Download Appendix C Ohm`s Law, Kirchhoff`s Laws and AC Circuits

Appendix C
Ohm’s Law, Kirchhoff ’s Laws
and AC Circuits
C.1
Introduction
This write–up deals with the behaviour of circuits consisting of resistances,
capacitances and inductances, and sinusoidal voltage sources. We assume
that the source has been turned on for a long time (compared to any characteristic time constants of the circuit), and that we are dealing with the
steady state circuit operation. (The transient operation of such circuits are
discussed in write–ups for individual experiments.)
C.1.1
Reactance and Impedance
A sinusoidal voltage V (t) can be written as:
V (t) = V0 cos (ωt + φ)
(C.1)
where V0 is the amplitude, ω is the angular frequency in rads/sec, and φ is
the phase angle in radians. The phase angle can be determined from V |(t=0) :
V |(t=0) = V0 cos φ
(C.2)
When such a source is used to drive an RLC circuit (one containing resistors, capacitors and inductors), the current and voltage associated with any
branch will also vary sinusoidally at the same frequency, but with a different
amplitude and possibly a different phase angle.
C-2
Ohm’s Law, Kirchhoff ’s Laws and AC Circuits
Resistors
For example, if Equation C.1 refers to the voltage applied to a simple resistor
R, Ohm’s Law tells us that:
V (t) = I(t)R
(C.3)
and thus the current is given by:
I(t) =
=
V (t)
R
V0
R
(C.4)
(C.5)
cos (ωt)
where we have taken the phase angle φ to be zero for convenience. We see
that the current through the resistor varies sinusoidally, and that it is in
phase with the applied voltage.
Capacitors
Let us analyze the situation when a voltage (Equation C.1) is applied to a
capacitor. The current flowing through the capacitor is, of course, the time
derivative of the charge stored in the capacitor:
I(t) =
dQ
dt
(C.6)
and, since
Q(t) = CV (t) = CV0 cos (ωt)
(C.7)
we have for the current:
I(t)
= C dVdt(t)
= −ωCV0 sin (ωt)
(C.8)
(C.9)
Using the fact that
π
− sin θ = cos θ +
2
we can also write the current as:
I(t) = ωCV0 cos ωt +
π
2
(C.10)
which shows that the current through the capacitor is also sinusoidal, but
we see that there is a phase difference of π/2 between the voltage and the
C.1 Introduction
C-3
current. Thus the current leads the voltage, or more commonly, the voltage
lags the current. Note that the phase difference of π/2 means that the voltage
across the capacitor is zero when the current is at its peak value. We can
also write Equation C.9 in a form which looks like Ohm’s Law:
π
V0 cos ωt +
2
1
I(t)
ωC
(C.11)
= XC I(t)
(C.12)
=
where
1
ωC
and XC is known as the Capacitive Reactance and is somewhat analogous
to resistance in DC circuits; ie. it is the ratio between the maximum voltage
across the device and the maximum current through the device. Note that
XC is frequency dependent.
XC =
Inductors
Lastly, consider what happens when a sinusoidal current flows through an
inductor. The voltage across an inductance L is given by:
V (t) = L
dI
dt
(C.13)
and if the current is given by:
I(t) = I0 cos (ωt)
(C.14)
the voltage is then:
V (t)
= −ωLI0 sin (ωt)
= ωLI0 cos ωt +
π
2
(C.15)
(C.16)
Thus the voltage across the inductor leads the current by a phase angle of
π/2. Analogous to the case above,
XL = ωL
and now the Inductive Reactance is XL .
C-4
C.1.2
Ohm’s Law, Kirchhoff ’s Laws and AC Circuits
Ohm’s Law and AC Circuits
The total of resistances and reactances in a circuit or a branch is called the
impedance Z, where
V = IZ
(C.17)
This is the AC analog of Ohm’s Law for DC circuits. It looks similar to
Ohm’s Law for DC circuits, but now the phases of I, V , and Z must be
taken into account. Since they have both a magnitude and a phase, then it
is clear that V , I, and Z are all vectors. In addition, I and V can be time
dependent (in general Z is not) and so V and I may be better represented
~ and I,
~ and Equation C.17 is better written as
as V
~ = IZ
~
V
C.1.3
(C.18)
Phasors
To determine the impedance of a circuit, (i.e. its resistance including both
magnitude and phase information), and also the voltages and currents, it
is very convenient to introduce the use of complex algebra. That is, we
represent voltages, currents and impedances by complex quantities, with the
implicit understanding that we take the real parts of final answers to compare
with measured values. This is an eminently sensible thing to do, since our
instruments are not capable of measuring anything but real quantities! With
this in mind, we can make use of the extremely important result:
e±jθ = cos θ ± j sin θ
where
j=
√
(C.19)
−1
~ Let us use Equa(Here j replaces i to avoid confusion between i and I.)
tion C.19 to represent a sinusoidal voltage:
V (t)
= V0 cos ωt
= < (V0 ejωt )
~
= <V
(C.20)
(C.21)
(C.22)
where the notation ‘<()’ means to extract the real part of the complex expression. What do you think ‘=()’ means when applied to a complex expression?
C.1 Introduction
C-5
We can rewrite Equation C.22 as
~ = V0 ejωt
V
(C.23)
Let us examine the capacitor again, this time using our complex notation.
With the applied voltage given by Equation C.23, the current through the
capacitor is:
~
I
~
= C ddtV
d
= C dt
(V0 ejωt )
= jωC (V0 ejωt )
~
= jωC V
Rearranging gives
1
~ =I
~
V
jωC
!
Notice the enormous simplification. If we define the capacitive impedance as
1
jωC
−j
= ωC
−1
= j ωC
ZC =
(C.24)
(C.25)
(C.26)
then the above equation becomes
~ = IZ
~
V
which is analogous to Ohm’s Law for DC circuits. Since
XC =
1
ωC
(C.27)
then we have
ZC = −jXC
(C.28)
Next, notice that
j = e(jπ/2)
so we can write
~ = ωCV0 ej(ωt+π/2)
I
(C.29)
C-6
Ohm’s Law, Kirchhoff ’s Laws and AC Circuits
and after extracting the real part of the current:
π
I(t) = ωCV0 cos ωt +
2
(C.30)
This is identical to our earlier result in Equation C.10. Similarly, if the
current flowing through an inductance L is represented as
~ = I0 ejωt
I
the voltage across the inductor will be given by
~ (jωL)
~ = jωL I0 ejωt = I
V
(C.31)
and if we define the inductive impedance as
ZL = jωL
(C.32)
XL = ωL
(C.33)
ZL = jXL
(C.34)
where
we have
Notice that the capacitive and inductive impedances are now imaginary.
This means that circuit impedances will be complex quantities (in general)
and for both capacitors and inductors, |Z| = X.
C.1.4
Kirchhoff ’s Laws and AC Circuits
Using complex numbers to reflect the vector nature of circuit parameters
results in the following formulation of Kirchhoff’s laws for AC circuits
X
~ =0
I
where now it is a vector sum
X
~ =
V
X
~
IZ
and this is a vector sum also.
We have gained an enormous mathematical advantage — the total impedance
of an AC circuit can be determined in the same way as resistors are combined
C.1 Introduction
C-7
Figure C.1: Sample Phasor Diagram
in DC circuits with the understanding that now all quantities are vectors,
and specifically
ZR = R = R ∠0
ZC = −jXC = −
1
j
=
∠ − 90◦
ωC
ωC
ZL = jXL = jωL = ωL∠90◦
As an example of how this works, consider a simple RL circuit. The
impedance Z of this circuit is given by:
Z
= R + ZL
= R + jXL
= R + jωL
(C.35)
(C.36)
(C.37)
Figure C.1 shows a Phasor Diagram representing the impedance of the
circuit. From the diagram, it should be clear to you that a phasor is nothing
more than a vector in the complex plane. As such, we are interested in two
quantities: the length of the phasor, which is the magnitude of Z, denoted
by |Z|, and the angle φ that the phasor makes with the Real axis. Clearly,
the magnitude of Z is just the square root of the real part squared plus the
C-8
Ohm’s Law, Kirchhoff ’s Laws and AC Circuits
imaginary part squared, which is obtained from:
|Z|2
= ZZ ∗
= (R + jωL) (R − jωL)
= R 2 + ω 2 L2
or
|Z| = R2 + ω 2 L2
1/2
(C.38)
(C.39)
(C.40)
(C.41)
where the notation Z ∗ denotes complex conjugation (multiply the imaginary part by −1). The phase angle φ is just
φ = arctan =(Z)
<(Z)
= arctan
ωL
R
(C.42)
(C.43)
This gives us yet another way to represent the complex impedance, namely
Z = |Z| ejφ
(C.44)
Now we can find the current in the circuit very easily:
~
~= V
I
Z
so
I(t)
V (t)
Z
V0 ejωt
R+jωL
(C.45)
ejωt
(C.47)
=
=
V0
|Z|ejφ
V0 j(ωt−φ)
e
|Z|
=
=
(C.46)
(C.48)
Figure C.2 shows a phasor diagram representing the input voltage and the
circuit current. Both phasors rotate counterclockwise as ωt increases. Notice
that the voltage leads the current, as expected for an inductive circuit, and
that the angle between the voltage and current phasors is the phase angle
φ determined from the complex impedance. A final point: the projection of
the voltage phasor on the real axis is what you actually measure with an AC
voltmeter; similarly for the current phasor (this is the correspondence between a physical measurement and the mathematical operation of extracting
the real part of a complex quantity).
C.1 Introduction
C-9
Figure C.2: Phasors Changing Over Time
Once the circuit current has been determined, the voltages across the
individual components are easily determined:
|VR (t)| = |I(t)| R
|VL (t)| = |I(t)| XL