Complex Impedances in Series

Complex Impedances in Series

When you see resistors, coils, and capacitors in series, each component has an impedance that can be represented as a vector in the RX plane. The vectors for resistors are constant, regardless of the frequency. But the vectors for coils and capacitors vary with frequency.

Pure Reactances

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Pure inductance and pure capacitance are represented by reactance vectors that point straight up and down.

Pure inductive reactances (XL ) and capacitive reactances (XC) simply add together when coils and capacitors are in series. Thus, X = XL + XC. In the RX plane, their vectors add, but because these vectors point in exactly opposite directions—inductive reactance upward and capacitive reactance downward (above figure)—the resultant sum vector inevitably points either straight up or straight down, unless the reactances are equal and opposite, in which case they cancel and the result is the zero vector.

Problem 1

Suppose a coil and capacitor are connected in series, with jXL = j200 and jXC = −j150. What is the net reactance?
Just add the values: jX = jXL + jXC = j200 + (−j150) = j(200 − 150) = j50. This is a pure inductive reactance, because it is positive imaginary.

Problem 2

Suppose a coil and capacitor are connected in series, with jXL = j30 and jXC = −j110. What is the net reactance?
Again, add the values: jX = j30 + (−j110) = j(30 − 110) = −j 80. This is a pure capacitive reactance, because it is negative imaginary.

Problem 3

Suppose a coil of inductance L = 5.00 μH and a capacitor of capacitance C = 200 pF are connected in series. Suppose the frequency is f = 4.00 MHz. What is the net reactance?
First, calculate the reactance of the inductor at 4.00 MHz. Proceed as follows:
jXL = j6.28f L
= j(6.28 × 4.00 × 5.00)
= j126

Next, calculate the reactance of the capacitor at 4.00 MHz. Proceed as follows:
jXC = −j[1/(6.28fC )]
= −j[1/(6.28 × 4.00 × 0.000200)]
= −j199

Finally, add the inductive and capacitive reactances to obtain the net reactance:
jX = jXL + jXC
= j126 + (−j199)
= −j 73

This is a pure capacitive reactance.

Problem 4

What is the net reactance of the aforementioned inductor and capacitor combination at the frequency f = 10.0 MHz?
First, calculate the reactance of the inductor at 10.0 MHz. Proceed as follows:
jXL = j 6.28f L
= j(6.28 × 10.0 × 5.00)
= j314

Next, calculate the reactance of the capacitor at 10.00 MHz. Proceed as follows:
jXC = −j[1/(6.28fC )]
= −j[1/(6.28 × 10.0 × 0.000200)]
= −j 79.6

Finally, add the inductive and capacitive reactances to obtain the net reactance:
jX = jXL + jXC
= j314 + (−j 79.6)
= j234

This is a pure inductive reactance. For series-connected components, the condition in which the capacitive and inductive reactances cancel is known as series resonance. We’ll deal with this in more detail in the next topics.

Adding Impedance Vectors

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When resistance is present along with reactance, impedance vectors point at angles; they are neither vertical nor horizontal.

In the real world, there is resistance, as well as reactance, in an ac series circuit containing a coil and capacitor. This occurs because the coil wire has some resistance (it’s never a perfect conductor). It can also be the case because a resistor is deliberately connected into the circuit.

Whenever the resistance in a series circuit is significant, the impedance vectors no longer point straight up and straight down. Instead, they run off toward the northeast (for the inductive part of the circuit) and southeast (for the capacitive part). This is illustrated in above figure.

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Parallelogram method of complex-impedance vector addition.

When two impedance vectors don’t lie along a single line, you must use vector addition to be sure that you get the correct net impedance. In above figure, the geometry of vector addition is shown. Construct a parallelogram, using the two vectors Z1 = R1 + jX1 and Z2 = R2 + jX2 as two adjacent sides of the figure. The diagonal of the parallelogram is the vector representing the net complex impedance. (Note that in a parallelogram, pairs of opposite angles have equal measures. These equalities are indicated by single and double arcs in above figure )

Formula for Complex Impedances in Series

Suppose you are given two complex impedances, Z1 = R1 + jX1 and Z2 = R2 + jX2. The net impedance, Z, of these in series is their vector sum, given by the following formula:
Z = (R1 + jX1) + (R2 + jX2)
= (R1 + R2) + j (X1 + X2)

Calculating a vector sum using the formula is easier than doing it geometrically with a parallelogram. The arithmetic method is also more exact. The resistance and reactance components add separately. Just remember that if a reactance is capacitive, then it is negative imaginary in this formula.