Everywhere read ? = w (omega i.e. angular frequency)

Resistors and Ohm's law in AC circuits

The voltage v across a resistor is proportional to the current i travelling through it. Further, this is true at all times: v = Ri. So, if the current in a resistor is

i = I_m . sin (wt) ,           we write:
v = R.i = R.I_m sin (wt)
v = V_m. sin (wt)            where
V_m = R.I_m

So for a resistor, the peak value of voltage is R times the peak value of current. Further, they are in phase: when the current is a maximum, the voltage is also a maximum. (Mathematically, phi = 0.)

What are impedance and reactance?

Circuits in which current is proportional to voltage are called linear circuits. (As soon as one inserts diodes and transistors, circuits cease to be linear, but that's another story.) The ratio of voltage to current in a resistor is its resistance. Resistance does not depend on frequency, and in resistors the two are in phase, as we have seen in the animation. However, circuits with only resistors are not very interesting. In general, the ratio of voltage to current does depend on frequency and in general there is a phase difference. So impedance is the general name we give to the ratio of voltage to current. It has the symbol Z. Resistance is a special case of impedance. Another special case is that in which the voltage and current are out of phase by 90°: this is an important case because when this happens, no power is lost in the circuit. In this case where the voltage and current are out of phase by 90°, the ratio of voltage to current is called the reactance, and it has the symbol X.

Capacitors and charging

The voltage on a capacitor depends on the amount of charge you store on its plates. The current flowing onto the positive capacitor plate (equal to that flowing off the negative plate) is by definition the rate at which charge is being stored. So the charge Q on the capacitor equals the integral of the current with respect to time. From the definition of the capacitance.
V_C = q/C, so

V_m = X_C.I_m.

Inductors and the Farady emf

An inductor is usually a coil of wire. In an ideal inductor, the resistance of this wire is negligibile, as is its capacitance. The voltage that appears across an inductor is due to its own magnetic field and Faraday's law of electromagnetic induction. The current i(t) in the coil sets up a magnetic field, whose magnetic flux ?_B is proportional to the field strength, which is proportional to the current flowing. (Do not confuse the phase ? with the flux ?_B.) So we define the (self) inductance of the coil thus:

?_B(t) = L.i(t)

Faraday's law gives the emf E_L = - d?_B/dt. Now this emf is a voltage rise, so for the voltage drop v_L across the inductor, we have:

V_m = X_L.I_m.

Impedance of components

The impedance is the general term for the ratio of voltage to current. Resistance is the special case of impedance when w = 0, reactance the special case when w = ± 90°. The table below summarises the impedance of the different components. It is easy to remember that the voltage on the capacitor is behind the current, because the charge doesn't build up until after the current has been flowing for a while.

C Series combinations

When we connect components together, Kirchoff's laws apply at any instant. So the voltage v(t) across a resistor and capacitor in series is just

v_series(t) = v_R(t) + v_C(t)

however the addition is complicated because the two are not in phase. The next animation makes this clear: they add to give a new sinusoidal voltage, but the amplitude is less than V_mR(t) + V_mC(t). Similarly, the AC voltages (amplitude times 2^1/2) do not add up. This may seem confusing, so it's worth repeating:

v_series = v_R + v_C        but
V_series > V_R + V_C.

All of the variables (i, v_R, v_C, v_series) have the same frequency f and the same angular frequency ?, so their phasors rotate together, with the same relative phases. So we can 'freeze' it in time at any instant to do the analysis. The convention I use is that the x axis is the reference direction, and the reference is whatever is common in the circuit. In this series circuit, the current is common. (In a parallel circuit, the voltage is common, so I would make the voltage the horizontal axis.)

From Pythagoras' theorem:

V²_mRC = V²_mR + V²_mC

If we divide this equation by two, and remembering that the RMS value V = V_m/2^1/2, we also get:

Now this looks like Ohm's law again: V is proportional to I. Their ratio is the series impedance, Z_series and so for this series circuit,

   

Note the frequency dependence of the series impedance Z_RC: at low frequencies, the impedance is very large, because the capacitive reactance 1/?C is large (the capacitor is open circuit for DC). At high frequencies, the capacitive reactance goes to zero (the capacitor doesn't have time to charge up) so the series impedance goes to R. At the angular frequency ? = ?_o = 1/RC, the capacitive reactance 1/?C equals the resistance R. We shall show this characteristic frequency on all graphs on this page.

Z_series² = R² + X² .

Ohm's law in AC. We can rearrange the equations above to obtain the current flowing in this circuit. Alternatively we can simply use the Ohm's Law analogy and say that I = V_source/Z_RC. Either way we get

   

where the current goes to zero at DC (capacitor is open circuit) and to V/R at high frequencies (no time to charge the capacitor).

From simple trigonometry, the angle by which the current leads the voltage is

    tan^-1 (V_C/V_R) = tan^-1 (IX_C/IR)
  = tan^-1 (1/?RC) = tan^-1 (1/2?fRC).

However, we shall refer to the angle ? by which the voltage leads the current. The voltage is behind the current because the capacitor takes time to charge up, so ? is negative, ie

? = -tan^-1 (1/?RC) = tan^-1 (1/2?fRC).   

At low frequencies, the impedance of the series RC circuit is dominated by the capacitor, so the voltage is 90° behind the current. At high frequencies, the impedance approaches R and the phase difference approaches zero. The frequency dependence of Z and ? are important in the applications of RC circuits. The voltage is mainly across the capacitor at low frequencies, and mainly across the resistor at high frequencies. Of course the two voltages must add up to give the voltage of the source, but they add up as vectors.

V²_RC = V²_R + V²_C.

At the frequency ? = ?_o = 1/RC, the phase ? = 45° and the voltage fractions are V_R/V_RC = V_C/V_RC = 1/2V^1/2 = 0.71.

RL Series combinations

In an RL series circuit, the voltage across the inductor is aheadof the current by 90°, and the inductive reactance, as we saw before, is X_L = ?L. The resulting v(t) plots and phasor diagram look like this.

It is straightforward to use Pythagoras' law to obtain the series impedance and trigonometry to obtain the phase. We shall not, however, spend much time on RL circuits, for three reasons. First, it makes a good exercise for you to do it yourself. Second, RL circuits are used much less than RC circuits. This is because inductors are always* too big, too expensive and the wrong value, a proposition you can check by looking at an electronics catalogue.

RLC Series combinations

Now let's put a resistor, capacitor and inductor in series. At any given time, the voltage across the three components in series, v_series(t), is the sum of these:

v_series(t) = v_R(t) + v_L(t) + v_C(t),

The current i(t) we shall keep sinusoidal, as before. The voltage across the resistor, v_R(t), is in phase with the current. That across the inductor, v_L(t), is 90° ahead and that across the capacitor, v_C(t), is 90° behind.

Look at the phasor diagram: The voltage across the ideal inductor is antiparallel to that of the capacitor, so the total reactive voltage (the voltage which is 90° ahead of the current) is V_L - V_C, so Pythagoras now gives us:

V²_series = V²_R + (V_L - V_C)²

Now V_R = IR, V_L = IX_L = ?L and V_C = IX_C= 1/?C. Substituting and taking the common factor I gives:

Z_series² = R² + X_total²
      = R² + (X_L- X_C)².

Remember that the inductive and capacitive phasors are 180° out of phase, so their reactances tend to cancel.

? = tan^-1 ((V_L - V_C)/V_R).

Substiting V_R = IR, V_L = IX_L = ?L and V_C = IX_C= 1/?C gives:

       

(Setting the inductance term to zero gives back the equations we had above for RC circuits, though note that phase is negative, meaning (as we saw above) that voltage lags the current. Similarly, removing the capacitance terms gives the expressions that apply to RL circuits.)

Resonance

Note that the expression for the series impedance goes to infinity at high frequency because of the presence of the inductor, which produces a large emf if the current varies rapidly. Similarly it is large at very low frequencies because of the capacitor, which has a long time in each half cycle in which to charge up. As we saw in the plot of Z_series? above, there is a minimum value of the series impedance, when the voltages across capacitor and inductor are equal and opposite, ie v_L(t) = - v_C(t) so V_L(t) = V_C, so

?L = 1/?C        so the frequency at which this occurs is

where ?_o and f_o are the angular and cyclic frequencies of resonance, respectively. At resonance, series impedance is a minimum, so the voltage for a given current is a minimum (or the current for a given voltage is a maximum).

This phenomenon gives the answer to our teaser question at the beginning. In an RLC series circuit in which the inductor has relatively low internal resistance r, it is possible to have a large voltage across the the inductor, an almost equally large voltage across capacitor but, as the two are nearly 180° degrees out of phase, their voltages almost cancel, giving a total series voltage that is quite small. This is one way to produce a large voltage oscillation with only a small voltage source. In the circuit diagram at right, the coil corresponds to both the inducance L and the resistance r, which is why they are drawn inside a box representing the physical component, the coil. Why are they in series? Because the current flows through the coil and thus passes through both the inductance of the coil and its resistance.

Bandwidth and Q factor

At resonance, the voltages across the capacitor and the pure inductance cancel out, so the series impedance takes its minimum value: Z_o = R. Thus, if we keep the voltage constant, the current is a maximum at resonance. The current goes to zero at low frequency, because X_C becomes infinite (the capacitor is open circuit for DC). The current also goes to zero at high frequency because X_L increases with ? (the inductor opposes rapid changes in the current). The graph shows I(?) for circuit with a large resistor (lower curve) and for one with a small resistor (upper curve). A circuit with low R, for a given L and C, has a sharp resonance. Increasing the resistance makes the resonance less sharp. The former circuit is more selective: it produces high currents only for a narrow bandwidth, ie a small range of w or f. The circuit with higher R responds to a wider range of frequencies and so has a larger bandwidth (indicated by the horiztontal bars on the curves) is defined as the difference between the two frequencies w₊ and w_- at which the circuit converts power at half the maximum rate.

Q  =  w_o/(delta)w.

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