Capacitors
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Capacitors
What is a capacitor? • • • •
Electronic component Two conducting surfaces separated by an insulating material Stores charge Uses – Time delays – Filters – Tuned circuits
Capacitor construction • Two metal plates • Separated by insulating material • ‘Sandwich’ construction • ‘Swiss roll’ structure • Capacitance set by...
A C=ε d
Defining capacitance • • • •
‘Good’ capacitors store a lot of charge… …when only a small voltage is applied Capacitance is charge stored per volt Capacitance is measured in farads F – Big unit so nF, mF and µF are used
Q C= V
Graphical representation Equating to the equation of a straight line
Q C= V Q = CV y = mx
Q
Gradient term is the capacitance of the capacitor
V
Charge stored is directly proportional to the applied voltage
Energy stored by a capacitor • By general definition E=QV – product of charge and voltage • By graphical consideration... Q
1 E = QV 2 V
Area term is the energy stored in the capacitor
Other expressions for energy • By substitution of Q=CV
1 E = QV 2 1 2 E = CV 2 2 1Q E= 2 C
Charging a capacitor • Current flow • Initially – High • Finally – Zero • Exponential model • Charging factors – Capacitance – Resistance
I
t
Discharging a capacitor • Current flow • Initially – High – Opposite to charging • Finally – Zero • Exponential model • Discharging factors – Capacitance – Resistance
t
I
Voltage and charge characteristics V or Q
V or Q
t • Charging
V = V0 (1 − e
t Discharging
−t
RC
)
Q = Q0 e
−t
RC
Time constant • Product of – Capacitance of the capacitor being charged – Resistance of the charging circuit – CR • Symbol τ ‘Tau’ • Unit seconds
Q V CR = × V Q ÷t CR = t
When t equals tau during discharge
Q = Q0 e Q = Q0 e Q = Q0 e
−t
RC
− RC
RC
−1
Q = Q0 × 0.37
• At t = tau the capacitor has fallen to 37% of its original value. • By a similar analysis tau can be considered to be the time taken for the capacitor to reach 63% of full charge.
Graphical determination of tau • V at 37% • Q at 37% • Compared to initial maximum discharge
t =τ t = RC C= t R
V or Q
t
Logarithmic discharge analysis • Mathematical consideration of discharge • Exponential relationship • Taking natural logs equates expression to ‘y=mx+c’ • Gradient is -1/Tau
V = V0 e
−t
RC
−t V = e RC V0
ln V − ln V0 = − t
RC
−1 ln V = t + ln V0 RC
Logarithmic discharge graph
lnV
Gradient term is the -1/Tau
t
www.search for... • Capacitors
What is a capacitor? • • • •
Electronic component Two conducting surfaces separated by an insulating material Stores charge Uses – Time delays – Filters – Tuned circuits
Capacitor construction • Two metal plates • Separated by insulating material • ‘Sandwich’ construction • ‘Swiss roll’ structure • Capacitance set by...
A C=ε d
Defining capacitance • • • •
‘Good’ capacitors store a lot of charge… …when only a small voltage is applied Capacitance is charge stored per volt Capacitance is measured in farads F – Big unit so nF, mF and µF are used
Q C= V
Graphical representation Equating to the equation of a straight line
Q C= V Q = CV y = mx
Q
Gradient term is the capacitance of the capacitor
V
Charge stored is directly proportional to the applied voltage
Energy stored by a capacitor • By general definition E=QV – product of charge and voltage • By graphical consideration... Q
1 E = QV 2 V
Area term is the energy stored in the capacitor
Other expressions for energy • By substitution of Q=CV
1 E = QV 2 1 2 E = CV 2 2 1Q E= 2 C
Charging a capacitor • Current flow • Initially – High • Finally – Zero • Exponential model • Charging factors – Capacitance – Resistance
I
t
Discharging a capacitor • Current flow • Initially – High – Opposite to charging • Finally – Zero • Exponential model • Discharging factors – Capacitance – Resistance
t
I
Voltage and charge characteristics V or Q
V or Q
t • Charging
V = V0 (1 − e
t Discharging
−t
RC
)
Q = Q0 e
−t
RC
Time constant • Product of – Capacitance of the capacitor being charged – Resistance of the charging circuit – CR • Symbol τ ‘Tau’ • Unit seconds
Q V CR = × V Q ÷t CR = t
When t equals tau during discharge
Q = Q0 e Q = Q0 e Q = Q0 e
−t
RC
− RC
RC
−1
Q = Q0 × 0.37
• At t = tau the capacitor has fallen to 37% of its original value. • By a similar analysis tau can be considered to be the time taken for the capacitor to reach 63% of full charge.
Graphical determination of tau • V at 37% • Q at 37% • Compared to initial maximum discharge
t =τ t = RC C= t R
V or Q
t
Logarithmic discharge analysis • Mathematical consideration of discharge • Exponential relationship • Taking natural logs equates expression to ‘y=mx+c’ • Gradient is -1/Tau
V = V0 e
−t
RC
−t V = e RC V0
ln V − ln V0 = − t
RC
−1 ln V = t + ln V0 RC
Logarithmic discharge graph
lnV
Gradient term is the -1/Tau
t
www.search for... • Capacitors
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