Chemistry 564 Electronics Handout: Use of Complex Impedance for RLC Circuit Analysis
It is often easiest to use the complex number
representation of capacitive and inductive reactance in circuit
analysis. Consider the circuit below.
This circuit may represent the input of an instrument where the capacitance, C, is due to the proximity of conductors, and the inductance, L, is that of the conducting wires. The capacitive reactance is XC=-i/wC and the inductive reactance XL=iwL. i is the square-root of -1, and w=2pf is angular frequency. The circuit is composed of two parallel series sub-circuits. Reactance and resistance are summed in series circuits and the inverse rule is used for parallel circuits. The complex impedance in the series sub-circuit with the capacitor is
and that containing the inductor is
The total impedance is obtained from the reciprical sum relationship for parallel circuits
The real impedance (the length of the ZTOT vector) is obtained from the square-root of the complex conjugate product,
|ZTOT|=(ZTOT*ZTOT)1/2
where the * indicates the complex conjugate (replace i by -i). It is relatively straightforward to show that
The phase angle may be obtained from the arc-tangent relationship, ATAN[Im{ZTOT}/Re{ZTOT}], where Re and Im are used to indicate the real and imaginary parts of ZTOT.