A. C. CIRCUITS

Introduction - What are A.C. Circuits?
A. C. Circuits are electrical circuits in which the source of electricity generates an Alternating Current. This generator produces a sinusoidal voltage (emf) with a corresponding sinusoidal current through a close circuit. These voltages and currents are usually generated by a rotating coil in a constant magnetic field with a constant angular velocity ω which induces in the coil an instantaneous voltage v expressed by the following equation:
v = V_m sin ωt, (1)
where V_m is the maximum voltage produced by the ac generator. The angular velocity is related to the frequency f of the wave by the equation
ω = 2πf = 2π/T, (2)
where T is the period of the wave, which is the time of one cycle of the sinusoidal voltage wave. The ac current exhibits a similar time variation. This kind of circuit is called alternating because the voltage and current in the circuit periodically reveses its polarity and direction, respectively.

A.C. Circuit with Resistance only.
Consider the simple ac circuit which consists of a resistor and an ac generator. At any instant, the algebraic sum of potential increases and decreases around the close loop of the circuit must be zero (kirchhoff's loop equation). Therefore, v - v_R = 0, or
v = v_R = V_m sin ωt, (3)
where v_R is the instantaneous voltage drop across the resistor. Therefore, the instantaneous current in the resistor is equal to
i_R = v_R/R = (V_m/R) sin ωt = I_m sin ωt, (4)
where R is the resistance of the resistor and I_m is the maximum current in the circuit, given by the equation
I_m = V_m/R. (5)
From equations (3) and (4) it is clear that V_m = I_mR and the instantaneous voltage drop across the resistor is
v_R = I_mR sin ωt. (6)

Since v_R and i_R both vary sinsusoidal as sin ωt and reach their maximum values at the same time, they are said to be in phase; they also reach their maximum and zero values at the same instant. Very often this phase relationship between the voltage and current is represented by a phasor diagram. In these diagrams, the alternating quantities is represented by an arrow, called a phasor, that rotates in a counterclockwise direction. The length of the arrow corresponds to the maximum value of the alternating quantity, and the projection of the arrow on the vertical axis gives the instantaneous value of the quantity. In the case of a single-loop resistive circuit, the current and voltage phasors lie along the same line, since v_R and i_R are in phase with each other. Note that although arrows are used for the phasors they are not vectors in the ordinary sense, because they do not represent a direction in space.

Note that the average value of the current over one cycle is zero. That is, the current is maintained in the same direction (say, the positive direction) over the same amount of time and at the same magnitude as in the opposite direction (the negative direction). Note that the direction of the current has no effect upon the resistor. Although the temperature in the resistor depends on the magnitude of the current, it does not depend on its direction. The rate at which electrical energy is converted into heat energy, which is called the power P, is given by
P = i²R, (7)
where i is the instantaneous current in the resistor. Since the heating effect of a electrical current is proportional to the square of the current, it makes no difference whether the current is direct or alternating; that is, whether the sign of the current is positive or negative. However, the heating effect produced by an alternating current having a maximum value I_m is not the same as that produced by a direct current of the same value. The alternating current is at its maximum value for only a brief instant during a cycle. What is important for the heating effect in an ac circuit is the average value of the current that is called the rms current. The term rms refers to the root mean square method of determining the average by taking the square root of the average of the square of the current. Since I² varies as sin² ωt, it can be shown that the average value of I² is (1/2)I_m².
Note first that the graph of cos² ωt versus time is identical to the graph of sin² ωt versus time, except that the graph is shifted on the time axis. Thus the time average of sin² ωt is equal to the time average of cos² ωt when taken over one or more complete cycles. That is,
(sin² ωt)_av = (cos² ωt)_av.
Using the trigonometic identity,
sin² θ + cos² θ = 1, we get
(sin² ωt)_av + (cos² ωt)_av = 2(sin² ωt)_av = 1, or
(sin² ωt)_av = 1/2.
Substituting this into the following equation, after taking the average of
i² = (I_m)² sin² ωt,
we get
(i²)_av = (I_m)² (sin² ωt)_av = I_m²(1/2).
Taking the square root, we get
I_rms = I_m/√2.
Therefore, the rms current I_rms is related to the maximum value of the alternating current I_m, as
I_rms = I_m/√2 = 0.707 I_m. (8)
Thus the average power dissipated in a resistor carrying an alternating current is
P_av = I_rms²R. (9)

Alternating voltages should also be discussed in terms of rms voltages, and the relationship is the same as for rms currents; that is, rms voltage, V_rms, is related to the maximum value of the voltage, V_m, as
V_rms = V_m/√2 = 0.707 V_m. (10)
In this paper the rms values shall be used when discussing alternating currents and voltages. One reason for doing this is that is that ac ammeters and voltmeters are designed to measure rms values. Another reason is that many of equations of ac circuits are similar to those in dc circuits, if they are written with rms values.

A. C. Circuit with Induction only.
Consider now an ac circuit consisting of only an inductor connected to the terminals of an ac generator. The instantaneous voltage drop across the inductor is v_L and applying Kirchhoff's loop rule to the circuit we get v + v_L = 0, or
v - L(di/dt) = 0.
Rearranging this equation, and substituting for v = V_msin ωt, we get
L(di/dt) = V_m sin ωt. (11)
Solving this equation for di, we get
di = (V_m/L) sin ωt dt,
and integrating this equation gives the current as a function of time:
i_L = (V_m/L)∫(sin ωt dt) = -(V_m/ωL) cos ωt. (12)
Using the trigonometric identity cos ωt = -sin (ωt - π/2), equation (12) can be written
i_L = (V_m/ωL) sin(ωt - π/2). (13)
Comparing this equation with equation (11) clearly shows that the current is out of phase with the voltage by π/2, or 90°. A plot of the voltage and current versus time shows that the voltage reaches its maximum value at a time that is one quarter of its period before the current reaches its maximum value. Thus we see that for a sinusoidal applied voltage, the current always lags behind the voltage across the inductor by 90°. The explanation of this is that since the voltage across the inductor is proportional to di/dt, the value of v_L is largest when the current is changing most rapidly. Since i versus t is a sinusoidal curve, di/dt (the slope) is a maximum when the curve goes through zero. This shows that v_L reaches its maximum value when the current is zero.

From equation (12) we see that the maximum current I_m is
I_m = (V_m/ωL) = V_m/X_L, (14)
where the quantity X_L is called the inductive reactance and is defined as
X_L = ωL. (15)
The rms current is given by equations similar to equation (14) with V_m replaced by V_rms. The term reactance is used so that it is not confused with resistance. In a resistive circuit, i and v are always in phase, whereas i lags behind v by 90° in a purely inductive circuit.
Using equations (11) and (14), the instantaneous voltage drop across the inductor can be found by following equation:
v_L = V_m sin ωt = I_mX_L sin ωt. (16)
This equation can be considered as Ohm's law for an inductive circuit. That is,
V_m = I_mX_L. (17)
Note that the reactance of an inductor increases with increasing frequency. This is because at higher frequencies, the current must change more rapidly, which in turn causes an increase in the induced emf associated with a given maximum current.

Inductance and Resistance.
An inductor cannot be entirely without resistance since resistance is an inherent property of the conductors in the coil of wires in the inductor. Thus the resistance of the inductor must be lumped with any resistance in series with the inductor. The same current will be present in all parts of a series circuit. Thus the circuit current is used as reference to show the phase between current and voltages in the resistance and inductance. The voltage across the resistance is in phase with the circuit current. The voltage across the inductance leads the circuit current by 90°. The voltage across the series combination of R and L will then lead the circuit current by some angle between 0° and 90°.

Because of the difference of phase angles of the voltages across the resistance and inductance, the voltages V_R and V_L cannot be added algebraically but must be added vectorally to give the circuit voltage V. That is, the voltage V across the whole external circuit is the resultant, or vector sum, of the voltages V_L and V_R. The phase angle φ will have a positive value less than 90°.

The potential drop across the resistance in series with the inductance is equal to the product IR, where R is the common resistance to the current in the external circuit. The inductance as a consequence of its opposition to the current in the circuit has a voltage difference across it. The oppositon is non-resistive because no power is dissipated in the inductance and thus is called reactance, X_L. This inductive reactance X_L is the ratio of the effective value of the potential, V_L, to the effective value of the current, I; that is,
X_L = V_L/I.
Thus the voltage drop across the inductance V_L is equal to the product, IX_L.

The inductive reactance of a coil is directly proportional to the inductance L and to the frequency f of the current in the circuit. This is because the rate of change of a given current increases with frequency. And the larger the inductance L, the larger the opposition to the change of the current at a given frequency. That is,
X_L = 2πfL = ωL,
where ω is called the angular velocity of the frequency and is equal to 2πf. The frequency is measured in hertz and inductance L in henrys, and the inductive reactance X_L in ohms. The 2π is the constant of proportionality between the inductive reactance and the product of the frequency and inductance.

The vector sum of the IX_L and the IR is equal to the voltage, V, applied to the external circuit. This voltage or potential difference across the external circuit is the product of the circuit current, I, and the combination of the effect of the inductive reactance, X_L, and the resistance, R, in the load. This combination or joint effect of the reactance and resistance in an ac circuit is called impedance, Z. It is measured in ohms. Hence, V is equal to the product IZ. If each voltage V_L, V_R and V are divided by the current I, then V_L/I = IX_L/I yields the inductive reactance X_L and V_R/I = IR/I yields the resistance R, and V/I = IZ/I yields the impedance Z.

The relationship between the inductive reactance X_L, resistance R and the impedance Z can be represented by a right triangle that is called a impedance diagram. In this right triangle that is called an impedance diagram, the short sides or legs represent X_L and R and the long side or hypotenuse of the right triangle represents the impedance Z. The acute angle opposite the side that represents the inductive reactance X_L represents the phase angle φ. The tangent of the phase angle φ is defined as the ratio of the side of the right triangle opposite to φ to the side adjacent to φ in the right triangle. Thus,
X_L/R = tan φ
This equation gives the ratio X_L/R as a function of φ. It also enables us to find φ as a function of X_L/R. That is, φ is an angle whose tangent is X_L/R. This phrase, "φ is an angle whose tangent is X_L/R" expresses what is called in trigonometry the inverse function to the tangent. The inverse function of the tangent is called the "arc tangent". The symbol for arc tangent is arc tan or tan^-1. The inverse function to the tangent of the phase angle in the above equation is commonly written:
φ = arc tan(X_L/R) = tan^-1(X_L/R).
In general, the impedance Z has a magnitude equal to √(R² + X²) and a phase angle φ (the direction of Z relative to R), whose tangent is X/R. This can be written in the polar form Z∠φ as
Z = √(R² + X²)∠X/R.
When the reactance is inductive, that is, X = X_L, and φ is a positive angle whose tangent is X_L, the magnitude of the impedance is given by the expression: √(R² + X_L²).

In this polar expression, √(R² + X²) is the magnitude and the symbol ∠ is interpreted to mean "at an angle of —°".
Ohm's law for dc circuits can be generalized into a form that applies to ac circuits. For the entire ac circuit,
E = IZ,
and for external circuit,
V = IZ.
The relation between the current I in the external circuit and the voltage V across it may be written as
I = VY,
where Y is called admitance and is defined as the reciprocal of impedance, that is,
Y = 1/Z,
so that,
V = IZ = I/Y.
Admittance is measured in mhos ("ohm" spelled backwards).

A. C. Circuit with Capitance only.
Let us now consider an ac circuit that consists of a capacitor connected across the terminals of an ac generator. The instantaneous voltage drop across the capacitor is v_C and applying Kirchhoff's loop rule to the circuit we get v - v_C = 0, or
v = v_C = V_m sin ωt. (18)
From the definition of capacitance, v_C = Q/C, which when substituted in equation (18), we get
Q = CV_m sin ωt. (20)
And since i = dQ/dt, differentiating equation (20) gives the instantaneous current:
i_C = dQ/dt = ωCV_m cos ωt. (21)
Again, we see that the current is not in phase with the voltage drop across the capacitor, given by equation (18). Using the trigonometric identity cos ωt = sin (ωt + π/2), equation (21) can be written
i_C = ωCV_m sin(ωt + π/2). (22)
Comparing this equation with equation (18), we see that the current is 90° out of phase with the voltage across the capacitor. A plot of the voltage and current versus time shows that the current reaches its maximum value at a time that is one quarter of its cycle sooner the voltage reaches its maximum value. Thus we see that for a sinusoidal applied emf, the current always leads the voltage across the capacitor by 90°.
From equation (22), we see that the maximum current in the circuit is given by
I_m = ωCV_m (V_m/X_C), (23)
where the quantity X_C is called the capacitive reactance and is defined as
X_C = 1/ωC. (24)
The rms current given by equations similar to equation (23) with V_m is replaced by V_rms.
Using equations (18) and (24), the instantaneous voltage drop across the capacitor can be found by following equation:
v_C = V_m sin ωt = I_mX_C sin ωt. (25)
This equation can be considered as Ohm's law for an capacitive circuit. That is,
V_m = I_mX_C. (26)
The unit for X_C is also the ohm. As the frequency of the circuit increases, capacitive reactance decreases. For a given maximum applied voltage, V_m, the current increases as the frequency increases. As the frequency approaches zero, the capacitive reactance approaches infinity. Therefore, the current approaches zero. This makes sense since the circuit approaches dc conditions as ω → 0. Of course, a capacitor passes no current under steady-state conditions.

Capactance and Resistance.
In an ac circuit the opposition to the current due to capacitance is nonresistive since no power is dissipated in the capacitor. This nonresistive opposition is called capactive reactance. The symbol for capacitive reactance is X_C and it is measured in ohms.

The inherent resistance in any real circuit containing capacitance can be represented as in series with X_C. The potential difference V across the external circuit is the vector sum of V_C and V_R; the series current is used as the reference. There is, of course, no electrons flowing through the capacitance; the capacitor is alternately charged, first in one sense and then in the other, as the changing current alternates. The phase angle φ is negative, which is characteristic of a lagging voltage.

The corresponding phasor diagram for a capacitve circuit can be produced by dividing each voltage phasor by the circuit current. Just as V_C is negative with respect to V_L, so X_C is plotted opposite to the direction of X_L.

Because V_C = Q/C, the higher the capactance the lower is the potential difference V_C that develops across the capacitor for a given charge Q. Thus the capacitive reactance in an ac circuit varies inversely with the capacitance in the circuit. As the frequency of a changing current is increased, a shorter time is available during each current cycle for the capacitor to charge and discharge. Smaller changes in voltage occur across the capacitor. Hence the capacitive reactance to the circuit current varies inversely with the current frequency. That is,
X_C = 1/(2πfC),
where f is measured in hertz, C in farads, the capacitance X_C is measured in ohms.

The impedance of a circuit, Z, containing capacitance and resistance in series, has a magnitude equal to √(R² + X_c²) and a phase angle, φ, whose tangent is -X_C/R.

When a capacitor in a dc circuit is charged to the applied voltage, it acts as an open switch and the current in the circuit is zero. While the capacitor remains in fully charged state, the capacitive reactance in the circuit is infinitely high regardless of the value of C. A capacitor in an ac circuit has a very different effect on the alternating current and voltage. Electrons flow in one direction during on half-cycle, charging the capacitor in one sense. The electron flow is in the opposite direction during the other half-cycle, reversing the charge on the capacitor. If the current has very low frequency, the capacitive reactance is very large, since
X_C = 1/(2πfC).
This large reactance effectively limits the circuit current. At higher frequencies, less change occurs in the charge on the capacitor during each half-cycle and the reactive opposition to the circuit current is lower. At extremely high frequencies, the change in capacitor charge becomes negligible and the capacitive reactance in the circuit becomes negligible factor in limiting the circuit current. The larger the capacitor in the circuit, the more rapidly the capacitive reactance decreases with increasing frequency. Note that in the above equation for X_C, the magnitude of X_C varies inversely with frequency f, with capacitance C, and with their product fC.

The RLC Series Circuit.
In the previous section, we have examined the effects that an inductor, a capacitor, and a resistor have when they are placed separately across the terminals of an ac voltage source. Now we shall consider what happens when they are placed in combinations across the ac source. This is a more realistic situation, because resistance always occurs in the external ciruit and/or in the internal circuit of the ac source.

We shall consider the circuit in which the resistor, inductor, and capacitor are connected in series across an ac-voltage source. Let us assume that the sinsusoidal current in the circuit has reached a steady-state value. As before, the applied voltage is assumed to vary sinusoidally with time. We will assumed that the applied voltage is
v = V_m sin ωt,
while the current varies as
i = I_m sin(ωt - φ),
where the quantity φ is called the phase angle between the current and the applied voltage. Our problem is to determine φ and I_m.

Note that since the circuit elements are in series, the current everywhere in the circuit must be same at any instant. That is, the ac current at all points in a series circuit must be the same amplitude and phase. The voltage across the different elements will have different amplitude and phases. In particular, the voltage across the inductor leads the current by 90°, and the voltage across the capacitor lags behind the current by 90°. Thus the instantaneous voltages across the three elements with their phase angles are
v_R = I_mR sin ωt = V_R sin ωt, (27)
v_L = I_mX_L sin(ωt + π/2) = V_L cos ωt, (28)
v_C = I_mX_C sin(ωt - π/2) = -V_C cos ωt, (29)
where V_R, V_L, and V_C are the maximum voltages across each element, given by
V_R = I_mR, (30)
V_L = I_mX_L, (31)
V_C = I_mX_C, (32)
so that the instantaneous voltage across the three elements is equal to the sum of the instantaneous voltage drops across the three circuit elements, R, L, and C, that is,
v = v_R + v_L + v_C. (33)
Assuming a sinusoidal impressed voltage, we can write equation (33) as
V_m sin (ωt - φ) = Ri + L(di/dt + q/C),
or, since i = dq/dt, we get
V_m sin (ωt - φ) = L(d²q/dt²) + R(dq/dt) + q/C,
The solution of this linear second order differential equation can be found similar to the solution of an equivalent differential equation for a forced mechanical oscillator. In the solution q is a function of time, and knowing q as a function of time, then the current as a function of time can be found.

Although this analytical approach can be used, it is simpler to obtain the sum by examining the rotating phasor diagram of maximum voltages. Because the current in each element is the same at any instant, the diagram of each phasor can be combined into a single diagram, where a single phasor I_m is used to represent the common current in each element. To obtain the vector sum of these voltages, the diagram must be redrawn as a right triangle where the phasor of the resistance element V_R, which is in phase with the current phasor I_m, is drawn along the horizontal leg and the phasors for V_L and V_C are drawn along the vertical leg of the right triangle. Since the voltage phasor V_L leads the current phasor I_m by 90° and V_C lags behind the current phasor by 90°, they can be drawn on the phasor diagram along the same vertial line. And since V_L and V_C are thus always along the same verticle line but in opposite directions, they can be combined in in a single phasor having the following magnitude:
V_X = V_L - V_C,
which is perpendicular to the phasor V_R. This single phasor V_X is represented as along the other leg of the right triangle. Thus the total voltage V_m is the hypotenuse of the right triangle, whose sides are V_R and V_X = V_L - V_C, and makes an phase angle φ with the current phasor I_m. Using the pythagorean theorem, we get
V_m = √[V_R² + (V_L - V_C)²] = √[(I_mR)² + (I_mX_L - I_mX_C)²], or
V_m = I_m √[R² + (X_L - X_C)²], (34)
where inductive reactance X_L = ωL and capactive reactance X_C = 1/ωC.
Therefore, solving equation (34) for I_m, the maximum current is
I_m = V_m/Z,
where Z is called the impedance of the circuit and is defined as
Z = √[R² + (X_L - X_C)²] = √[R² + X²], (35)
where X = X_L - X_C is called the total reactance of the circuit and may be either positive or negative (X² is always positive).
Therefore, equation (34) can be written as
V_m = I_mZ. (36)
Impediance also is measured in ohms. Equation (36) may be regarded as a generalization of Ohm's law as applied to an ac circuit. Note that the current in the circuit depends upon the resistance, the inductance, the capacitance, and the frequency since the reactances are frequency dependent.
By removing the common factor I_m from the our redrawn phasor diagram, we can construct an impedance triangle. From the redrawn phasor diagram, we see that the phase angle φ between the current and voltage is
tan φ = (V_L - V_C)/V_R = (IX_L - IX_C)/IR = X/R,
or
tan φ = (X_L - X_C)/R = X/R. (37)
For example, when X_L > X_C (which occurs at high frequencies), the phase angle is positive, signifying that the current lags behind the applied voltage. On the othe hand, X_L < X_C, the phase angle is negative, signifying that the current leads the applied voltage. Finally, when X_L = X_C the phase angle is zero. In this case, the ac impedance equals the resistance and the current has its maximum value, given by V_m/R. The frequency at which this occurs, is called the resonance frequency.

Series Resonance.
In a series RLC circuit the voltages across an inductance and a capacitance are of opposite polarity. If X_L is larger than X_C, the circuit is inductive and the voltage across the circuit leads the current. If X_C is the larger reactance, the circuit is capacitive and the circuit voltage lags behind the current. In either instance, the impedance, Z, has a magnitude
√[R² + (X_L - X_C)²]
and a phase angle, φ, whose tangent is
(X_L - X_C)/R.

The frequency of the ac voltage applied to a series circuit with fixed values of L, R, and C determines the reactance of the circuit. The reactance of the circuit will be inductive over a range of high frequencies and capacitive over a range of lower frequencies. At some intermediate frequency, the inductive and capacitive reactances will be equal; then the reactance of the circuit X = X_L - X_C, equals zero. At this frequency, the impedance, Z, is at its minimum value for the series circuit. It is equal to the inherent circuit resistance, R. As is true of all circuits with pure ohmic resistance, the voltage across the circuit and the current in the circuit will be in phase. This special circuit condition is known as series resonance. The frequency at which resonance occurs is called the resonant frequency, f₀. Series resonance is a condition in which the impedance of an RLC series circuit is equal to the circuit resistance, and the voltage across the circuit is in phase with circuit current. That is, at resonance, Z = R since X_L = X_C.
To find the resonant frequency, f₀, solve this last relation for ω₀, since X_L = ωL and X_C = 1/ωC. That is,
ω₀L = 1/(ω₀C), or ω₀² = 1/(LC), or ω₀ = 1/√(LC).
Since ω = 2πf, we get
f₀ = 1/[2π√(LC)].

At frequencies below the resonance point, the series circuit is capacitive, performing as one with an X_C and R in series. At the resonant frequency, f₀, the circuit performs as one containing pure resistance. Above the resonant frequency, it is inductive, performing as one with an X_L and R in series.

The current in a series RLC circuit reaches its maximum value when the frequency of the generator reaches ω₀; that is, when the "driving" frequency matches the resonance frequency.

Selectivity in Series Resonance.
A resonant circuit responds to impressed voltages of different frequencies in a selective manner. The circuit presents a high impedance to a signal voltage of a frequency above or below the resonant frequency of circuit. The resulting current is correspondingly small. The same circuit presents a low impedance, consisting of the circuit resistance, to a signal voltage at the resonant frequency. The current is correspondingly large.

The principle of resonance is widely used in electronic as a "tuned" circuit to select a particular frequency from a band of frequencies that may be impressed on a circuit. A tuned circuit thus discrimates among signl voltages of different frequencies. This property of signal discrimation is known as the circuit's selectivity. The lower the resistance of the series circuit, the higher the resonant current and more sharply selective is the circuit. The usual circuits used for signal discrimation consists of an inductor and a capacitor. The only resistance is that inherent in the circuit, which is almost all entirely the resistance of the coil of the inductor. The characteristics of the series resonant circuit depends primarly on the ratio of the inductive reactance to the circuit resistance, X_L/R. That is,
Q₀ ≈ ω₀L/R.
This ratio is called the Q, or quality factor, of the circuit. It is essentially a design characteristic of the inductor. The higher the Q, the more sharply selective is the series resonant circuit.

Resonant circuits can be tuned over a range of frequencies by making either the inductance or the capacitance variable. The position of a powdered-iron core in a coil can be varied to change the coil's inductance. A capacitor can be made variable by providing a means of changing the plate area or the plate separation. A variable capacitor of the type used in the tuning circuits of small radios uses a set of rotating plates which move between a set of stationary plates. The rotor plates are attached to a shaft by which they can rotated between the stator plates, varying the amount of effective plate area of the capacitor. Resonance is particularly important in communication circuits.

Power in an AC Circuit.
In a series RLC circuit, the instantaneous power is
p = iv = [I_m sin (ωt - φ)][V_m sin ωt] = I_mV_m sin ωt sin (ωt - φ)
Let us find the average power of p over one or more cycles. Using the trigonometic identity,
sin (ωt - φ) = sin ωt cos φ - cos ωt sin φ,
and substituting this into the previous equation, we get
p = I_mV_m sin ωt cos φ - I_mV_m cos ωt sin φ.
Now let us take the time average of p over one or more cycles, noting that I_m, V_m, φ and ω are constants. The time average of the first term on the right involves the average value of sin ωt which is 1/2. The time average of the second term on the right is identically zero, because
sin ωt cos ωt = (1/2)sin 2ωt, whose average value is zero.
Therefore, the aveage power is
P_av = (1/2)I_mV_m cos φ.
This can be expressed in terms of the rms voltage and the rms current defined by equations (8) and (9) above. Thus we get
P_av = I_rmsV_rms cos φ,
where the quantity cos φ is called the power factor.
The maximum voltage drop across the resistor is
V_R = I_m cos φ = I_mR, or
cos φ = I_mR/V_m.
Using this expression for cos φ and equations (8) and (9),
we get an equivalent expression for the average power:
P_av = I_rms²R.
The average power delivered by the generator to a series RLC circuit is dissipated as heat in the resistor. There is no power loss in an ideal inductor or capacitor.
The rms current (or current amplitude) for a series RLC circuit is
I_rms = V_rms/√[R² + (X_L - X_C)²], where V is the rms applied voltage.

The Transformer and Power Tranmission.
A transformer is a device designed to raise (or lower) an ac voltage and current without causing an appreciable change in the product of the current and voltage, IV. In its simplest form, it consists of a primary coil of N₁ turns and a secondary coil of N₂ turns, both wound on a common soft iron core. When a voltage V₁ is applied to the primary, the voltage V₂ across the secondary is given by
V₂ = (N₂/N₁)V₁.
When the N₂ is greater than N₁, the output voltage V₂ will be greater than the input voltage V₁. The transformer is referred to as step-up transformer. When the N₂ is less than N₁, the output voltage V₂ will be less than the input voltage V₁. The transformer is referred to as step-down transformer.

An ideal transformer is one in which the energy losses in the transformer windings and core can be neglected. In an ideal transformer, the power supplied by the generator I₁V₁ is equal to the power in the secondary circuit, I₂V₂. That is,
I₁V₁ = I₂V₂.
In an ideal transformer, the power delivered by the generator must equal the power dissipated at the load. If a load resistor R is connected across the secondary coil, this means that
I₁V₁ = I₂V₂ = V₂²/R.
In a real transformer, the power in the secondary is typically between 90% and 99% of the primary power. The energy losses are due mainly to hysteresis losses in the transformer core, and thermal energy losses from currents induced in the core and the coil winding themselves. Power losses due to hysteresis occur because the core is magnetized cyclically.

Resistance, Inductance, and Capacitance in Parallel.
An A.C. parallel circuit is defined by the characteristic that
The same voltage occurs across all parallel branches.
Thus in preparing a phasor diagram for any parallel circuit, the common voltage is used as the reference phasor. If we are given the numerical value of this voltage, then the currents in the individual branches can be determined by Ohm's law. Since
I_R = (E∠0°)/(R∠0°),
then I_R is in phase with the reference phasor.
Since
I_L = (E∠0°)/(X_L∠+90°),
then I_L lags the reference voltage by 90°.
Since
I_C = (E∠0°)/(X_C∠-90°),
then I_C leads the reference voltage by 90°.
The total current in a parallel circuit then becomes the vector sum of branch currents. That is, in complex notation,
I_T = I_R - jI_L + jI_C = I_R - j(I_L - I_C). (38)
Converting this equation to polar coordinates, we get
I_T = √(I_R² + I_X²) ∠tan^-1(I_X/I_R),
where I_X = I_L - I_C is called the net reactive current. Let us now determine the equivalent impedance of the parallel circuit by using Ohm's law for ac circuits,
Z_e = E/I_T. (39)
This method for solving for equivalent impedance of a parallel ac circuit is called the total current method. Even if the exact value of the applied emf is not known, any convenient value may be assumed to solve for the calculation of equivalent impedance.
First step. Calculate the value of the impedances, if they are not known, from the frequency and the value of the inductance L and capacitance C.
X_L = ωL = 2πfL, and
X_C = 1/ωC = 1/2πfC.
Second step. Assuming a convenient value of the applied emf, compute the value the current in each branch by one of the following formulas.
I_R = E/R,
I_L = E/X_L,
I_C = E/X_C,
Third Step. Using equation (38), compute the total current I.
Fourth Step. Using equation (39), compute the magnitude of the equivalent impedance.
Fifth step. Calculate the phase angle φ between the total current and voltage from
φ = arc tan[(I_X)/(I_R)].

Another method computing the equivalent impedance, which does not require the computation of the total current, makes use of the quantities called conductance and admittance. These quantities are the reciprocal of the resistance and impedance, respectively. Conductance is a measure of ability of a circuit of pure resistance to pass electric current and is the reciprocal of resistance; that is,
G = 1/R,
where G is the symbol of conductance and is measured in mhos.
Admittance is a measure of the over-all ability of ac circuit to pass an electric current and is the reciprocal of the impedance of an ac circuit; that is,
Y = 1/Z.
where Y is the symbol of admittance and is measured in mhos.
Equation (38) can be written as
E/Z = E/R - j[(E/X_L) - (E/X_C)].
And since the voltage across each component in a simple parallel circuit is the same, then voltage can be divided out of the equation and it becomes
1/Z = 1/R - j[(1/X_L) - (1/X_C)], or
Y = G - j[B_L - B_C],
where is B_L and B_C are the reciprocal of the inductive reactance and capacitive reactance, respectively, and is called susceptance. Susceptance is the ability of a pure inductance or pure capacitance to pass an alternating current. Note that when a vector quantity with a +90° angle is divided into 1∠0°, the quotient has a -90° angle and vice versa. Therefore, although inductive reactance is a +j quantity, inductive susceptance is a -j quantity. Similarly, capacitive susceptance is a +j quantity. Admittance can be expressed in polar coordinates as
Y = √(G² + B²)∠arc tan(B/G),
where B = B_C - B_L and is called net equivalent susceptance.

This method for solving for equivalent impedance of a parallel ac circuit is called the admittance method. The following are the steps of this method.
First step. Convert all resistances R's to conductances G's and sum all the conductances.
Second step. Convert all inductive reactances X_L to inductive susceptances B_L and sum all the inductive susceptances obtaining B_L.
Third step. Convert all capacitive reactances X_C to capacitive susceptances B_C and sum all the capacitive susceptances obtaining B_C.
Fourth step. Calculate the net equivalent susceptance B by combining the sums of inductive and capacitive susceptances. The inductive susceptances B_L are considered negative.
Fifth step. Calculate the magnitude of the admittance Y from
Y = √(G² + B²),
and, if needed, the magnitude of total impediance Z using
Z = 1/Y.
Sixth step. Calculate the phase angle φ between the total current and voltage from
φ = arc tan(B/G).

Parallel Resonance.
When inductance L and capacitance C are in parallel and the inductive reactance X_L is equal to the capacitive reactance X_C, the parallel circuit is in resonance, and the reactive branch currents, I_L and I_C, are equal and opposite, canceling each other to produce a minimum current in the main line. Since the line current is minimum, the impedance is maximum. These relations are based on the resistance R_s in series with the inductance, being very small compared to X_L at resonance.

Minimum Line Current at Parallel Resonance.
Since this is a parallel ac circuit, the capacitive current leads by 90° while the inductive current lags by 90°, compared with the reference angle of the generator voltage, which is applied across the parallel branches. Therefore, the opposite currents are 180° out of phase, canceling each other in the main line. The net current I in the line is, then, the difference between the currents in the branches. As the frequency increases toward resonance, the capacitive branch current increases because of the lower value of X_C, while the inductive branch current decreases with the higher value of X_L. As a result, there is less net line current as the two branch currents become nearly equal. At the resonant frequency, both reactance are equal and the branch currents are equal, canceling each other completely. Above the resonant frequency, there is more current in the capacitive branch than in the inductive branch and the net line current increases above the minimum value at resonance.

The in-phase current due to R_s in the inductive branch can be ignored off resonance because it is so small compared with the reactive line current. But at the resonant frequency, when the reactive currents cancel, the resistive component is the entire line current. Its value at resonance is small and is the minimum value of the line current at parallel resonance.

Maximum Line Impedance at Parallel Resonance.
The minimum line current resulting from parallel resonance is uselful because it corresponds to maximum impedance in the line across the ac generator. Therefore, an impedance that is high value for just one frequency but low impedance at other frequencies, either below or above resonance, can be obtained by using a parallel LC circuit at the desired frequency. This is another way of selecting one frequency by resonance. The response curve of line impediance versus frequency shows how the impedance rises to a maximum for parallel resonance.

The total impedance of the parallel ac circuit is calculated by dividing the generator voltage by the total line current. That is,
Z_T = E/I_T.
If a meter is inserted in series with the main line to indicate the total line current I_T, it will dip sharply to the minimum value of the line current at the resonant frequency. With the minimum current in the line, the impedance across the line is maximum at the resonant frequency. The maximum impedance at parallel resonance corresponds to a high value of resistance, without reactance, since the line current is then resistive with zero phase angle.