CIRCUIT THEORY EE411
INDEX
Euler’s identity.................. 6
frequency domain .............. 6
horsepower ........................ 7
impedance ......................... 6
impedance triangle ............ 7
inductor ............................. 6
LC tank circuit .............. 4
voltage and current........ 4
integrating amplifier .......... 3
inverting amplifier............. 3
LC circuits......................... 4
LC tank circuit................... 4
loop ................................... 5
magnetically coupled coils 8
maximum power transfer... 7
mesh .................................. 5
mesh current ...................... 8
motor efficiency ................ 7
motors
electric........................... 7
N transformer turns ratio .. 8
natural response................. 5
Neper frequency ................ 5
3-phase power ................... 7
admittance ......................... 6
amplifier
difference ...................... 3
differentiating................ 3
integrating ..................... 3
inverting ........................ 3
noninverting .................. 3
summing........................ 3
average power.................... 7
branch................................ 5
capacitor
voltage and current........ 4
coils
magnetically coupled .... 8
complex power .................. 7
critically damped ............... 5
current division.................. 3
delta circuit........................ 7
difference amplifier ........... 3
differentiating amplifier..... 3
essential branch ................. 5
essential node .................... 5
node................................... 5
noninverting amplifier....... 3
Norton equivalent.............. 4
one port network ............... 4
op amps ............................. 3
overdamped....................... 5
parallel RLC circuits ......... 5
path ................................... 5
phasor notation.................. 6
phasor transform ............... 5
power ................................ 7
average.......................... 7
complex ........................ 7
reactive.......................... 7
real ................................ 7
power factor ...................... 7
power factor correction ..... 7
power transfer ................... 7
power triangle ................... 7
reactance ........................... 6
reactive power ................... 7
rectangular notation........... 6
resonant frequency ............ 5
RLC circuits...................... 5
rms .................................... 2
root mean square ............... 2
series RLC circuits ............ 5
sinusoidal analysis ............ 5
step response ..................... 5
summing amplifier ............ 3
susceptance ....................... 6
T equivalent circuit ........... 8
tank circuit ........................ 4
Thèvenin equivalent.......... 4
transformers ...................... 8
trig identies ....................... 5
turns ratio .......................... 8
two-port circuits................ 1
underdamped..................... 5
wye circuit......................... 7
wye-delta transform .......... 7
z parameters ...................... 1
τ time constant ................. 4
TWO-PORT CIRCUITS
V1
I1
circuit
network
z=
1
y
I1 = y11V1 + y 22V2
I 2 = y 21V1 + y 22V2
I 2 = y12V2 + y11V1 ?
I2
V1 = z11 I1 + z 22 I 2
V2 = z 21 I1 + z 22 I 2
V1 = z12 I 2 + z11 I1 ?
V2
V1 = h11 I1 + h22V2
I 2 = h21 I1 + h22V2
I1 = h12 I 2 + h11V1 ?
To calculate the z parameters in a resistive network the
equations above are manipulated to the following form,
where one of the currents is held to zero. Various
manipulations are carried out to find values for the V and I
quantities using the known values of the resistors.
z11 =
V1
I1
z12 =
I2 = 0
V1
I2
z 21 =
I1 = 0
V2
I1
z 22 =
I2 =0
V2
I2
I1 = 0
h parameters are used for transistor specifications
y parameters may be easier to find than z parameters
and may be added when networks are paralleled.
Tom Penick
tom@tomzap.com
www.teicontrols.com/notes
5/12/2003 Page 1 of 8
RMS
rms stands for root mean square. To obtain the rms
value of a periodic function, first square the function,
then take the mean value, and finally the square root.
rms by definition:
X rms =
rms value of AC voltage:
root
mean
1
T
∫ ( f ( x ))
T
0
Vrms =
2
dt
Vmax
2
square
⟨ ⟩
p2
f ( t )rms =
f (t )
2
The plot below shows a sine wave and its rms value, along
with the intermediate steps of squaring the sine function
and taking the mean value of the square. Notice that for
this type of function, the mean value of the square is ½ the
peak value of the square.
Tom Penick
tom@tomzap.com
www.teicontrols.com/notes
5/12/2003 Page 2 of 8
Op Amps
INVERTING AMPLIFIER
DIFFERENCE AMPLIFIER
Rf
Rs
Vs
Rf
Vo = −
Rs
+Vcc
-
Vo
+
Vs
Rf
Rs
Va
Ra
Vb
-Vcc
Vn
-
Vp
+
+Vcc
Vo
-Vcc
Rb
INVERTING SUMMING AMPLIFIER
Rf
Ra
Va
Vn = V p = Vb
Rb
Vb
Rc
Vc
+Vcc
-
Vo
Vn − Va Vn − Vo
+
=0
Rs
Rf
Rb
Ra + Rb
INTEGRATING AMPLIFIER
Cf
+
-Vcc
Rs
Vs
Rf
Rf 
 Rf
V o = −
Va +
Vb +
V 
Rb
Rc c 
 Ra
+
NONINVERTING AMPLIFIER
Rf
Rs
Vo = −
+Vcc
-
Vo
Rg
Vg
+
Vo =
Rs + R f
Rs
1
Rs C f
t
∫V
to
s
+Vcc
Vo
-Vcc
dτ + Vo (t o )
CURRENT DIVISION
-Vcc
IS
I1
I2
R1
R2
Vg
R2
I
R1 + R2 S
R1
I2 =
I
R1 + R2 S
I1 =
DIFFERENTIATING AMPLIFIER
Rf
Cs
-
Vs
Vo = − R f Cs
dv S
dt
+
+Vcc
Vo
-Vcc
Tom Penick
tom@tomzap.com
www.teicontrols.com/notes
5/12/2003 Page 3 of 8
THÈVENIN AND NORTON EQUIVALENTS
LC CIRCUITS
A one-port network (circuit presenting 2 external terminals)
may be represented by either a Thèvenin or Norton
equivalent. Note that REQ has the same value in both the
Thèvenin and Norton equivalents.
NORTON EQUIVALENT
THÈVENIN EQUIVALENT
+
v L
-
LI o 2 (1 − e −2 t /τ )
Time Constant: τ = L / R
also: w =
R
Voltage:
REQ
IN
VTH +-
REQ
1)
Find the Thèvenin voltage (the open-circuit voltage) or the
Norton current (the short-circuit current).
2)
To find Req, first “Turn off” the independent sources, i.e.
voltage sources go to zero which means they are shorted
and current sources also go to zero which means they are
opened. Calculate the equivalent resistance of the circuit.
This is Req.
3)
4)
+
v C
-
Energy (joules):
w = 12 Ce 2
∫
1
2
∫
dv
dt
Current:
ic ( t ) = C dv
dt
LC Tank Circuit
C
Resonant frequency:
L
f =
Current: i( t ) = I f + ( I o − I f )e
4Ω
1
2π LC
−t/τ
Voltage: v ( t ) = V f + (Vo − V f )e
Given this circuit:
- 12 v
4Ω
RL
V TH = 6 v
4Ω
+
- 12 v
4Ω
4Ω
−t/τ
Power: p = I o R e −2 t /τ
where I0 is initial current [A]
If is final current [A]
t is time [s]
τ is the time constant; τ = RC for capacitive circuits,
τ = R/L for inductive circuits [s]
V0 is initial voltage [V]
Vf is final voltage [V]
p is power [W]
R is resistance [Ω]
2
4Ω
+
The Thèvenin resistance is
the equivalent resistance
with the independent voltage
source shorted and the load
disconnected.
1 t
v dτ + I o
L 0
Equations Common to L & C Circuits
THÈVENIN/NORTON EXAMPLE
The Thèvenin voltage is
the open circuit voltage,
i.e. with the load
disconnected.
Current: I L ( t ) =
CVo 2 (1 − e −2 t /τ )
Time Constant: τ = RC
1 t
Voltage: Vc ( t ) =
i dτ + Vo
C 0
R
P = Cv
1
2
v L ( t ) = L dtdi
also: w =
Power:
If there are no independent sources (dependent sources
may be present) then VTH = IN = 0 and the circuit reduces
to an equivalent resistance.
If there are independent and dependent sources, turn off
the independent sources and apply a test source (VTEST = 1
or ITEST = 1) to the port. Calculate the unknown parameter
VTEST or ITEST at the port and find REQ using
VTEST = I TEST REQ
w = 12 Li 2
Energy (joules):
R EQ = 2 Ω
2Ω
The Thèvenin equivalent
circuit can now be written
as:
+
- 6v
RL
VTH = I N RTH
And the Norton equivalent
can be written as:
3a
2Ω
Tom Penick
RL
tom@tomzap.com
www.teicontrols.com/notes
5/12/2003 Page 4 of 8
Sum of node currents in a Parallel RLC circuit:
C
dv
dt
v 1 t
+ +
v dτ + I o = 0
R L 0
∫
which differentiates to:
C ddt 2v +
2
1
R
dv
dt
+
1
v=0
L
RLC CIRCUITS -- Series
Sum of voltages in a Series RLC circuit:
L
di
dt
1 t
+ Ri +
i dτ + Vo = 0
C 0
∫
which differentiates to:
L dtd 2i + R dtdi +
2
1
i=0
C
d 2x
⇔ ( jω )2 X
2
dt
dx
⇔ jω X
dt
RLC CIRCUITS – solving second order equations
α the Neper frequency (damping coefficient) [rad/s]:
Parallel
circuits:
α=
Series
circuits:
1
2RC
α=
R
2L
ω the Resonant frequency [rad/s]:
ωo =
1
used in
underdampe
d calculations
ω d = ω o2 − α 2
LC
s1 , s2 the roots of the characteristic equation [rad/s]:
s1 = −α + α 2 − ω o 2
s2 = − α − α 2 − ω o 2
Overdamped α > ω
2
2
(real and distinct roots)
X ( t ) = X f + A1 ' e s1t + A2 ' e s2t
X ( 0 ) = X f + A1 '+ A2 '
Underdamped α < ω
2
2
X ( t ) = X f + B1 ' e
dx
dt
X ( 0) = X f + B1 '
2
2
−α t
= −α B1 '+ wd B2 '
product
sum
jφ
V = Vme = P{Vm cos( ω t + φ )}
v ( t ) = A cos( ω t + φ° ) ⇔ A ∠φ°
sin ω t = cos( ω t − 90° )
equivalent of two parallel impedances
( 0 ) = s1 A1 '+ s2 A2 '
dx
dt ( 0 )
Critically Damped α = ω
X ( 0) = X f + D2 '
where Vm and Im are maximums
Phasor Transform:
cos ω d t + B2 ' e − α t sin ω d t
2
SINUSOIDAL ANALYSIS
π × degrees = 180 × radians
ω = 2π f [rad / s] = 360 f [deg / s]
1
resonant frequency ω o =
LC
v ( t ) = Vm cos( ω t + φ )
i( t ) = I m cos( ω t + φ )
(complex roots)
−α t
X ( t ) = X f + D1 ' te
In an overdamped circuit, α > ω and the voltage or current
approaches its final value without oscillation.
2
2
In an underdamped circuit, α < ω and the voltage or current
oscillates about its final value.
2
2
In a critically damped circuit, α = ω and the voltage or
current is on the verge of oscillating about its final value.
When an expression is integrated, it may be necessary to add
in initial values for the constant of integration even if they
have been taken into account within other terms.
Natural response is the behavior of a circuit without external
sources of excitation.
Step response is the behavior of a circuit with an external
source.
A node is a point where two or more circuit elements join.
An essential node is a node where three or more circuit
elements join.
A path is a trace of adjoining basic elements with no elements
included more than once.
A branch is a path that connects two nodes.
An essential branch is a path which connects two essential
nodes without passing through an essential node.
A loop is a path whose last node is the same as the starting
node.
A mesh is a loop that does not enclose any other loops.
2
RLC CIRCUITS -- Parallel
Inverse Phasor Transform
=
P −1{Vm e jφ } = R{Vm e jφ e jωt }
A smaller φ causes a right shift of the sinusoidal graph.
(repeated roots)
+ D2 ' e − α t
dx
dt
(0) = D1 '−α D2 '
Some Trig Identities
A cos ω t + B sin ω t =

 − B 
A2 + B 2 cos cot + tan 

 A  

c ± jθ = cos θ ± j sin θ Euler identity
sin ω t = cos( ω t − 90° )
Tom Penick
tom@tomzap.com
www.teicontrols.com/notes
5/12/2003 Page 5 of 8
SINUSOIDAL ANALYSIS
Element:
Resistor
Capacitor
Inductor
Impedance (Z):
R (resistance)
− j / ωC
jω L
Reactance (X)
--
−1/ ωC
ωL
G (conductance)
jω C
1 / jω L
--
ωC
−1/ ω L
IR
I / jω C
jω L I
( I m / ω C ) ∠ ( θ V − 9 0° )
ω L I m ∠( θV + 90° )
jω C V
V / jω L
(Vm / ω C ) ∠( θ V + 90° )
(V m / ω L ) ∠(θ V − 90° )
Admittance (Y):
Susceptance:
Voltage:
V/R
Amperage:
PHASOR and RECTANGULAR NOTATION
Note: Due to the way the calculator works, if X is negative, you
The phasor is a complex number that carries the
amplitude and phase angle information of a sinusoidal
function. The Phasor concept is rooted in Euler’s
identity, which relates the exponential function to the
trigonometric function:
To convert from phasor to rectangular (j) notation:
M ∠φ°
Phasor form:
e± jθ = cos θ ± jsin θ
The use of phasor notation may be referred to as
working in the phasor domain or the frequency
domain. Note that the phasor notation M∠φ is
must add 180° after taking the inverse tangent. If the result
is greater than 180°, you may optionally subtract 360° to
obtain the value closest to the reference angle.
X (real) Value: M cos φ
Y (j or imaginary) Value:
M sin φ
In conversions, the j value will have the same sign as the θ
value for angles having a magnitude < 180°.
equivalent to Mejφ , where φ is in radians.
Rectangular Notation: X ± jY where X represents the
horizontal or real coordinate
and Y
the vertical or
de
itu M
imaginary coordinate.
Use
gn
a
Y
M
this form for addition and
subtraction by separately
θ
adding and subtracting the
X
real
and
imaginary
components. Be careful with
the sign of the j term:
Phasor impedance
diagram for series
circuits, where +j is
inductive and -j is
capacitive:
XL
Phasor voltage
diagram for series
circuits, where +j is
inductive and -j is
capacitive:
VL
R
IC
VR
I
XC
Phasor amperage
diagram for parallel
circuits, where +j is
capacitive and -j is
inductive:
IR
I
VC
V
IL
( A + jB) + ( C − jD) = ( A + C ) + j[B + ( − D)]
Phasor Notation: M ∠φ° , where M is the magnitude of the
phasor and f is the angle CCW from the X axis. Use this
form for multiplication and division.
E ∠θ E
( E ∠θ )( F ∠φ ) = EF ∠(θ + φ )
= ∠(θ − φ )
F ∠φ F
A negative magnitude may be converted to positive by
adding or subtracting 180° from the angle.
To convert from rectangular to phasor notation:
Rectangular form: X ± jY
Magnitude:
Angle φ:
M=
tan φ =
X 2 + Y2
Y
X
(Caution: The Y will be
negative is the j value is being
subtracted from the real.)
Tom Penick
tom@tomzap.com
www.teicontrols.com/notes
5/12/2003 Page 6 of 8
POWER
3-PHASE POWER
Average Power or real power (watts)
Phase and line voltage relationships in a Wye Circuit
(positive sequence - clockwise)
P=
Vm I m
cos( θ v − θ i )
2
= Vrms I rms cos( θ v − θ i )
Positive P means the load is
absorbing average power,
negative means delivering
or generating.
A
Vm I m
sin(θ v − θ i )
2
= Vrms I rms sin(θ v − θ i )
B
C
Positive Q means the load is
absorbing magnetizing
vars (inductive), negative
means delivering
(capacitive).
conjugate of"
I aA
Phase
Angle
θ
True or Average Power [watts]
θ = cos
−1
VBC
I CA
I CA
I bB
B
I cC
Lagging: Inductive, current lags (-j), +Q
Leading: Capacitive, current leads (+j), -Q
I AB
Z
C
30°
I BC
I bB
I aA = I AB 3
I aA
I BC
Wye-Delta Transform (for balanced circuits only)
]
ms
[oh
e
c
an
ped
Im
Z
Phase
Angle
VBN
I AB
Z
1
V 2 rms
= Vmax I *max =
2
Z*
Q
Z
Phase and line current relationships in a Delta Circuit
(positive sequence - clockwise)
A
I cC
Power and Impedance triangles
S
VAN
VAB = VAN 3 × (1 ∠ 30° )
* means "the complex
]
VA
er [
w
Po
ent
Reactive
par
p
power in
A
[VARS]
30°
N
V AB = V AN 3 or
Power Factor (ratio of true power to apparent power)
pf =
cos( θ v − θ i )
VAB
VCA
Z
VBC
-
VCN
-
Z
= Vrms I *rms
VAB
+
Complex Power (VA)
S = P + jQ
= Vrms I rms (θ v − θ i )
Z
+
Reactive Power (VARS)
Q=
VAN
+
θ
X
Reactance
[ohms]
Resistance [ohms]
P
R
pf The power triangle is geometrically identical
to the impedance triangle.
ZD
3
Vline− to− line = Van 3
V
I aA = an
Zφ
Single-phase Equivalent
Circuit:
a
+
n
A
+
+
I aA
Van
VAN
-
-
N
Motor Ratings
Maximum Power Transfer
Maximum power transfer occurs when the
load impedance is equal to the complex
conjugate of the source impedance. Under
these conditions, the maximum value of
average power absorbed is
ZY =
2
Pmax
VTH
=
4 RL
P = 3 VL I L cos( θ v − θ i ) =
hp × 746
efficiency
where:
P is the power input in watts
cos( θ v − θ i ) is the power factor
efficiency is expressed as a decimal value
Power Factor Correction
Q=
VARS
( 460 / 3 )2
=
3
x c = −1 / ω C
where:
VARS is a negative value for the amount of correction
460 is the line voltage
C is the value of the capacitor in Farads
Tom Penick
tom@tomzap.com
www.teicontrols.com/notes
5/12/2003 Page 7 of 8
TRANSFORMERS
Ideal Transformer
1:a
ZS
+
-
V1
VS
a=
V2
ZL
N1
N2
V1 =
Z IN =
V1
1 V
1
= 2 2 = 2 ZL
I1 a I 2
a
V2
a
I1 = aI 2
ZIN is the load seen by the source.
Transformer Turns Ratio
V1
I1
V1
V
= 2
N1 N 2
V2
N1 N2
I2
N 1 I1 = N 2 I 2
Moving a dot results in a
negative sign on one side of
each equation.
Magnetically Coupled Coils
V1
di1
di
+M 2
dt
dt
di
di
V2 = L2 2 + M 1
dt
dt
V2
M
I1
V1 = L1
I2
M = k L1 L2 where k is
L1 L2
the coefficient of coupling
T Equivalent Circuit
L1 -M
L2 -M
M
This circuit is equivalent to the
magnetically coupled coils
above. We are not concerned
about the orientation of the dots.
Mesh Current Equations involving mutual inductors
A mesh is a loop that does not enclose other loops in the
circuit.
1. Draw current loops emanating from positive voltage
sources if present and label I1, I2, I3, etc. for each interior
path of the circuit.
2. For each loop form an equation in the form: Voltage or 0
if there is no source in the loop = R1 × (sum of amperages
passing through R1) + L1
d
dt
× (sum of amperages
passing through L1) + . . .
3. Amperages are positive in the direction of loops
regardless of the location of dots on inductors in the loop.
However, the sign of an amperage through a mutual
inductor is positive iff it enters the mutual inductor at the
same end (i.e. dotted or undotted) at which the reference
current loop enters the reference inductor.
Tom Penick
tom@tomzap.com
www.teicontrols.com/notes
5/12/2003 Page 8 of 8