage may be changed by these elements. Their action can be described as follows.

Resistance            R=V/I or V=IR,                                       (1)

This is Ohms law, where V is voltage (V), I is current (Amp) and R is resistance (Ohm). In a circuit with only resistors, all current and voltages are in phase. Halving the resistance means doubling the current when the voltage over the resistance remains the same. A daily analogue is taking a second water hose to sprinkle the garden. Then total flow of water is doubled since the pressure of the water supplying system is constant.

Resisters dissipate heat, caused by electrical “friction”. This can be expressed as power P (in Watt):

  P = IV = I²R = V²/R (W).                                                      (2)

Fig. 1   Circuit to charge the capacitance C of the capacitor with source voltage V_S. sw is switch.

Capacitance         dV/dt = (1/C)dQ/dt = I/C,     (3)

where C is the capacitance (F, farad), Q the instantaneous charge (in coulombs, Q) of the capacitor and t time. When Q is constant in time (a charged capacitor, also a battery) it holds that C = Q/V.

For a circuit with a constant voltage source (so called DC voltage from direct current) and comprising of a resistor and capacitor in series (Fig. 1), the voltage across the capacitor cannot exceed the voltage of the source. Suppose that before the start the circuit is switched off and the charge of the capacitor is zero. Next, the circuit is switched on (closed). Then a current provided by the voltage source is charging the capacitor until an equilibrium is reached:  the voltage across the capacitor V_C finally becomes the source voltage V_S and the charging current becomes zero. For this reason, it is commonly said that capacitors block direct current (DC). At the start the current is maximal and at the end of the charging process minimal, whereas the voltage then reaches the highest values. So, one could say that the voltage over the capacitor lags the current. The current diminishes exponentially with a time constant (see Halftime and time constant and Linear first order system) τ = RC.

When the voltage of the source changes periodically the current will also change periodically, called alternating current (AC). Then the current through a capacitor reverses its direction periodically. That is, the alternating current alternately charges the plates: first in one direction and then in the other. With the exception of the instant that the current changes direction, the capacitor current is non-zero at all times during a cycle. For this reason, it is commonly said that capacitors "pass" AC. However, at no time do electrons actually cross between the plates, unless the dielectric breaks down. Such a situation would involve physical damage to the capacitor.

With sine waves as input voltage, the voltage across the capacitor V lags the current 90^o, since the voltage is proportional to the integral of the current (see (3)). That is, the voltage and current are 'out-of-phase' by a quarter cycle. So, the phase is –90^o, the minus sign denotes lag (notice sin(x – 90) = cosx). The amplitude of the capacitor voltage depends on the amplitude of the current divided by the product of the frequency f of the current with the capacitance, C, so   V ~ I/fC.

For a more formal description we need the quantity impedance (Z), which reflects in a network the action on amplitude and phase. It is the ratio of the voltage (of unit amplitude and phase zero) across a circuit element over the current through that element.  It comprises a real part, a resistance, and imaginary part, due to capacitance (or inductance, the action of a coil).

  Z = R + jX.                                                                            (4)

For a capacitor, the impedance is given by:

  Z_c = V_c /I _c = Vj/(2πfC) = Vj/(ωC) = –jX _c,                         (5)

where f is the frequency (Hz) of a sinusoidal voltage, ω ( = 2πf) is the angular frequency and X _c ( = –1/(ωC) is called the capacitive reactance, the quantity denoting the imaginary part of the impedance. j is the imaginary unit (j = √V1, in mathematics and often in physics denoted by i). It denotes the phase lag of 90^o between voltage and current. Since a pure capacitance has no resistance, the real part of the impedance is not present in (5).

Ideal capacitors dissipate no energy.

Inductance òVdt = LI,                                                (6)

where L is self inductance. The action of an inductance can be called reciprocal of that of a capacitance with respect to amplitude: the inductive reactance is X _L =ωL. Consequently, an inductance (in hardware a coil) blocks AC (high frequency) due to its virtual high resistance. It passes DC, since then X _c = 0. With respect to phase, its action causes sign reversal. So, there is a phase lead of 90^o. Equation (6) shows that directly since V is proportional to the derivative of I.

Ideal inductances dissipate no energy, but the intrinsic resistance of coils (the winding of the wire) of coarse do, the reason why operating coils are always warm.

The formal definition of inductance is:

  L= Φ/I.                                                                                  (7)

Where Φ is magnetic flux (in H, henry = weber/ampere). In linear systems (6) is sufficient, but when electromagnetic fields are generated the  theory of electromagnetic fields are of importance and so (7). The description of this theory is beyond the scope of this compendium, with exception of the small contribution Lorentz force.

In conclusion:

  Z_R = V_R/I_R;                                                                            (8a)

  Z_C = V_C/ I_C = –1/jωC;                                                          (8b)

  Z_L = V_L/ I_L = jωL.                                                                  (8c)

In the complex Z-plane it can be visualized as in Fig. 2.

Fig. 2.  Impedances in the complex impedance plane.

Application

It needs no further explanation  that in all electric instruments etc. resistors and capacitors are applied, as individual pieces or incorporated in integrated circuits.

Coils can be applied in for instance specific electronic filters (see Linear first order system, Linear second order system and System analysis), in dynamo’s to generate current, in transformers, for spark ignition, in mechanical current and voltage meters and in liquid current meters. Further, they are applied in (medical) equipment to refract or change electromagnetic radiation (Mass spectrography and MRI machines, particle accelerators needed for PET, Electron Spin Resonance machines).

More info

In hardware the resistor is the circuit element to provide the action of resistance. A resistance always gives some inductance: current flowing through a cable produces a small magnetic field. It has also some capacitance. Both are very dependent on the design (material, configuration) of the resistor.

Similar considerations hold for the capacitor, the physical realization of the capacitance. It has some resistance (the dielectricum does not isolate perfect) and there is some self inductance.

In general these imperfections can be avoided by choosing the right type of resistor and capacitor (their numerical value as well as technical design). However, these imperfections generally only play a role with extreme high frequencies.

The coil is the realization of the self inductance. It has always a resistance, and generally this is so large that one should consider it in circuit designs and calculations. It also has some capacitance.

Resistors

Resistors, say R₁, R₂ etc., in a parallel configuration have the same potential difference (voltage).

Fig. 2    Resistors in parallel.

Their total equivalent resistance (R_eq):

 1/R_eq = 1/R₁ + 1/R₂ + .. 1/R_n.                                               (9a)

The parallel property can be represented in equations by two vertical lines "||" (as in geometry) to simplify equations. For two resistors,

   1/R_eq = R₁ || R₂ = R₁R₂/(R₁ + R₂).                                      (9b)

Fig. 3    Resistors in series.

The current through resistors in series stays the same, but the voltage across each resistor can be different. The sum of the individual voltage differences is equal to the total voltage. Their total resistance is:

   R_eq = R₁ + R₂ + .. R_n.                                                          (9c)

Capacitors

For capacitors in parallel hold:

   C_eq = C₁ + C₂ + .. C_n.                                                         (10a)

And in series:

   1/C_eq =  1/C₁ + 1/C₂ + .. 1/C_n.                                           (10b)

Coils

For reasons given by the electromagnetic filed theory, in electric circuits mimicking linear systems, coils are generally not put in series or parallel. The above simple rules about parallel and serial generally do not hold. 

Analogues of R, C and L

Mechanical:          R_M (Ns/m), mechanical resistance, due to friction;

                                C_M (m/N), mechanical compliance;

                                mass (kg).

Fluid dynamics:   R_F  (Ns/m⁵), fluid resistance;

                                C_F   (m⁵/N), compliance ;

                                M_F   Ns/m⁴), inertance.

Acoustical :           R_A  (Ns/m⁵), acoustic resistance;

                                C_A   (m⁵/N), acoustic compliance ;

                                M_A   Ns/m⁴), acoustic inertance.