AcademyAlternating Current
Academy
Resistance and Reactance
Level 1 - Physics topic page in Alternating Current.
Principle
In AC circuits, resistors oppose current with resistance, while capacitors and inductors oppose current with frequency-dependent reactance.
Notation
\(R\)
resistance
\(\mathrm{\Omega}\)
\(X_C\)
capacitive reactance
\(\mathrm{\Omega}\)
\(X_L\)
inductive reactance
\(\mathrm{\Omega}\)
\(C\)
capacitance
\(\mathrm{F}\)
\(L\)
inductance
\(\mathrm{H}\)
\(\omega\)
angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(Z\)
impedance magnitude
\(\mathrm{\Omega}\)
Method
Derivation 1: Resistor phase
For a resistor, voltage and current are in phase. The ratio of rms voltage to rms current is \(R\).
Resistor
\[V_R=IR\]
Phase
\[\phi_R=0\]
Derivation 2: Capacitive reactance
For a capacitor, current leads voltage by \(90^\\circ\). The reactance decreases as frequency increases.
Capacitive reactance
\[X_C=\frac{1}{\omega C}\]
Capacitor rms relation
\[V_C=IX_C\]
Derivation 3: Inductive reactance
For an inductor, voltage leads current by \(90^\\circ\). The reactance increases as frequency increases.
Inductive reactance
\[X_L=\omega L\]
Inductor rms relation
\[V_L=IX_L\]
Rules
Resistor
\[V_R=IR\quad\text{in phase}\]
Capacitor
\[X_C=\frac{1}{\omega C},\qquad V_C=IX_C\]
Inductor
\[X_L=\omega L,\qquad V_L=IX_L\]
Frequency trend
\[X_C\downarrow\text{ as }\omega\uparrow,\qquad X_L\uparrow\text{ as }\omega\uparrow\]
Examples
Question
Find \(X_L\) for
\[L=0.20\,\mathrm H\]
at \[f=50\,\mathrm{Hz}\]
Answer
\[\omega=2\pi f=314\,\mathrm{rad\,s^{-1}}\]
\[X_L=\omega L=(314)(0.20)=62.8\,\Omega\]
Checks
- Reactance has units of ohms.
- Capacitors pass high frequencies more easily than low frequencies.
- Inductors oppose high-frequency changes more strongly.
- Do not add \(R\), \(X_L\), and \(X_C\) as simple scalar voltages unless they are in phase.