Lecture 20, Oct. 23, 2000

Recall from last lecture:

An ac voltage source has a sinusoidally varying output: v = V_msin 2p f t
RMS voltage and current: V_r = v_m / sqrt(2), I_r = I_m / sqrt(2)
See the summary after section 21.3.

21.3 Inductors in an ac Circuit

And finally, consider what happens when an inductor is put in a circuit with an ac voltage source. Similar arguments apply as for the capacitor, but this time we find that the current leads the voltage by a quarter cycle (90°), and that the rms current is related to the rms voltage across the solenoid by:

V_L = I_r X_L

The quantity X_L is called the inductive reactance, given by:

X_L = 2p f L

Example: P21.15

A 2.40mF capacitor is connected across an alternating voltage with an rms value of 9.00V. The rms current in the capacitor is 25.0mA. (a) What is the source frequency? (b) If the capacitor is replaced by an ideal coil with an inductance of 0.160H, what is the rms current in the coil?

(a) From the rms voltage and current, we can determine the capacitive reactance, and this is related to the capacitance and the frequency of the source.
X_C = V_r / I_r = 9.00V / 25.0mA = 360W.
f = 1 / (2p C X_C) = 1 / (2p(2.40mF)(360W)) = 184Hz.

At this frequency, the inductive reactance is
X_L = 2p f L = 2p (184Hz)(0.160H) = 185W.
The rms current can be found from the rms voltage and reactance:
I_r = V_r / X_L = 9.00V / 185W = 48.6mA.

Summary of the properties of ac circuit elements.
	Resistor	Capacitor	Inductor
units	ohm, W = V / A	farad, F = C / V	henry, H = V s / A
symbol	R	C	L
reactance	X_R = R	X_C = 1 / (2p f C)	X_L = 2p f L
phase	0°	-90°	90°

21.4 The RLC Series Circuit

Now we want to examine what happens when a resistor, capacitor, and inductor are combined in an ac circuit. We begin by considering the circuit consisting of an ac voltage source, a resistor, a capacitor, and an inductor in series (sketch circuit like Figure 21.8). What should we expect to happen in this circuit, that is, what will be the relation between voltage and current? This is a series circuit, so the current will be the same through all the elements. Although the current is the same, the voltages across each element will in general be different in magnitude and in phase. The result is that the voltage from the ac source need not be in phase with the current, and the relative phase can be essentially anything.

Now let's work this out quantitatively. If we let the current be

i = I_m sin 2p f t

then the voltage drop across the resistor is

DV_R = i R,

the voltage drop across the capacitor is

DV_C = i X_C,

and the voltage drop across the inductor is

DV_L = i X_L.

Note that the frequency of the current is the same as the frequency of the voltage source. From the previous sections we know that the voltage drop across the resistor is in phase with the current, the voltage drop across the capacitor lags the current by 90°, and the voltage drop across the inductor leads the current by 90°. We can represent all of this with a phasor diagram. A phasor diagram is like a vector diagram, but representing voltages (or reactances) and the relative phases between them (draw phasor diagram for voltages). From this, we get the following triangle for the voltage sum (diagram like Figure 21.11). The hypotenuse of the triangle is the total voltage of the voltage source, and the angle it makes with the x-axis is the phase of the voltage with respect to the current:

DV = Sqrt{DV_R² + (DV_L - DV_C)²}

tan f = (DV_L - DV_C) / DV_R

Because the voltages are related to the reactances by a common factor, the current, we can also make a phasor diagram for the reactances. The hypotenuse of the reactance diagram is called the impedance,

Z = Sqrt(R² + (X_L - X_C)²}.

tan f = (X_L - X_C) / R

In terms of the impedance, we can write Ohm's law for ac circuits:

DV = I Z

where V and I can be either rms values, or peak values, but don't mix them and choose one of each!

Example: P21.20

A 50.0W resistor, a 0.100H inductor, and a 10.0mF capacitor are connected in series to a 60.0Hz source. The rms current in the circuit is 2.75A. Find the rms voltages across (a) the resistor, (b) the inductor, (c) the capacitor, and (d) the RLC combination. (e) Sketch the phasor diagram for this circuit.

Since the rms voltage is related to the peak voltage by a constant, we can use the Ohm's law relations for rms voltages and currents.
(a) V_R(rms) = I_r R = (2.75A)(50.0W) = 138V_rms.
(b) X_L = 2p f L = 2p (60.0Hz)(0.100H) = 37.7W
V_L(rms) = I_r X_L = (2.75A)(37.7W) = 104V_rms.
(c) X_C = 1/2p f C = 1 / (2p(60.0Hz)(10.0mF) = 265W
V_C(rms) = I_r X_C = (2.75A)(265W) = 729V_rms

(d) The rms voltage drop across the RLC combination can be determined using the expression for the hypotenuse of the voltage triangle.
DV = Sqrt{V_R² + (V_L - V_C)²} = Sqrt{(138V)² + (104V - 729V)²} = 640V.

21.5 Power in an ac Circuit

As mentioned previously (see the summary table at the beginning of the lecture of Oct. 20, 2000), capacitors and inductors store energy, but do not dissipate energy. Only resistors dissipate energy. Therefore, the power dissipated in an ac circuit depends only on the resistance in the circuit

P_av = I² R.

Since R = DV_R / I, the average power can also be written:

P_av = I DV_R.

Finally, we can express DV_R = DV cosf, as can be seen from the voltage triangle of the previous section. This gives us a third form for the average power:

P_av = I DV cosf. The cos f factor is known as the power factor.

Example: P21.26

In a certain RLC circuit, the rms current is 6.0A, the rms voltage is 240V, and the current leads the voltage by 53°. (a) What is the total resistance of the circuit? (b) Calculate the total reactance, X_L - X_C. (c) Find the average power dissipated in the circuit.

Begin by drawing a phasor diagram of the rms voltage. Because the current leads the voltage, the voltage lags the current, so the phase angle is f = -53°. (a) V_R is the component of the voltage along the x-axis, and is related to the resistance by R = V_R / I_r.
V_R = DV cosf = (240V) cos-53° = 144V
R = V_R / I_r = (144V) / (6.0A) = 24W

(b)X_L - X_C = R tanf = 24W tan-53° = -32W