Inductive Circuits and Using Kirchhoff’s Current Law
Most people recognize the name of H.G. Wells but few can say what Edward Page Mitchell accomplished. Regardless of his name recognition, Edward Page Mitchell has a place in history as the first author to write about a machine that enabled time travel. In Mitchell’s short story “The Clock That Went Backward,” a professor and two cousins not only encounter a grandfather clock that causes time to flow backwards but also become subject to a time paradox.
Good Times Are Coming (for Inductive Circuits)!
If we could take a time machine and travel back to our entry into the world of electronics, we might choose to stop at inductive circuits. While inductors are passive circuit elements, the components provide a very necessary function by storing energy in a magnetic field adjacent to the coils of the inductor. Because of that functionality, inductors work within LC tank circuits that assist with processing radio frequency (RF) and intermediate frequency (IF) signals. Inductors also couple IF amplifier inter-stages in receivers and serve as a load impedance for RF amplifiers.
As we consider those applications, let’s step back for a moment and review a few basic principles that affect the operation of inductive circuits.
The flow of current through a wire produces a magnetic field.
According to the right-hand rule, the magnetic field circles the wire.
Tightly winding a wire in a helix allows flux linkages to exist between the loops.
The combination of voltage across the inductor, current through the inductor, and the value of inductance controls the behavior of the inductor in a circuit.
The current flowing through an inductor must charge up over time and cannot change instantly.
It’s Just a Phase
Any circuit energized by a single ac voltage is a single-phase circuit. With just a voltage source and a resistance in the circuit, current remains in phase with the voltage and the ratio of voltage to current equals the resistance. Replacing the resistance with an inductance changes everything. If a single-phase circuit contains only an inductance, the resulting current leads the voltage by exactly 90o and the ratio of voltage to current equals the reactance.
When we study inductive circuits, we can use Kirchoff’s Voltage and Current Laws (KVL and KCL) to analyze closed loop behavior in AC circuits. Since the laws apply at any instant of time in an AC circuit if we account for the voltage and current magnitudes as well as phase relationships, we can express those as instantaneous equations or:
KVL = ∑v = 0
KCL = ∑i = 0
For sinusoidal voltages in phasor form, the phasor sum of all voltages around a closed loop equals zero.
Let’s go a step further with the basic principles. When we place inductors in series and in parallel, we can find an equivalent inductance. In the following parallel combination of inductors, the same voltage applies across all the inductors.
Using Kirchoff’s Current Law, we also find that inductors in parallel combine like resistors in parallel or:
i = 1/L1 + 1/L2 + 1/L3 + … + 1/Ln
Using Kirchhoff's Voltage Law around the loop, we see that a series combination of inductors combines the same as resistors in series or:
v = vl + v2 + v3 + … + vn
Circuit Membership is Open
We can use a simple RLC circuit, an applied sinusoidal voltage, and Kirchoff’s Voltage Law to illustrate the behavior of the circuit.
The voltage across the resistor equals the product of the current multiplied by the resistance (IR) and remains in phase with the current. Because the circuit has a combination of resistance and reactance, the total opposition to current is the impedance (Z). Impedance is a ratio of the phasor values of volts to the phasor value of amperes and measures in ohms or:
Z = V/I
with V and I representing the phasor values.
The voltage across the inductor equals the product of the current multiplied by the reactance presented by the inductor or IXL. In turn, the voltage across the capacitor equals the product of the current multiplied by reactance presented by the capacitor or IXC. Because the voltage across the inductor leads the current by 90o and the voltage across the capacitor lags the current by 90o, the inductive and capacitive voltages are out of phase with each other by 180o.
The total impedance or Zt equals:
R + j(XL - XC) = R + jXt
Xt represents the total reactance. As we analyze circuits, we express a phasor as the sum of horizontal and vertical components. To distinguish between the components, the symbol j signifies the vertical component. The j operator--or multiplying factor--used in the equation indicates that the phasor quantity rotates through a 90o counterclockwise rotation.
If the reactance presented by the inductor is greater than the reactance presented by the capacitor, we have an inductive circuit. The voltage leads the current and the circuit appears electrically equivalent to a single resistor in series with the inductor.
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