The Sinusoidal Source
A sinusoidal voltage source (dependent or independent) produces a voltage that varies as a sine wave with time. A sinusoidal current source (dependent or independent) produces a current that varies with time. The sinusoidal varying function can be expressed either with the sine function or cosine function. Either works equally as well; both functional forms cannot be used simultaneously. Using the cosine function throughout this article, the sinusoidal varying voltage can be written as:To aid discussion of the parameters of the sinusoidal voltage equation, below is the Fig. 1.1 of the voltage versus time plot
A cycle per second is referred to as a hertz, or abbreviated as Hz. In the sinusoidal voltage equation, the coefficient t, contains the value of T or f. Omega (
This angular frequency equation is based on the fact that the cosine (or sine) function passes through a complete set of values each time its argument,
The angle
One more important characteristic of the sinusoidal voltage (or current) is its rms value. This value is defined as the square root of the mean value of the squared function. From Eq. 1.1, the rms value of
The quantity under the radical sign in Eq. 1.5 reduces to
This value of the sinusoidal voltage depends only on the maximum amplitude of
The Phasor
The phasor is a complex number that carries the amplitude and phase angle information of a sinusoidal function. This idea of a phasor is rooted in Euler's identity, which relates the exponential function to the trigonometric function:This equation is important because it gives another way of expressing the cosine and sine functions. The cosine function can be thought of as real part of the exponential function and the sine function as the imaginary part of the exponential function; this gives:
Where
v=VmR{ejωtejϕ}
The coefficient
v=R{Vmejϕejωt}
In Eq. 1.11, the quantity
This phasor transform transfers the sinusoidal function from the time domain to the complex-number domain, or frequency domain. One more thing to add regarding Eq. 1.12 is in order. The periodic occurrenceof the exponential function
Passive Circuit Elements in the Frequency Domain
First, a relationship must be made between the phasor current and phasor voltage at the terminals of the passive circuit elements. Second, a version of Kirchhoff's laws must be created pertaining to the phasor domain. From Ohm's law, if the current in the resistor varies as a sine wave with time, that is,Where
But
Eq. 1.16 is the relationship between phasor voltage and phasor current for a resistor, which states that the phasor voltage at the terminals of a resistor is simply the resistance times the phasor current. Figure 1.1 shows the circuit diagram for a resistor in the frequency domain.
V-I relationship for an Inductor
The relationship between the phasor current and phasor coltage at the terminals of an inductor can be derived by assuming a sinusoidal current and usingEq. 1.17 denotes that the phasor voltage at the terminals of an inductor equals
\textbf{V}=\omega LI_{m}\angle (\theta _{i} + 90)^{\circ} (1.18)
V-I relationship for a Capacitor
The relationship between the phasor current and phasor voltage at the terminals of a capacitor from the derivation of Eq. 1.17. Note that for a capacitor thatThe above equation denotes that the equivalent circuit for the capacitor in the phasor domain. The voltage across the terminals of a capacitor lags behind the current by exactly
Figure 1.3 shows the phase relationship between the current and voltage at the terminals (\theta _{i}=60^{\circ})
Impedance and Reactance
Concluding this discussion of passive circuit elements in the frequency domain with one more important observation. Comparing Eqs. 1.16, 1.17, and 1.19, are all of the formIn Eq. 1.21 Z denotes the impedance of the given circuit element. Solving for Z, it can be shown that the impedance is the ratio of a circuit element's voltage phasor to its current phasor. Noting that although impedance is a complex number, it is not a phasor. Recalling, a phasor is a complex number that shows up as the coefficient of
Impedance in the frequency domain is the quantity analogous to resistance, inductance, and capacitance in the time domain. The imaginary part of the impedance is called reactance. The values of impedance and reactance for each of the component values are summarized in Table 1.1.
Circuit Element | Impedance | Reactance |
Resistor | R | -- |
Inductor | ||
Capacitor |
Solving a circuit in the Frequency Domain
A 400 Hz sinusoidal voltage with a maximum amplitude of 100 V at t=0 is applied across the terminals of an inductor. The maximum amplitude of the steady-state current in the inductor is 20 A.A) What is the frequency of the inductor current?
The frequency of current in the inductor is also the same as the frequency of the voltage across the inductor terminals. Therefore, the frequency of the inductor current is 400 HzB) If the phase angle of the voltage is zero, what is the phase angle of the current?
The phase angle of inductor voltage:Since the current through the inductor lags behind the voltage by exactly
C) What is the inductive reactance of the inductor?
The phasor form of voltage applied across the inductor is,The phasor form of current in the inductor is,
Determining the impedance of the inductor:
The Expression for impedance of the inductor is:
Therefore, the inductive reactance,
D) What is the inductance of the inductor in millihenrys?
The expression for impedance of the inductor is:The expression for the inductor, L:
Substituting
Therefore, the inductance of the inductor in millihenrys is
E) What is the impedance of the inductor?
Therefore, the impedance of the inductor,
No comments:
Post a Comment