Alternating Current

RCL Series Circuits (cont'd)

Phasor Diagrams

We can understand AC circuits in terms of phasor diagrams. Here a vector of length V_o rotates with angular velocity . Its projection onto the vertical axis is

v = V_o sin t

and that is just the AC voltage from a generator or a power supply.

The current i_R through a resistor R is in phase with the voltage v_R across the resistor so we may show that on a phasor diagram by drawing a vector I_R which lines up with the voltage V_R.

The current i_C through a capacitor C is ahead of the voltage v_C across the capacitor by 90 degrees so we may show that on a phasor diagram by drawing a vector I_C which leads the voltage V_C by 90 degrees.

The current i_L through an inductor L is behind the voltage v_L across the inductor by 90^o so we may show that on a phasor diagram by drawing a vector I_L which lags the voltage V by 90 degrees.

At any given time, the sum of the voltages across the three components, R, C, and L, must equal the voltage of the AC source. This is just energy conservation or Kirchoff's Loop Rule,

v(t) = v_R(t) + v_C(t) + v_L(t)

or

v = v_R + v_C + v_L

Remember that the phase relationship between the AC voltage v and each of the other voltages differs for each of the voltages. We can handle this graphically with the following phasor diagram:

or

For convenience, we may want to rotate this diagram,

Now we can see that the voltages form the sides of a right triangle with the voltage of the AC source forming the hypotenuse. That means

V = I SQRT[ R² + (X_L - X_C)² ]

or, since V = I Z, we can now write the total impedence for this series RCL circuit as

Z = SQRT[ R² + (X_L - X_C)² ]

where R is the resistance, X_C is the capacitive reactance, and X_L is the inductive capacitance.

[ Of course, the "SQRT" needs to be replaced with a properly written radical sign in the equations].

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(c) Doug Davis, 2002; all rights reserved