Design of Complex Integrated Circuits
\[ \Delta T = Q \cdot R_\mathrm{th} \tag{49}\]
\[ R_\mathrm{th,series} = \sum_{i} R_{\mathrm{th},i} \qquad \frac{1}{R_\mathrm{th,parallel}} = \sum_{i} \frac{1}{R_{\mathrm{th},i}} \tag{50}\]
\[ \Delta T = R_\mathrm{th,ja} \cdot P_\mathrm{diss} \tag{51}\]
Figure 56: Thermal equivalent network of a packaged IC: the dissipated power \(P_\mathrm{diss}\) acts as a current source injecting heat into the junction node.
Figure 57: Equivalent circuit of a single package connection from the PCB to the on-chip pad: the bond wire and package trace contribute a series inductance and resistance, while the package trace and the bond pad add shunt capacitances.
\[ L' \approx 0.2 \ln \left( \frac{2h}{r} \right) \; \mathrm{nH/mm} \tag{52}\]
\[ L' \approx \frac{1.6}{0.72 + W/d} \; \mathrm{nH/mm} \tag{53}\]
\[ L_\mathrm{m}' \approx 0.1 \ln \left[ 1 + \left( \frac{2h}{d} \right)^2 \right] \; \mathrm{nH/mm} \tag{54}\]
Note 4: Example Calculation of the Bond-Wire Inductance
Let us assume a bond wire with a diameter of 25 µm (i.e., \(r = 12.5\,\mu\text{m}\)) and a length of \(l = 2\,\text{mm}\), running at a height of \(h = 300\,\mu\text{m}\) above the ground plane. With Equation 52, the inductance per length is
\[ L' \approx 0.2 \ln \left( \frac{2 \cdot 300\,\mu\text{m}}{12.5\,\mu\text{m}} \right) \; \text{nH/mm} = 0.2 \ln (48) \; \text{nH/mm} = 0.77\,\text{nH/mm} \]
and the total inductance of the bond wire is
\[ L = L' \cdot l = 0.77\,\text{nH/mm} \cdot 2\,\text{mm} = 1.55\,\text{nH} \]
Note that the height enters only logarithmically: doubling \(h\) to \(600\,\mu\text{m}\) increases the inductance by just 18 % to \(1.83\,\text{nH}\). At a frequency of \(f = 1\,\text{GHz}\), this bond wire already shows a reactance of
\[ X_L = 2 \pi f L = 9.7\,\Omega \]
Example Calculation of the Bond-Wire Inductance
If this bond wire carries the supply current of a digital block, a current step of \(\Delta i = 10\,\text{mA}\) within \(\Delta t = 100\,\text{ps}\) causes a supply bounce of
\[ V = L \frac{\Delta i}{\Delta t} = 1.55\,\text{nH} \cdot \frac{10\,\text{mA}}{100\,\text{ps}} = 155\,\text{mV} \]
which is significant for a core supply voltage of about 1.2 V.
To reduce the supply bounce, a second bond wire of the same dimensions is placed in parallel at a distance of \(d = 300\,\mu\text{m}\). With Equation 54, the mutual inductance per length is
\[ L_\mathrm{m}' \approx 0.1 \ln \left[ 1 + \left( \frac{2 \cdot 300\,\mu\text{m}}{300\,\mu\text{m}} \right)^2 \right] \; \text{nH/mm} = 0.1 \ln (5) \; \text{nH/mm} = 0.16\,\text{nH/mm} \]
Example Calculation of the Bond-Wire Inductance
resulting in a mutual inductance of \(L_\mathrm{m} = L_\mathrm{m}' \cdot l = 0.32\,\text{nH}\) between the two bond wires. The magnetic coupling factor is
\[ k = \frac{L_\mathrm{m}}{\sqrt{L_1 L_2}} = \frac{L_\mathrm{m}}{L} = \frac{0.32\,\text{nH}}{1.55\,\text{nH}} = 0.21 \]
As both bond wires carry currents in the same direction, the mutual inductance adds to the self-inductance of each wire, and the effective inductance of the parallel connection is
\[ L_\mathrm{eff} = \frac{L + L_\mathrm{m}}{2} = \frac{L}{2} (1 + k) = 0.94\,\text{nH} \]
Example Calculation of the Bond-Wire Inductance
Instead of halving the inductance to \(0.77\,\text{nH}\), the second bond wire reduces it only by 40 %, and the supply bounce for the same current step drops from \(155\,\text{mV}\) to \(94\,\text{mV}\). A larger distance between the bond wires reduces \(k\) and brings \(L_\mathrm{eff}\) closer to \(L/2\). Conversely, if the two bond wires carry unrelated signals, the same coupling causes crosstalk: the current step of \(10\,\text{mA}\) in \(100\,\text{ps}\) in one bond wire induces a voltage of \(L_\mathrm{m} \, \Delta i / \Delta t = 32\,\text{mV}\) in the other.
| Test (examples) | Standard | Motivation & check |
|---|---|---|
| Pre-conditioning (PC) | JESD22-A113 | Moisture soak and reflow simulate storage and board assembly before the other stress tests. |
| Moisture sensitivity level (MSL) | J-STD-020 | Classification of how long a package may be exposed to ambient moisture before soldering. |
| Temperature cycling (TC) | JESD22-A104 | Repeated heating and cooling must not cause failures (thermal expansion mismatch, see CTE). |
| Temperature humidity bias (THB) / biased HAST | JESD22-A101 / A110 | Corrosion and leakage under humidity, temperature, and applied voltage. |
| Unbiased highly accelerated stress test (UHAST) | JESD22-A118 | Exposure of package and die to high temperature and humidity without bias. |
| Test (examples) | Standard | Motivation & check |
|---|---|---|
| High-temperature storage life (HTSL) | JESD22-A103 | Storage at high temperature for thousands of hours (intermetallic growth). |
| Wire bond pull and shear | MIL-STD-883 (2011) / JESD22-B116 | Mechanical strength of the bond wires and their connections. |
| Solder ball shear | JESD22-B117 | Mechanical strength of the solder balls of BGA and WLCSP packages. |
| Drop test | JESD22-B111 | The assembled PCB is dropped repeatedly under standardized conditions. |
| Temperature cycling on board (TCoB) | IPC-9701 | The assembled PCB is temperature-cycled to check the solder-joint reliability. |