Chapter 24: Application of Complex Numbers to Series AC Circuits
24.1 Introduction
AC circuits may be analysed by using complex numbers for simplified phasor diagrams.[cite: 1040-1042] This can solve complicated circuits.[cite: 1042]
24.2 Series AC Circuits
Pure Resistance
In pure resistance the circuit in polar equation is given by:[cite: 1043-1044]
Pure Inductance
In pure inductance the current lags the applied voltage by 90°.[cite: 1055-1059]
Where XL is inductive reactance, XL = ωL = 2πfL ohms.[cite: 1065]
Pure Capacitance
The voltage lags the current by 90° in this circuit.[cite: 1075-1076]
Where XC is 1 / ωC.[cite: 1092-1093] A notable equation converts the imaginary representation:[cite: 1094]
R-L Series Circuit
The current is said to be lagging even though the phase is +90° as the current is behind the voltage.[cite: 1123-1125]
Z = R + jXL[cite: 1117]
R-C Series Circuit
The relations for the voltage and impedance triangles are:[cite: 1142-1143]
Z = R – jXC[cite: 1162]
R-L-C Series Circuit
The voltage and total impedance are defined as:[cite: 1159, 1177]
|Z| = √(R2 + (XL – XC)2), φ = tan-1((XL – XC) / R)[cite: 1177]
Z = R + j(XL – XC) = |Z|∠φ[cite: 1178]
General Series Circuit
In an a.c. circuit containing several impedances connected in series, say Z1, Z2, Z3 … Zn, the total equivalent impedance is given by:[cite: 1179-1184]
Chapter 25.3: Parallel AC Networks
For a circuit containing parallel impedances Z1, Z2 and Z3, the potential difference is the same and equal to the supply voltage V.[cite: 1187-1191]
If ZT is the total impedance, then:[cite: 1199]
In general for impedances connected in parallel, the total admittance YT is:[cite: 1201]
For Two Impedances Connected in Parallel
Some useful current equations:[cite: 1208]
I1 = I(Z2 / (Z1 + Z2))[cite: 1211]
I2 = I(Z1 / (Z1 + Z2))[cite: 1208]