The 10 Percent Rule: This rule is a standard method for selecting R 1 and R 2 that takes into account
2.27 Circuit with Sinusoidal Sources
arg(Z) (or phase θ) represents the argument of Z or the phase angle (θ). For example, if Z = 3 + j4, then:
Re Z = 3 Im Z = 4 |Z| = (3)2+(4)2 =5 arg(Z) = arctan 4 3
= 53.1°
This alternative approach makes use of what are called complex impedances—some- thing that will use the complex numbers we talked about in the last section.
2.27.1 Analyzing Sinusoidal Circuits with Complex Impedances To make solving sinusoidal circuits easier, it’s possible to use a technique that enables you to treat capacitors and inductors like special kinds of resistors. After that, you can analyze any circuit containing resistors, capacitors, and inductors as a “resis- tor” circuit. By doing so, you can apply all the dc circuit laws and theorems that were presented earlier. The theory behind how the technique works is a bit techni- cal, even though the act of applying it is not hard at all. For this reason, if you do not have the time to learn the theory, we suggest simply breezing through this section and pulling out the important results. Here’s a look at the theory behind complex impedances.
In a complex, linear, sinusoidally driven circuit, all currents and voltages within the circuit will be sinusoidal in nature after all transients have died out. These cur- rents and voltages will be changing with the same frequency as the source voltage (the physics makes this so), and their magnitudes will be proportional to the mag- nitude of the source voltage at any particular moment in time. The phase of the cur- rent and voltage patterns throughout the circuit, however, most likely will be shifted relative to the source voltage pattern. This behavior is a result of the capacitive and inductive effects brought on by the capacitors and inductors.
As you can see, there is a pattern within the circuit. By using the fact that the voltages and currents will be sinusoidal everywhere, and considering that the fre- quencies of these voltages and currents will all be the same, you can come up with a mathematical trick to analyze the circuit—one that avoids differential equations.
The trick involves using what is called the superposition theorem. The superposi- tion theorem says that the current that exists in a branch of a linear circuit that con- tains several sinusoidal sources is equal to the sum of the currents produced by each source independently. The proof of the superposition theorem follows directly from the fact that Kirchhoff’s laws applied to linear circuits always result in a set of linear equations that can be reduced to a single linear equation with a single unknown.
The unknown branch current thus can be written as a linear superposition of each of the source terms with an appropriate coefficient. (Figure 2.158 shows the essence of superimposing of sine waves.)
What this all means is that you do not have to go to the trouble of calculating the time dependence of the unknown current or voltage within the circuit because you know that it will always be of the form cos (ωt + φ). Instead, all you need to do is calculate the peak value (or RMS value) and the phase, and apply the superposition theorem. To represent currents and voltages and apply the superposition theorem, it would seem obvious to use sine or cosine functions to account for magnitude, phase, and frequency. However, in the mathematical process of superimposing (adding, multiplying, etc.), you would get messy sinusoidal expressions in terms of sines and cosines that would require difficult trigonometric rules and identities to convert the answers into something you could understand. Instead, what you can do to represent amplitudes and phase of voltages and currents in a circuit is to use complex numbers.
Recall from the section on complex numbers that a complex number exhibits sinu- soidal behavior—at least in the complex plane. For example, the trigonometric form of a complex number z1 = r1 cos θ1 + jr1 sin θ1 will trace out a circular path in the complex plane when θ runs from 0 to 360°, or from 0 to 2π radians. If you graph the real part of z versus θ, you get a sinusoidal wave pattern. To change the amplitude of the wave pattern, you simply change the value of r. To set the frequency, you simply multiply θ by some number. To induce a phase shift relative to another wave pattern of the same frequency, you simply add some number (in degrees or radians) to θ. If you replace θ with ωt, where (ω = 2πf), replace the r with V0, and leave a place for a term to be added to ωt (a place for phase shifts), you come up with an expression for the voltage source in terms of complex numbers. You could do the same sort of thing for currents, too.
Now, the nice thing about complex numbers, as compared with sinusoidal func- tions, is that you can represent a complex number in various ways, within rectangular, polar- trigonometric, or polar- exponential forms (standard or shorthand versions).
Having these different options makes the mathematics involved in the superimpos- ing process easier. For example, by converting a number, say, into rectangular form, you can easily add or subtract terms. By converting the number into polar- exponential
FIGURE 2.158 (a) Shows two sine waves and the resulting sum—another sine wave of the same frequency, but shifted in phase and amplitude. This is the key feature that makes it easy to deal with sinusoidally driven linear circuits containing resistors, capacitors, and inductors. Note that if you were to try this with sine waves of different frequencies, as shown in (b), the resultant waveform isn’t sinusoidal. Superimposing nonsinusoidal waveforms of the same frequency, such as squarewaves, isn’t guaranteed to result in a similar waveform, as shown in (c).
form (or shorthand form), you can easily multiply and divide terms (terms in the exponent will simply add or subtract).
It should be noted that, in reality, currents and voltages are always real; there is no such thing as an imaginary current or voltage. But then why are there imaginary parts? The answer is that when you start expressing currents and voltages with real and imaginary parts, you are simply introducing a mechanism for keeping track of the phase. (The complex part is like a hidden part within a machine; its function does not show up externally but does indeed affect the external output—the “real,” or important, part, as it were.) What this means is that the final answer (the result of the superimposing) always must be converted back into a real quantity. This means that after all the calculations are done, you must convert the complex result into either polar- trigonometric or polar- exponential (shorthand form) and remove the imagi- nary part. For example, if you come across a resultant voltage expressed in the fol- lowing complex form:
V(t) = 5 V + j 10 V
where the voltages are RMS, we get a meaningful real result by finding the mag- nitude, which we can do simply by converting the complex expression into polar exponential or shorthand form:
ej
(5.0 V)2+(10.0 V)2 (63.4 )° = (11.2 V) ej(63.5°) = 11.2 V ∠ 63.5°
Whatever is going on, be it reactive or resistive effects, there is really 11.2 V RMS present. If the result is a final calculation, the phase often isn’t important for practical purposes, so it is often ignored.
You may be scratching your head now and saying, “How do I really do the super- imposing and such? This all seems too abstract or wishy- washy. How do I actually account for the resistors, capacitors, and inductors in the grand scheme of things?”
Perhaps the best way to avoid this wishy- washiness is to begin by taking a sinusoidal voltage and converting it into a complex number representation. After that, you can apply it individually across a resistor, a capacitor, and then an inductor to see what you get. Important new ideas and concrete analysis techniques will surface in the process.
2.27.2 Sinusoidal Voltage Source in Complex Notation
Let’s start by taking the following expression for a sinusoidal voltage:
V0 cos (ωt) (ω = 2πf) and converting it into a polar- trigonometric expression:
V0 cos (ωt) + jV0 sin (ωt)
What about the jV0 sin (ωt) term? It is imaginary and does not have any physi- cal meaning, so it does not affect the real voltage expression (you need it, how- ever, for the superimposing process). To help with the calculations that follow, the
polar-trigonometric form is converted into the polar- exponential form using Euler’s relation ejθ = r cos (θ) + jr sin (θ):
V0ej(ωt) (2.67)
In polar- exponential shorthand form, this would be:
V0 ∠ (ωt) (2.68)
Graphically, you can represent this voltage as a vector rotating counterclockwise with angular frequency ω in the complex plane (recall that ω = dθ/dt, where ω = 2πf).
The length of the vector represents the maximum value of V—namely, V0—while the projection of the vector onto the real axis represents the real part, or the instantaneous value of V, and the projection of the vector onto the imaginary axis represents the imaginary part of V.
Now that you have an expression for the voltage in complex form, you can place, say, a resistor, a capacitor, or an inductor across the source and come up with a com- plex expression for the current through each component. To find the current through a resistor in complex form, you simply plug V0ej(ωt) into V in I = V/R. To find the capacitor current, you plug V0ej(ωt) into I = C dV/dt. Finally, to find the inductor cur- rent, you plug V0ej(ωt) into I = 1/L
∫
Vdt. The results are shown in Fig. 2.160.Comparing the phase difference between the current and voltage through and across each component, notice the following:
Resistor: The current and voltage are in phase, φ = 0°, as shown in the graph in Fig.
2.160. This behavior can also be modeled within the complex plane, where the voltage and current vectors are at the same angle with respect to each other, both of which rotate around counterclockwise at an angular frequency ω = 2πf.
Capacitor: The current is out of phase with the applied voltage by +90°. In other words, the current leads the voltage by 90°. By convention, unless otherwise stated, the
FIGURE 2.159
phase angle φ is referenced from the current vector to the voltage vector. If φ is positive, then current is leading (further counterclockwise in rotation); if φ is neg- ative, current is lagging (further clockwise in rotation).
Inductor: The current is out of phase with the applied voltage by −90°. In other words, the current lags the voltage by 90°.
We call the complex plane model, showing the magnitude and phase of the volt- ages and currents, a phasor diagram—where the term phasor implies phase compari- son. Unlike a time- dependent mathematical function, a phasor provides only a snap- shot of what’s going on. In other words, it only tells you the phase and amplitude at a particular moment in time.
Now comes the important trick to making ac analysis easy to deal with. If we take the voltage across each component and divide it by the current, we get the following (see Fig. 2.161):
FIGURE 2.160
As you can see, the nasty V0ej(ωt) terms cancel, giving us resistance, capacitive reac- tance, and inductive reactance in complex form. Notice that the resulting expressions are functions only of frequency, not of time. This is part of the trick to avoiding the nasty differential equations.
Now that we have a way of describing capacitive reactance and inductive reac- tance in terms of complex numbers, we can make an important assumption. We can now treat capacitors and inductors like frequency- sensitive resistors within sinusoi- dally driven circuits. These frequency- sensitive resistors take the place of normal resistors in dc circuit analysis. We must also replace dc sources with sinusoidal ones.
If all voltages, currents, resistances, and reactances are expressed in complex form when we are analyzing a circuit, when we plug them into the old circuit theorems (Ohm’s law, Kirchhoff’s law, Thevinin’s theorem, etc.) we will come up with equa- tions whose solutions are taken care of through the mathematical operations of the complex numbers themselves (the superposition theorem is built in).
For example, ac Ohm’s law looks like this:
V(ω) = I(ω) × Z(ω) (2.69)
What does the Z stand for? It’s referred to as complex impedance, which is a generic way of describing resistance to current flow, in complex form. The complex
FIGURE 2.161