The Parallel RLC Circuit is the exact opposite to the series circuit we looked at in the previous tutorial although some of the previous concepts and equations still apply.

1. Series-Parallel
ac networks
R-L-C series
circuit

2. Series-Parallel ac networks: R-L-C series circuit
Series configuration
• Current is the same through each element
• Current is determined by Ohm’s law
Total impedance of a system is the sum of the individual impedances

3. Series-Parallel ac networks: R-L-C series circuit
Series configuration
Total impedance Current
Voltage across each element and
KVL or
Power θT is the phase angle between E and I

4. Series-Parallel ac networks: R-L series circuit
Resistance and Inductance (R-L) in series,
Z1 = R and Z2 = j XL = j ωL
Then, ZT = R + j XL = R + j ωL
So, the magnitude of the total impedance, ZT = √(R2
+ XL
2
)
and the phase angle, θT = tan-1 (XL/R)
Therefore, (Impedance)2
= (Resistance)2
+ (Reactance)2
and tan (θT) = (Reactance) / (Resistance)

5. Series-Parallel ac networks: R-C series circuit
For a series resistance and capacitance (R-C) circuit,
Z1 = R and Z2 = 1/(jXC) = -j/ωC
Then, ZT = R - j XC = R - j /ωC
So, the magnitude of the total impedance, ZT = √(R2
+ XC
2
)
and the phase angle, θT = tan-1 (-XC/R)
Therefore, (Impedance)2
= (Resistance)2
+ (Reactance)2
and tan (θT) = (- Reactance) / (Resistance)

6. Series-Parallel ac networks: R-L-C series circuit
Series resistance, inductance and capacitance (R-L-C) circuit,
Z1 = R, Z2 = j XL = j ωL and Z3 = 1/(jXC)
= -j/ωC
Then, ZT = R + j XL - j XC = R + j ωL - j /ωC
= R + j (XL - XC) = R + j (ωL - 1/ωC)
So, the magnitude of the total impedance, ZT = √(R2
+ (XL -
XC)2
and the phase angle, θT = tan-1 [(XL – XC)/R]
Therefore, (Impedance)2
= (Resistance)2
+ (Total Reactance)2

7. Series-Parallel ac networks: R-L-C series circuit
Ex: Determine the input impedance to the series network of Fig. 12(a).
Draw the impedance diagram.

8. Series-Parallel ac networks: R-L series circuit
Total impedance,
,

9. Series-Parallel ac networks: R-L series circuit
Total impedance,
,

10. Series-Parallel ac networks: R-L series circuit
,
Power factor PF = cos 53.13° = 0.6 lagging
where 53.13° is the phase angle between E and I