Alternating Current Circuits

• Consider Inductor and Capacitor connected in series
• make them ideal so that there is no resistance in the circuit
• start with the capacitor fully charged
• connect to inductor at t = 0 s.
• The total energy will be conserved, inductor and capacitor do not dissipate energy.
• at t = 0, current through the circuit is zero, charge on capacitor is max.
• all of the energy is stored in the electric field of the capacitor
• at t > 0, the capacitor will discharge, current will flow in the circuit
• less energy is stored in the electric field of the capacitor, some energy is stored in the magnetic field of the inductor. The sum of these two energies is constant and equal to the total energy stored in the circuit.
• at t > 0, the capacitor continues to discharge, current flow will increase until the capacitor is completely discharged and current is a maximum.
• no energy is stored in the electric field of the capacitor, all of the energy is stored in the magnetic field of the inductor.
• the inductor keeps current flowing in the original direction and the capacitor begins to charge with polarity opposite to what it had at t = 0.
• less energy is stored in the magnetic field of the inductor, some energy is stored in the electric field of the capacitor. The sum of these two energies is constant and equal to the total energy stored in the circuit.
• the capacitor is now fully charged with polarity opposite to what it had at t = 0, no current is flowing in the circuit.
• all of the energy is stored in the electric field of the capacitor
• the capacitor will discharge, current will flow in the circuit opposite the direction it had above
• less energy is stored in the electric field of the capacitor, some energy is stored in the magnetic field of the inductor. The sum of these two energies is constant and equal to the total energy stored in the circuit.
• the capacitor continues to discharge, current flow will increase until the capacitor is completely discharged and current is a maximum.
• no energy is stored in the electric field of the capacitor, all of the energy is stored in the magnetic field of the inductor.
• the inductor keeps current flowing in this direction and the capacitor begins to charge with polarity identical to what it had at t = 0.
• less energy is stored in the magnetic field of the inductor, some energy is stored in the electric field of the capacitor. The sum of these two energies is constant and equal to the total energy stored in the circuit.
• the capacitor is now fully charged with polarity equal to what it had at t = 0, no current is flowing in the circuit.
• all of the energy is stored in the electric field of the capacitor
• the process repeats itself indefinitely because no energy is lost, "oscillation"
• What is the frequency of this oscillation?
• total energy in the circuit
• 
• U is constant, if we take the derivative of the total energy with respect to time, this must be equal to zero.



• 



• but the current is just the time derivative of the charge and the above is



• 



• solutions to this equation have the form



• 



• A real circuit will always have some resistance and to maintain the current, we will need a driving force.
• 

• Consider some very simple circuits,
• a resistor connected to an ac source
• applying Kirchhoff's Loop rule to the circuit



• 



• current through the resistor and voltage across the resistor are in phase.
• a plot of current and resistance vs time would look like: (red=current, green= voltage)
  

• a capacitor connected to an ac source

• the voltage across the capacitor will be a maximum when the capacitor is fully charged, at this point, there will not be any current flowing through the circuit
• the current will be a maximum when the capacitor has no charge and no potential difference across it.
• the current and the voltage will be out of phase.
• 
• the charge across the capacitor is Cv_c
• the current through the circuit



• 



• the current through a purely capacitive circuit leads the voltage by 90^o






• the current equation has the form, I = V/R if
• 



• X_C is called the capacitive reactance and has units of Ohms.



• an inductor connected to an ac source
• the voltage across the inductor will be a maximum when the current is changing at its largest rate. This occurs when the current is zero, (think about a sine function), the voltage across the inductor will be zero when the current is not changing, which will be at its max and min values, (again think about the slope of the sine function at its highest and lowest points.
• the voltage and current will be out of phase.






• 
• the current through a purely inductive circuit lags the voltage by 90^o


• the current equation has the form, I = V/R if
• 



• X_L is called the inductive reactance and has units of Ohms.



• How do we combine all three together in a series combination and determine the total current through the series combination and the voltage across the entire circuit and across individual components? The voltages across each of the components will be out of phase.

• Phasors

• represent the voltage across any component as a "vector" rotating in a counter clockwise direction with angular velocity, omega. The projection of this "vector" on the y-axis at any instant in time gives the instantaneous value of the voltage. We have to pick an instant in time to examine the voltages, so we'll pick a time when the current is zero and increasing. Then, the arrow representing the current will be along the positive x-axis as will the arrow representing the voltage across the resistor. The voltage across the inductor leads the current so it will be directed along the positive y-axis and the voltage across the capacitor lags the current so it will be directed in the negative y-direction.

• Using the laws of vector addition, we can determine the magnitude of V and the size of phi.



• 



• Z is called the impedance of the circuit and has units of ohms.



• 



• Both the inductive reactance and the capacitive reactance are frequency dependent.

• as frequency increases, the inductive reactance increases while the capacitive reactance decreases. There will be a frequency where the two are equal. We can determine this frequency



• 
• This is the resonance frequency for the circuit, at which the impedance is a minimum, the current is a maximum, and the voltage of the source and the current are in phase.
• The power is given by



• and is a maximum at resonance.