Kirchhoff's laws of voltage and current also apply in AC circuits in the frequency domain when voltages and currents are represented as phasors. This allows techniques such as impedance combination, nodal analysis, mesh analysis, and source transformation to be used to analyze AC circuits. Impedance combination rules show that resistors in series add their impedances together, while resistors in parallel calculate an equivalent impedance using current division. Delta-wye and wye-delta transformations can also be used with impedances. AC bridges use these principles to measure inductance and capacitance by balancing the bridge at resonance. Phase shifting circuits made of RC networks can be analyzed using impedance concepts.

1. Kirchoff’s Law in the
Frequency Domain

2. • For KVL, let 𝑣1, 𝑣2 … , 𝑣𝑛 be the voltages around a closed loop. Then
• In the sinusoidal steady state, each voltage may be written in cosine form,
so that it becomes

3. • 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
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 many things, such as impedance combination, nodal and mesh
analyses, superposition, and source transformation.

4. Impedance Combinations
• Consider the N series-connected impedances shown in the figure. The
same current I flows through the impedances. Applying KVL around the
loop gives

5. Given the circuit:
• If N = 2 as shown in the figure, the current through the impedances is
• Since 𝑉1 = 𝑍1𝐼 and𝑉2 = 𝑍2𝐼, 𝑡ℎ𝑒𝑛
• which is the voltage-division relationship. In the same manner, we can obtain the equivalent
impedance or admittance of the N parallel-connected impedances shown in the next figure.. The
voltage across each impedance is the same. Applying KCL at the top node,

6. N-parallel Impedances

7. • which is the current-division principle.

8. • The delta-to-wye and wye-to-delta transformations that we
applied to resistive circuits are also valid for impedances. With
reference to the figure below, the conversion formulas are as
follows.

9. • A delta or wye circuit is said to be balanced if it has equal impedances in all three branches. When
a ∆ − 𝛾circuit is balanced,
• Where 𝑍𝛾 = 𝑍1 = 𝑍2 = 𝑍3 𝑎𝑛𝑑 𝑍∆ = 𝑍𝑎 = 𝑍𝑏 = 𝑍𝐶

10. • Example: Find the input impedance of the circuit in the figure. Assume that the circuit
operates at 𝜔 = 50
𝑟𝑎𝑑
𝑠
.

11. • Determine 𝑣𝑜 𝑡 in the circuit

13. Find current I in the circuit

14. Application:
• A phase-shifting circuit is often employed to correct an
undesirable phase shift already present in a circuit or to
produce special desired effects. An RC circuit is suitable
for this purpose because its capacitor causes the circuit
current to lead the applied voltage.

15. Example: For the RL circuit shown in the figure, calculate the amount of
phase shift produced at 2 kHz.

16. Application:
• AC Bridges - An ac bridge circuit is used in measuring the
inductance L of an inductor or the capacitance C of a capacitor.
It is similar in form to the Wheatstone bridge for measuring an
unknown resistance and follows the same principle. To measure
L and C, however, an ac source is needed as well as an ac
meter instead of the galvanometer. The ac meter may be a
sensitive ac ammeter or voltmeter.
• Consider the general ac bridge circuit displayed in the figure.
The bridge is balanced when no current flows through
the meter. This means that V1 = V2. Applying the
voltage division principle,

17. Specific ac bridges for measuring L and C
are shown in the figure, where Lx and Cx
are the unknown inductance and
capacitance to be measured while LS and
CS are a standard inductance and
capacitance (the values of which are known
to great precision). In each case, two
resistors, R1 and R2 are varied until the ac

19. Example: The ac bridge circuit of the figure balances when Z1 is a 1 kΩ
resistor, Z2 is 4.2 kΩ a resistor, Z3 is a parallel combination of a 1.5 MΩ
resistor and a 12-pF capacitor, and f = 2 kHz. Find: (a) the series
components that make up Zx and (b) the parallel components that make up
Zx.

20. a) We assume that ZX is made up of series connected components,

21. 1. Find the I0 in the circuit.
2. Given that vs (t) = 20 sin (100t -40°), determine
ix(t).
3. Determine the equivalent impedance of the circuit
Assignment: This will be done by pair. Write neatly in a yellow pad.
Show your solution. Make sure to box in your final answer. Submit on
Monday (Feb.26,2024)