Universty Of Jordan
Faculty of Engineering and Technology
Electrical Engineering Department
Summery for RL, RC, and RLC circuits
Series R-L circuit

Impedance of a Series R-L Circuit:
The impedance of an RL circuit is the total opposition to AC current flow caused by the resistor (R)
and the reactance of the inductor (XL).
XL = 2πfL
The equation for the impedance of an RL circuit is: Z = √
where:
Z = the total impedance in ohms
XL = the inductive reactance in ohms
R = the resistance in ohms

Voltages in a Series R-L Circuit:
The total voltage in a series RL circuit is given by this
equation:
VS = √
where:
VT = total voltage
VR = voltage across resistor R
VL = voltage across inductor L
When VR= I x R and VL = I x XL
VS = √
So,
I=
√
Eng.enaam Al-khatib

Phase shift in Series R-L Circuits
It is a basic property of resistors and inductors that their phase angles in a series circuit are
always give by:
ӨR = 0º , ӨL = 90º
However, there are two equations you can use for defining the total phase angle for a series RL
circuit. The total phase shift can be determined by the equation:
ӨS = - tan-1(
)
VS leads IS
where:
ӨS = total phase shift in degrees or radians
XL= inductive reactance in ohms
R = resistance in ohms
The total phase angle is also determined by the equation:
ӨS = - tan-1 (
)
Note that : 0 < Ө < - 90
where:
ӨS = total phase angle in degrees or radians
VL= AC (RMS) voltage drop across the inductor in volts
VR = AC (RMS) voltage drop across the resistor in volts

Frequency response for series R-L circuit :
You must know if the source frequency increase then :






XL : will be increase
VL : increase
VR : decrease
Is : decrease
Ө: increase(-ve) ( Is with respect to Vs )
Power factor : cos ӨS (lagging)
Eng.enaam Al-khatib
Series R-C circuit

Impedance of a Series R- C Circuit:
The impedance of an R-C circuit is the total opposition to AC current flow caused by the resistor (R)
and the reactance of the capacitor (Xc).
Xc =
The equation for the impedance of an R-C circuit is:
Z= √
where:
Z = the total impedance in ohms
Xc= the capacitive reactance in ohms
R = the resistance in ohms

Voltages in a Series R-C Circuit:
The total voltage in a series RL circuit is given by this
equation:
VS = √
where:
VT = total voltage
VR = voltage across resistor R
Vc = voltage across capacitor C
When VR= I x R and VC = I x XC
VS = √
So,
I=
√
Eng.enaam Al-khatib

Phase Angle in Series R-C Circuits
It is a basic property of resistors and capacitors that their phase angles in a series circuit are
always give by:
ӨR = 0º
ӨC = 90º
However, there are two equations you can use for defining the total phase angle for a series RC
circuit. The total phase angle can be determined by the equation:
ӨS = tan-1(
)
VS lags IS
where:
ӨT = total phase angle in degrees or radians
XC = capacitive reactance in ohms
R = resistance in ohms
The total phase angle is also determined by the equation:
ӨT = tan-1 (
)
Note that : 90 < Ө <0
where
ӨT = total phase angle in degrees or radians
VC= AC (RMS) voltage drop across the capacitor in volts
VR = AC (RMS) voltage drop across the resistor in volts

Frequency response for R-C circuit:
You must know if the source frequency increase then :






Xc : will be decrease
Vc : decrease
VR : increase
Is : increase
Ө: decrease(+ve) ( Is with respect to Vs )
Power factor : cos ӨT (leading)
Eng.enaam Al-khatib
Parallel R-L circuit

Impedance of a parallel RL Circuit:
The impedance of an RL circuit is the total opposition to AC current flow caused by the resistor (R)
and the reactance of the inductor (XL).
XL = 2πfL
The equation for the impedance of an RL circuit is:
Y=√
where:
Y= the total admittance (1/Z) in siemens (1\Ω)
BL= the inductor susceptance (1/XL) in siemens .
G = conductance (1/R) in siemens.

Currents in a parallel R-L Circuit:
The total current in a parallel RC circuit is given by this equation:
Is = √
where:
Is = total current
IR= current through resistor R
IL = current through inductor L
Eng.enaam Al-khatib

Phase Angles in Parallel RL Circuits
It is a basic property of resistors and inductors that their phase angles in a parallel circuit are
always give by:
ӨR = 0º
ӨL = 90º
The total phase angle can be determined by the equation:
ӨT = tan-1(
)
Note that : - 90 < Ө < 0
VT leads IT
where:
ӨT = total phase angle in degrees or radians
XL= inductive reactance in ohms
R = resistance in ohms.

Frequency response for parallel R-L circuit :
You must know if the source frequency increase then :






XL : will be increase
IL : decrease
IR : constant
Is : decrease
Ө: decrease (-ve) ( Is with respect to Vs )
Power factor ; cos ӨT (lagging)
Eng.enaam Al-khatib
Parallel R-C circuit

Impedance of a parallel R-C Circuit:
The impedance of an RL circuit is the total opposition to AC current flow caused by the resistor (R)
and the reactance of the capacitor(Xc).
Xc =
The equation for the impedance of an RC circuit is:
Y=√
where:
Y= the total admittance (1/Z) in siemens (1\Ω)
Bc = the capacitor susceptance (1/XC) in siemens
G = conductance (1/R) in siemens.

Currents in a parallel R-C Circuit:
The total current in a parallel RC circuit is given
by this equation:
IS = √
where:
IS= total current
IR= current through resistor R
IC = current through capacitor C
Eng.enaam Al-khatib

Phase Angles in Parallel R- C Circuits
It is a basic property of resistors and capacitor that their phase angles in a parallel circuit are
always give by:
ӨR = 0º
Өc = 90º
The total phase angle can be determined by the equation:
ӨT = = tan-1(
)
Note that : 0 < Ө <90
IT leads VT
where:
ӨT = total phase angle in degrees or radians
XC= capacitive reactance in ohms
R = resistance in ohms

Frequency response for parallel R-C circuit :
You must know if the source frequency increase then :






XC : will be decrease
IC : increase
IR : constant
Is : increase
Ө: increase(+ve) ( Is with respect to Vs )
Power factor ; cos ӨT (leading)
Eng.enaam Al-khatib
Series R-L-C circuit

Impedance of a Series R-L-C Circuit:
The impedance of an RLC circuit is the total opposition to AC current flow caused by the resistor (R),
the reactance of the inductor (XL), and the reactance of the capacitor (Xc).
XL = 2πfL
Xc =
The equation for the impedance of an RLC circuit is:
Z=√
where:
Z = the total impedance in ohms
XL = the inductive reactance in ohms
XL = the capacitive reactance in ohms
R = the resistance in ohms

Voltages in a Series R-L-C Circuit:
The total voltage in a series RLC circuit is given by this
equation:
VS = √
where:
VS = total voltage
VR = voltage across resistor R
VL = voltage across inductor L
VC = voltage across capacitor C
I=
√
Ө=
or Ө=
Eng.enaam Al-khatib

Resonance:
When the voltages across capacitor and inductor are equal and opposite
VL (t)= -VC(t) so VL = VC
So, XL= XC
And , jωL=
so the frequency at which this occurs is
ωr =
fr =
√
√
where : ωr ,fr are the angular and cyclic frequencies of resonance, respectively .
Now ,

* If f < fr then XC > XL , so the circuit acts as capacitive, and the total current will be increase .

* If f = fr then XC = XL , so the circuit acts as resistive, and the total current reach its maximum value .

* If f > fr then XL > XC , so the circuit acts as inductive, and the current will be decrease .
XC < XL
XC > XL
Eng.enaam Al-khatib

Phase Angles in Series R-L-C Circuits
It is a basic property of resistors and inductors and capacitors that their phase angles in a series
circuit are always give by:
ӨR = 0º ӨL = 90º
ӨC = -90o
However, there are two equations you can use for defining the total phase angle for a series RL
circuit.
The total phase angle can be determined by the equation:
ӨT =
Note that : 90 < Ө < -90
where:
ӨT = total phase angle in degrees or radians
XL= inductive reactance in ohms
Xc = capacitive reactance in ohms
R = resistance in ohms
We must know :



If f < fr then ӨT will be decrease(+ve) (IT leads VT)
If f = fr then ӨT = 0o (pure resistive) (IT in-phase with VT)
If f > fr then ӨT : will be increase(-ve) (VT leads IT)
Eng.enaam Al-khatib

Frequency response for series R-L-C circuit :
You must know if the source frequency increase then :

XL : increase
Xc : decrease

VL : increase
Vc : decrease (VL = VC at fr )

VR : increase until f = fr , then will be dcrease .

Is : increase until f = fr, then will be decrease .

Ө: decrease (+ve) until f = fr, then will be increase (-ve).

Power factor : At f < fr the circuit acts as capacitive so P.F is leading
(XC = XL at fr )
At f > fr the circuit acts as inductive so P.F is lagging
f < fr
f = fr
f > fr
Eng.enaam Al-khatib
Parallel R-L-C circuit

Impedance of a parallel R-L-C Circuit:
The impedance of an RLC circuit is the total opposition to AC current flow caused by the resistor (R),
the reactance of the inductor (XL), and the reactance of the capacitor (Xc).
XL = 2πfL
Xc =
The equation for the impedance of an RLC circuit is:
Y=√
where:
Y= the total admittance (1/Z) in siemens (1\Ω)
Bc = the capacitor susceptance (1/XC) in siemens
BL = the inductor susceptance (1/XL) in siemens.
G = conductance (1/R) in siemens.

Currents in a parallel R-L-C Circuit:
The total current in a parallel RLC circuit is given by this equation:
IS = √
where:
IS= total current
IR = current through resistor R
IL = current through inductor L
IC = current through capacitor C
V S = IS X (
Ө=
)
or Ө=
Eng.enaam Al-khatib

Resonance:
When the currents across capacitor and inductor are equal and opposite :
IL (t)= -IC(t) so IL = IC
So, XL= XC
And , jωL=
so the frequency at which this occurs is
ωr =
fr =
√
√
where ωr ,fr are the angular and cyclic frequencies of resonance, respectively .
Now ,

* If f < fr then XC > XL , so the circuit acts as inductive, and the total current will be decrease .

* If f = fr then XC = XL , so the circuit acts as resistive, and the total current reach its minimum value .

* If f > fr then XL> XC , so the circuit acts as capacitive, and the current will be increase .

Eng.enaam Al-khatib

Phase Angles in parallel R-L-C Circuits
It is a basic property of resistors and inductors and capacitors that their phase angles in a series
circuit are always give by:
ӨR = 0º ӨL = 90º
ӨC = -90o
However, there are two equations you can use for defining the total phase angle for a series RL
circuit.
The total phase angle can be determined by the equation:
ӨT =
Note that : -90 < Ө < 90
where:
ӨT = total phase angle in degrees or radians
XL= inductive reactance in ohms
Xc = capacitive reactance in ohms
R = resistance in ohms
We must know :
* if f < fr then
ӨT will be decrease (-ve)
* if f = fr then ӨT = 0o (pure resistive)
* if f > fr then ӨT : will be increase(+ve)
Eng.enaam Al-khatib

Frequency response for parallel R-L-C circuit :
You must know if the source frequency increase then :






XL : increase
Xc : decrease (XC = XL at fr )
IL : decrease
Ic : increase (IL = IC at fr )
IR : constant .
Is : decrease until f = fr, then will be increase .
Ө: decrease (-ve) until f = fr, then will be increase (+ve).
Power factor : At f < fr the circuit acts as inductive so P.F is lagging.
At f > fr the circuit acts as capacitive so P.F is leading.
f < fr
f = fr
f > fr
Eng.enaam Al-khatib