Figure 9.17

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Figure 9.17

For Practice Prob. 9.9.

This can be written as

$$\text{Re}(V_{m1}e^{j\theta_1}e^{j\omega t}) + \text{Re}(V_{m2}e^{j\theta_2}e^{j\omega t}) + \dots + \text{Re}(V_{mn}e^{j\theta_n}e^{j\omega t}) = 0$$

or

$$\text{Re}[(V_{m1}e^{j\theta_1} + V_{m2}e^{j\theta_2} + \dots + V_{mn}e^{j\theta_n})e^{j\omega t}] = 0 \tag{9.53}$$

If we let Vk = Vmkej^θk, then

$$Re[(V_1 + V_2 + \dots + V_n)e^{j\omega t}] = 0$$

\n(9.54)

Because ejω^t ≠ 0,

$$V_1 + V_2 + \dots + V_n = 0 \tag{9.55}$$

indicating that Kirchhoff’s voltage law holds for phasors.

By following a similar procedure, we can show that Kirchhoff’s current law holds for phasors. If we let i1, i2, … , in be the current leaving or entering a closed surface in a network at time t, then

$$i_1 + i_2 + \dots + i_n = 0 \tag{9.56}$$

If I1, I2, … , In are the phasor forms of the sinusoids i1,i2, … ,in, then

$$\mathbf{I}_1 + \mathbf{I}_2 + \dots + \mathbf{I}_n = 0 \tag{9.57}$$

which is Kirchhoff’s current law in the frequency domain.

Once we have shown that both KVL and KCL hold in the frequency domain, it is easy to do man y things, such as impedance combination, nodal and mesh analyses, superposition, and source transformation.