What you always wanted to know about

PHASORS and IMPEDANCES

but were afraid to ask

    We have learned that the voltage  across an inductor is proportional to the time derivative of the current through the inductor and the voltage across a capacitor is proportional to the integral of the current through the capacitor. As a result, in the presence of inductors and/or capacitors Kirchhoff's current and voltage laws are  expressed in the form of differential equations, even in the simplest cases. In case of transient response or an arbitrary time dependence of voltage or current sources, circuit analysis involves solution of ordinary differential equations.

    There is, however, one very important special case when differential equations can be replaced by complex analysis. This is the case of alternating currents and voltages used for industrial and household applications all over the world. In this case the currents and voltages are sinusoidal functions of time. A sinusoidal function can always be represented in the form of

V(t) = Vp cos (w t + q v)                                                                                                                               

for voltages and

i(t) = ip cos (w t +q i)                                                                                                                                                                                              for   currents where t is time, Vp and ip are the peak values of voltage and current, respectively, q v and q i are the initial phases of voltage and current, respectively, and w is the angular frequency of the signals generated by voltage and/or current sources.

     By definition,  any complex number Z is represented as

Z = Re{Z} + j Im{Z}           (rectangular form),

or

Z = M (cos a + j sin a)      (polar form)

where j = sqrt(-1), Re{Z} is the real part and Im{Z} is the imaginary part of the complex number Z, M is its magnitude, and a is the angle between the real axis and the vector leading from the origin to the point in the complex plane that defines the given complex number. According to Euler's theorem, the complex exponential function is defined as

exp (j a) = cos a + j sin a

It follows from here that the polar form of any complex number can be represented as

Z = M exp (j a)

and our functions can be rewritten as real parts of complex numbers:

V(t) = Re{Vp exp[j (w t + q v)]}                                                                                                                           

and

i(t) = Re{ip exp[j (w t +q i)]}

Let us rewrite our functions once again:

V(t) = Re{Vp exp (jq v) exp (jwt)} = Re{V exp (jw t )}                                                                                           

and

i(t) = Re{ip exp(jqi) exp (jwt)} = Re{i exp (jw t )}

where

V =  Vp exp (jq v)

and

i = ip exp (jq i)

are the voltage and current phasors, respectively. The phasors are complex numbers. Their mission is to help us find the real functions V(t) and i(t). Indeed, if we know the phasors, all we have to do is to multiply them by exp (jwt) and then take the real part of the resulting complex number.

Note that the phasors are independent of time. Therefore, if we differentiate V(t) and i(t) with respect to time, the result is simply

dV/dt = Re{jwV exp (jw t )}

and

di/dt = Re{jwi exp (jw t )} ,

i.e., in the complex domain differentiation with respect to time is replaced with multiplication by jw. As a result, our differential equations become simple algebraic equations, similar to those that we have when we deal with resistive circuits. The only difference is that all voltages and currents are now phasors and all resistances, capacitances and inductances are replaced with their complex impedances ZR, ZL, and ZC, respectively,  where

ZR = R

ZL = jwL

ZC = 1/(jwC ) = -j/(wC)

The impedance of a resistor is simply its resistance (a real value) but the impedances of a capacitor or an inductor are both imaginary. While resistances, capacitances, and inductances are constant values, impedances are functions of frequency. The unit of impedance is ohm.

Circuit analysis in the complex domain is the same as  for purely resistive circuits but we must deal with phasors and impedances. When we have elements in series, their impedances are added to provide the equivalent impedance. For parallel elements, the reciprocals of the impedances are added to provide the reciprocal of the equivalent impedance. All methods learned for resistive circuits can be applied in the complex domain as well. After we find the phasor currents and voltages, we multiply them by  exp(jwt) and then take the real part of the resulting complex number which is our real function in the time domain.

All this seems to be rather abstract for some students. After a little practice, however, you will find this method easy and convenient. Let us consider a simple example. A source provides the voltage V(t) = 100 cos (500 t) to a series RLC circuit (R = 100 ohm, L = 1 H, C = 5 microfarad capacitor). What is the current in this circuit? Please try to solve the problem yourself and then compare your solution with the one that follows here.

The voltage phasor is V = 100 V. The impedance is

Z = 100 + j 500*1 - j/(500*5*10-6) = 100 + j(500 - 400) = 100 (1 + j) ohm.

We find the current phasor as i = V/Z = 100/[100(1 + j)] =(1 - j)/[(1 + j)(1 - j)] = (1 - j)/2 = [sqrt(2)/2] exp (-jp/4) A

The answer is i(t) = Re{0.707 exp (-jp/4 +j500t)} = 0.707 cos  (500t - p/4) A

If you could solve this problem on your own, you understand the basic idea about phasors and impedances. Congratulations!

Finally, we must emphasize that this powerful approach can only be used for sinusoidal currents and voltages. If the time-dependence is an arbitrary function of time, we must deal with differential equations!

 

    Please send me feedback about this tutorial. What do you think about it? Any questions?

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