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Physics · Ch 7 — Alternating Current

Reactance and Impedance

7.7

Reactance and Impedance

Reactance -- the opposition of LL or CC alone. Sections 7.5 and 7.6 each produced a relation of the Ohm's-law shape, i0=v0/(some constant)i_0=v_0/(\text{some constant}), for a pure inductor and a pure capacitor respectively. These constants are called reactances, and, exactly like resistance, are measured in ohms (Ω\Omega) -- but, unlike resistance, they represent an opposition to current that arises purely from the CHANGING nature of the current, not from any actual dissipation of energy as heat.

Inductive reactance.

XL=ωL=2πfLX_L = \omega L = 2\pi f L

XLX_L is directly PROPORTIONAL to frequency: at very low frequency (approaching DC, f→0f\to0), XL→0X_L\to0, so an inductor offers essentially no opposition to a slowly changing or steady current; at high frequency, XLX_L grows without bound, so an inductor increasingly BLOCKS rapidly alternating current. This is why an inductor is often called a 'choke' for high-frequency signals.

Capacitive reactance.

XC=1ωC=12πfCX_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}

XCX_C varies INVERSELY with frequency: at low frequency, XCX_C is very large, so a capacitor almost completely blocks a slowly changing current (and blocks true DC entirely, since f=0f=0 gives XC→∞X_C\to\infty); at high frequency, XC→0X_C\to0, so a capacitor offers almost no opposition to a rapidly alternating current, behaving almost like a plain wire. …

Table 1Opposition to current and phase angle in every circuit type covered in this chapter

| Circuit | Opposition to current | Phase angle ϕ\phi | Current vs voltage |

|---|---|---|

| Pure RR | Z=RZ=R | ϕ=0\phi=0 | In phase |

| Pure LL | Z=XL=ωLZ=X_L=\omega L | ϕ=π/2\phi=\pi/2 | Current lags |

| Pure CC | Z=XC=1/(ωC)Z=X_C=1/(\omega C) | ϕ=π/2\phi=\pi/2 | Current leads |

| Series LRLR | Z=R2+XL2Z=\sqrt{R^2+X_L^2} | tan⁡ϕ=XL/R\tan\phi=X_L/R | Current lags |

| Series CRCR | Z=R2+XC2Z=\sqrt{R^2+X_C^2} | tan⁡ϕ=XC/R\tan\phi=X_C/R | Current leads | …