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Delta and Star Delta Transformations

Delta and Star Transformations

Delta and Star, Delta Transformation

34.1 Introduction

Kirchhoff’s laws, mesh current analysis, nodal analysis, or the superposition theorem are means that can be applied to analyse circuits.[256-259] Thevenin’s and Norton’s theorems can be used to analyse circuits as well and result in massive reductions of time spent.[260] Star-delta and delta-star transformations may be applied to certain types of circuits to simplify them before application of circuit theorems.[261-263]

34.2 & 34.3 Delta-Star Transformations

A delta-connected network can be redrawn as a Pi (π) or mesh-connected network, as it is otherwise called.[286, 287] A star-connected network can be redrawn as a T-connected network.[289]

It is possible to replace the delta connection by an equivalent star connection.[304] The equivalent star network will have the same power factor and consume the same power as the delta network. A delta-star transformation may also be called a Pi to T (π to T) transformation.[306-309]

Summarising, the star network is equivalent to the delta network when:

Z1 = (ZAZB) / (ZA + ZB + ZC)[352]
Z2 = (ZBZC) / (ZA + ZB + ZC)[356]
Z3 = (ZAZC) / (ZA + ZB + ZC)[357]

The impedance of any branch of the star network is given by the product of the two impedances of the delta network joined at that node, divided by the sum of all three impedances of the delta network.[354, 358-363]

Example Calculation

Given a delta network with ZA = 2Ω, ZB = 3Ω, and ZC = 5Ω:[365-369]

Z1 = (2 × 3) / (2 + 3 + 5) = 0.6Ω[373]
Z2 = (3 × 5) / (2 + 3 + 5) = 1.5Ω[374]
Z3 = (5 × 2) / (2 + 3 + 5) = 1.0Ω[375]

Worked Problems

Problem 2

Find the current I and power dissipated across the 10Ω resistor.[380]

Delta elements: ZA = j10Ω, ZB = j15Ω, ZC = j25Ω.[386, 388, 393, 395] Converting to Star:

Z1 = (j10 × j15) / (j10 + j15 + j25) = j3Ω[408]
Z2 = (j15 × j25) / j50 = j7.5Ω[408]
Z3 = (j10 × j25) / j50 = j5Ω[408]

Total impedance evaluation yields:

Supply Current = 40V / 3.54∠45° = 11.3∠-45° A[430]
I10Ω = 11.3∠-45° × (-j5) / (10 + j5 – j5) = 5.65∠-135° A[436-438]
Power = I2 × R = 5.652 × 10 = 319 W[438]

Problem 3 (Bridge Network)

Find the single equivalent resistance that replaces the bridge network between nodes A and B.[444-446]

Upper Delta: 15Ω, 4Ω, 8Ω.[450, 453, 454] Using Delta-Star Transform:

Z1 = (4 × 8) / (15 + 4 + 8) = 32 / 27 ≈ 1.185Ω
Z2 = (15 × 8) / 27 = 120 / 27 ≈ 4.44Ω
Z3 = (15 × 4) / 27 = 60 / 27 ≈ 2.22Ω

Note: Based on simplified schematic values processed on page 7.

34.4 Star-Delta Transformation

It is possible to replace a Star section with an equivalent Delta section.[660]

ZA = (Z1Z2 + Z2Z3 + Z3Z1) / Z2[679, 700]
ZB = (Z1Z2 + Z2Z3 + Z3Z1) / Z3[680, 702]
ZC = (Z1Z2 + Z2Z3 + Z3Z1) / Z1[680, 704]

Worth noting that the denominator of the expression for ZA is Z2, which is connected to terminal 2. The denominator for ZB is Z3, connected to terminal 3. The denominator for ZC is Z1, connected to terminal 1.[681-684]

Example (Star to Delta)

Given a Star network: Z1 = 10Ω, Z2 = 20Ω, Z3 = j5Ω.[716-718]

Numerator: Z1Z2 + Z2Z3 + Z3Z1 = (10×20) + (20×j5) + (j5×10) = 200 + j150[705, 727]
ZA = (200 + j150) / 10 = 20 + j15 Ω[709-710]
ZB = (200 + j150) / j5 = 30 – j40 Ω[712, 729]
ZC = (200 + j150) / 20 = 10 + j7.5 Ω[711]

3.5 Maximum Power Transfer Theorems and Impedance Matching

Why it is important to understand: This states that maximum power transfer occurs when the impedance of the source equals the impedance of the load or vice versa.[757-759] It is also called Jacobi’s Law.[760]

In electronics, impedance matching is the practice of matching the input impedance of an electrical load (or the output impedance of its corresponding signal source) to maximize power or minimize reflections from the load.[762-766]