The impedance of an RC circuit is the total opposition to AC current flow caused by the resistor (R) and the reactance of the capacitor (XC).

The equation for the impedance of a series RC circuit is: 

where: 

Z = the total impedance in ohms 
X_C = the capacitive reactance in ohms 
R = the resistance in ohms

It is no accident that the equation for impedance looks like the equation for calculating the hypotenuse of a right triangle. Impedance in series circuit is, in fact, often portrayed as a vector diagram where the horizontal leg is the resistance, the vertical leg is the reactance, and the hypotenuse is the resulting impedance. 

Ans: 180 W

The sum of voltages in a series RC circuit is always greater than the sum of the voltages across the resistive and capacitive components.

Ans: 15.6 V

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Although you can use an ordinary ohmmeter to measure resistance, there are no common lab instruments for directly measuring reactance and impedance. For all practical purposes, then, you must calculate reactance and impedance from other circuit values that are more readily available. 

In practical RC circuits, you can readily determine or directly measure the values of VT, f, R, and C. There is more than enough information among these items to calculate the values of XC and Z.

Step 1. Calculate the value of X_C from the known values of f and C: 

X_C  = 1 / (2pfC)

Step 2. Calculate Z from the know value of R and the value of X_C calculated in Step 1: 

Ans: (a) X_C = 16 kW, (b) Z = 25.6 kW

However, there are two equations you can use for defining the total phase angle for a series RC circuit. 
 

The total phase angle is also determined by the equation:  qT = -tan-1(VC /  VR)   where:  qT = total phase angle in degrees or radians  VC= AC voltage drop across the capacitor in volts  VR = AC voltage drop across the resistor in volts

The  tan^-1 expression is the inverse tangent function which is used for calculating angle q for a right triangle, given the lengths of the two sides. 

So there are two different equations for calculating the total phase angle of a series RC circuit. Which should you use? Use the one that is simpler with the information you have at hand: 

• If know R and X_C, use the reactance version.
• If you know V_R and V_C, use the voltage version.

Ans:  0º, -90º, -51.30º

Remember that q_R = 0º and q_C = -90º in a series RC circuit, then use the equation based on values of R and X_C:  q_T = -tan^-1(X_C /  R)  

Ans:  10V, -36.9º

Since you know the voltages, use this equation for calculating the total phase shift:  q_T = -tan^-1(V_C /  V_R)

A complete analysis of a series RC circuit begins with the following known values: 

The value of resistor, R in ohms The value of capacitor, C in farads The amount of applied voltage, VT in volts The frequency, f in hertz

The challenge is to use these known values in a series of equations and apply a few principles to determine: 
 

Some of these remaining properties are determined from the nature of the components, themselves, and do not have to be calculated. For example, the phase angle for R is always 0º, and the phase angle for C is always -90º. Other values must to be calculated by means of various equations. 

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Here is the procedure for doing a complete analysis of a series RC circuit, given the values of R, C, f, and V_T. 

Step 1. Calculate the value of X_C: 

X_C = 1 / (2pfC)

Step 2. Calculate the total impedance:

Step 3. Use Ohm's Law to calculate the total current: 

I_T = V_T / Z

Step 4. Calculate the currents through R and C. Since this is a series circuit: 

I_R = I_T  I_C = I_T

Step 4. Calculate the voltages across R and C. By Ohm's Law: 

V_R = RI_R V_C = X_CI_R

Step 5. Determine the phase angles for R and C. Phase angles for these components are always: 

q_R = 0º 
q_C =  -90º

Step 6. Calculate the total phase angle for the circuit: 

q_T = tan^-1(X_C/ R)