Introduction:

In prior analysis of dc circuits all currents and voltages were constant, not differing with time. There was no profile or waveform related with the source driving an electric circuit. Therefore the question of vectors having phase and magnitude did not occur. Though, in case of analysis of ac circuits, the concept of currents and voltages as vectors is basic as the sources driving circuits will essentially be sinusoidal or at least time-varying with the periodic properties. Furthermore, the circuit elements are not purely resistive, however will comprise inductors and capacitors which have the property of impedance. This signifies that they behave in a way that interacts with the phase and magnitude of the voltages and currents existed in a circuit and modify the magnitude and phase of both. It is so of interest to characterise the current-voltage relationships of such components in the context of ac electricity.  

Resistive Circuit Elements:

In case of a resistor, under steady-state situations, if a sinusoidal voltage is positioned across it, then the current which flows via the resistor is as well sinusoidal and has exactly similar phase as the applied voltage signal. On the contrary, when a sinusoidal current is passed via a resistor the resultant voltage that is developed across it is sinusoidal and has similar phase as the source current. The ac voltage and current related with a resistor are thus always in phase, as shown in figure below.

Figure: Sinusoidal Current and Voltage related with a Resistor

v_R = V_m sin ωt

Then the current is provided as:    

i_R = (V_m/R) sin ωt = I_m sin ωt

The resistance states the current-voltage relationship as:

R = v_R/i_R = V_m sin ωt/I_m sin ωt = V_m/I_m

When the voltage across the resistor is stated as a general complex vector then:

v_R = V_m e^jωt

Then the current will contain similar form of complex vector and hence:

i_R = (V_m/R) e^jωt = Im e^jωt

and,

R = v_R/i_R = V_m e^jωt /Im e^jωt = V_m/I_m

This can be seen that the resistance is a scalar quantity that doesn’t vary with either time or frequency. The value of resistance is as shown in table below for a range of values of current and voltage.

Table: Values of Resistance for a range of Voltages and Currents

Capacitive Reactance:

In case of a capacitor, though, the situation is somewhat dissimilar. The voltage developed among the plates of a capacitor is as an effect of charge accumulated over time as current flows to supply or eliminate charge on the plates. This has the characteristic relationship shown in figure below.

i_C = C (dv_C/dt)

Figure: A Capacitor and its Characteristic Equation

When the voltage across the capacitor is carried out as sine function then:

v_C = V_m sin ωt

and,

i_C = C V_m ω cos ωt

Or, i_C = C V_m ω sin (ωt + ?/2)

This exhibits that in a capacitor with sinusoidal steady-state voltage excitation, the current leads the voltage by 90º. This is illustrated in figure below where the voltage is taken as a sine function and the resultant current becomes a cosine function:

Figure: The Relationship between Current and Voltage in a Capacitor

When the voltage across the capacitor is explained as a complex vector then:

v_C = V_m e^jωt = V_m (cos ωt + j sin ωt)

and hence,

i_C = C (dv_C/dt) = - C V_m ω sin ωt + jCV_m ω cos ωt

i_C = ω CV_m (j cos ωt - sin ωt)

i_C = ω CV_m (j cos ωt + j2 sin ωt)

i_C = j ω CV_m ejωt = j I_m e^jωt

The existence of j term in the outcome points out a phase advancement of 90^o in the complex plane. This is steady with the outcome obtained above where the exciting voltage was treated as singular sine function. The relationship among the current and voltage in a reactive element is stated as the impedance of the element, designated Z_C. This is thus stated for the capacitor as:

And hence:

Z_C = 1/j ωC = -j (1/ ωC)

It must be noted that the impedance is a complex parameter and has both phase and magnitude. The j term is related with the impedance and points out the fact that current leads voltage by 90o or that voltage lags current by 90o. This is frequently a point of confusion in the Electronic Engineering.
 
The property of reactance is as well stated for a capacitor as:

X_C = |v_C|/|i_C| = V_m/I_m = V_m/(CV_mω) = 1/ωC

Note: The reactance of a capacitor is a scalar quantity and consists units of Ohms. This is not complex or imaginary. Additionally, reactance of the capacitor is dependent on the frequency of sinusoidal excitation. The value of reactance reduces with increasing frequency as shown in figure below. Consequently, for a given excitation voltage a higher current will flow via the capacitor at higher frequencies. The values of capacitive reactance for ranges of frequency and capacitance are given in table shown below.

ω → 0  Xc → ∞

ω → ∞  Xc → 0

Figure: Capacitive Reactance as a Function of Frequency.

Table: Values of Reactance for a range of Frequency and Capacitance.

Inductive Reactance:

In case of an inductor, whenever current flows via it, a back emf is developed in a direction that opposes a change in the current flow. This back emf is as well opposite to the direction of potential drop across the inductor caused by the external applied electric field that gives mount to the current flow. The characteristic relationship among voltage and current in an inductor is given in figure shown below.

v_L = L (di_L/d_t)

Figure: An Inductor and its Characteristic Equation.

This time if the current is taken as a negative cosine function then:

i_L = - I_m cos ωt

v_L = L I_m ω sin ωt

and hence, v_L = L I_m ω cos (ωt + ?/2)

This exhibits that in an inductor the voltage leads the current by 90º or the current lags voltage by 90º beneath steady state sinusoidal excitation conditions. This is shown in figure below where the voltage is as sine function for comparison with the situation of capacitor.

Figure: The Relationship between Current and Voltage in an Inductor.

If the current via the inductor is explained as a complex vector then:

i_L = I_m e^jωt

and hence,

v_L = L (di_L/dt) = LI_m j ω e^jωt

v_L = j ω LI_m e^jωt = j Vm e^jωt

The existence of the j term in the outcome points out a phase advancement of 90^o in the complex plane. This is steady with the outcome obtained above as it exhibits the voltage to be advanced by 90^o in phase as compared with the current. Alternatively, the current can be considered to lag behind the voltage by 90o. The relationship between the voltage and current in the inductive element is again stated as its impedance, designated ZL and given as:

Z_L = v_L/i_L = (j V_m e^jωt)/(Im e^jωt) = (j ω LI_m e^jωt)/ (I_m e^jωt)

And hence,

Z_L = jωL

This is noted again that the impedance is a complicated parameter containing both phase and magnitude. The j term is related with the impedance and points out in this case that current lags the voltage by 90o or that voltage leads the current by 90^o.

The property of reactance can as well be defined for an inductor as:

X_L = |v_L|/|i_L| = V_m/I_m = (L I_m ω)/I_m = ωL

The reactance of an inductor is as well a scalar quantity and consists of units of Ohms. This is not imaginary or complex. The reactance of inductor is too dependent on the frequency of sinusoidal excitation. In case of the inductor, though, the magnitude of reactance rises with frequency and hence for a given excitation voltage a smaller current will flow at higher frequencies as shown in figure below. The values of inductive reactance for ranges of frequency and inductance are given in table shown below.

ω → 0 and XL → 0

ω → ∞ and XL → ∞

Figure: Inductive Reactance as a function of Frequency

Table: Values of Reactance for Ranges of Inductance and Frequency

A suitable method to remember the phase relationships between current and voltage in the capacitor and the inductor is given as:

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