NCEA Level 3 Physics AC Electricity in RC circuits AS91526

1. Capacitors in AC
Circuits

2. Capacitors
• A capacitor is used to
rapidly store and release
charge.
• Capacitors differ from
resistors in that; resistors
convert electrical energy to
heat (and often light), while
capacitors only store the
charge (electrical energy) in
the electric field between
their plates, then release it
again.

3. Capacitors in DC c.f. AC
• Once the capacitor is
fully charged the bulb
does not glow
~
• The bulb glows
continuously but more
dimly than without the
capacitor
Although no charge flows between the plates in either circuit,
the AC means that the current is constantly changing
direction, making the bulb glow.

4. A Capacitor in AC
• Capacitors react against a
change in voltage by either
supplying current (discharging)
or drawing current (charging).
• In an AC circuit a capacitor
alternates between charging and
discharging.
• As a result, capacitors behave in
a similar way to resistors in that
they oppose the current in a
circuit as they charge and
discharge. This property is called
reactance and has the symbol XC
and units Ohms

5. Voltage and Current Phase Differences
V
~
C
Step 1 - At point a (see diagram) the
voltage is zero and the capacitor is
uncharged. Initially, the voltage
increases quickly. The voltage
across the capacitor matches the
power supply voltage, so the
current is large to build up charge
on the capacitor plates. The closer
the voltage gets to its peak, the
slower it changes, meaning less
current has to flow. When the
voltage reaches a peak at point b,
the capacitor is fully charged and
the current is momentarily zero.
Note that;
Unlike a resistor where V and I
are in phase in a capacitor V lags
behind I by ¼ of a cycle (90)

6. Voltage and Current Phase Differences
V
~
C
Step 2 - After reaching a peak,
the voltage starts dropping.
The capacitor must discharge
now, so the current reverses
direction. When the voltage
passes through zero at point
c, it's changing quite rapidly;
to match this voltage the
current must be large and
negative.

7. Voltage and Current Phase Differences
V
~
C
Step 3 - Between points c and d,
the voltage is negative. Charge
builds up again on the
capacitor plates, but the
polarity is opposite to what it
was in step one. Again the
current is negative, and as the
voltage reaches its negative
peak at point d the current
drops to zero.
Step 4 - After point d, the voltage
heads toward zero and the
capacitor must discharge.
When the voltage reaches zero
it's gone through a full cycle so
it's back to point a again to
repeat the cycle.

8. Capacitors in AC
The larger the capacitance of the capacitor, the more
charge has to flow to build up a particular voltage on
the plates, and the higher the current will be. The
higher the frequency of the voltage, the shorter the
time available to change the voltage, so the larger the
current has to be. The current, then, increases as the
capacitance increases and as the frequency increases.
Usually this is thought of in terms of the effective
resistance of the capacitor, which is known as the
capacitive reactance, measured in ohms. There is an
inverse relationship between current and resistance, so
the capacitive reactance is inversely proportional to the
capacitance and the frequency:

9. Voltage and Current in a Capacitor
~
VC
7
6
5
4
3
2
1
Capacitor Voltage and Current
• In an AC circuit the current can be altered with a variable
resistor
• When the voltage and current are plotted on a graph they
show a linear relationship
Remind you of Ohm’s Law (V=IR)?
A
6V AC
200F
0
0 0.1 0.2 0.3 0.4
Capacitor Voltage (V)
Current (A)

10. Reactance (XC ) -the maths
• In the same way that V=IR the “opposing” of
AC current by a capacitor, reactance (XC ) can
be calculated by;
and so;
C C V  IX
V
X C
I
C 

11. Examples
1. Find the voltage of a capacitor with a reactance
of 1.2 and a current of 0.80A
0.96V
2. A capacitor with 8V AC across it has a reactance
of 45. Calculate the current of the circuit.
0.17A
3. Calculate the reactance of a capacitor with a
RMS voltage of 6V and a current of 1.8A
3.3

13. Worksheet

14. Factors Affecting Reactance (XC )
• Increasing the size of the capacitor means that more current is
required to charge and discharge the capacitor (decreasing XC)
1
C
XC

• Increasing frequency increases current (decreasing reactance).This is
because more frequent charging and discharging means more
current must flow to charge the capacitor in less time
1
f
XC

• The reactance of a capacitor with a supply frequency f;
XC C  
C
X
fC
1
2
1
 or 
The reactance Xc is large at low frequencies and small at high
frequencies. For steady DC which is zero frequency, Xc is infinite
(total opposition), hence the rule that capacitors pass AC but
block DC.

15. Examples
1. A 200F capacitor is connected to a 6V 50Hz
AC supply.
a) Calculate the reactance of the capacitor
16
b) The RMS current in the circuit
0.38A
2. What size capacitor is needed to give an
reactance of 50 in a 12V 60Hz circuit?
5 3F

17. RC Phase Differences
• In an AC circuit with a
resistor and a capacitor
(RC circuit) the voltages
across each component
are out of phase by ¼ of
a cycle
1.5
1
0.5
0
-0.5
-1
-1.5
Resistor and Capacitor Phase
Differences
0 200 400 600 800
Voltage (mV)
Time (ms)
Resistor
Capacitor

18. The Effect of Phase Differences in RC Circuits
• In DC circuits the voltages
across components in a
circuit add up to the supply
voltage
• In AC circuits the same does
not appear to apply (at first
glance)
6.0
VS
75 200F
VR VC
5.9 1.2

19. The Effect of Phase Differences in RC Circuits
• However if we consider the
phase differences, we see
that this is a vector problem
VS
75 200F
VR VC
~ ~ ~
VC
VR
V  V 
V
S R C VC
VR
VS
6.0
5.9 1.2
From
Pythagorus;
2 2 2
C  A 
B
2 2
V  V 
V
S R C VS

20. The Effect of Phase Differences in RC Circuits
In an RC circuit;
• At any instant
Note the graph
• But when considering
the rms voltages the
phase differences are
important
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
Supply Voltage of Resistor/Capacitor
Circuit
0 200 400 600 800
Voltage (mV)
Time (ms)
Resistor
Capacitor
Supply
Voltage
S R C V V V
~ ~ ~
V  V 
V
S R C

21. Exercises
1. Find the AC supply voltage of an RC circuit where the
resistor voltage is 3.4V and the capacitor voltage is
1.5V
3.7V
2. Calculate the voltage across the resistor in an AC
circuit with a supply voltage of 8.5V and a capacitor
voltage of 2.4V
8.2V
3. Calculate the voltage across the capacitor in an 12V
AC circuit with a voltage of 8.5V across the resistor.
8.5V
4. Find the supply voltage of an 60Hz AC circuit with a
120V across a 2k resistor and a capacitor voltage of
0.80V
120V

23. The Effect of RC Circuits on Current
• In an RC circuit both the
resistor and the capacitor
oppose the current so
V=IR wont work
• Any calculation of the
current will have to
involve both resistance
(R) and reactance (XC
)and allow for the phase
differences between
them
VS
75 200F
VR VC
A

24. Impedance in an RC Circuit
• Impedance is a measure of
the combined opposition
to alternating current of
the components of a
circuit.
• It describes not only the
relative amplitudes of the
voltage and current, but
also the relative phases the
components in the circuit.
• Impedance has the symbol
Z and units Ohms
1.5
1
0.5
0
-0.5
-1
-1.5
Resistor and Capacitor Phase Differences
0 200 400 600 800
Voltage (mV)
Time (ms)
Resistor
Capacitor
V  IZ

25. Impedance –the maths
• Impedance in an RC circuit involves both the
resistance and the reactance
• because of the phase differences they must
be added as vectors so;
From Pythagorus;
2 2 2
C  A 
B
2 2
Z  R 
X
C XC
R
Z

26. Examples
1. Calculate the impedance of an RC circuit with a
resistance of 75 and a reactance of 15 
76 
2. An RC circuit has an impedance of 65  and has
a resistance of 24 . What is the reactance of
the circuit?
60 
3. Find the resistance of an RC circuit with 25 
impedance and 12  reactance.
22 

28. Exercises
ESA Pg 269
Activity 16B,16C,16D
ABA
Pg 180-185